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algebrite/sources/integral.coffee

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1###
Table of integrals
3
4The symbol f is just a dummy symbol for creating a list f(A,B,C,C,...) where
5
A	is the template expression
7
B	is the result expression
9
C	is an optional list of conditional expressions
11###
12
13itab = [
# 1
"f(a,a*x)",
# 9 (need a caveat for 7 so we can put 9 after 7)
"f(1/x,log(x))",
# 7
"f(x^a,x^(a+1)/(a+1))",
# 12
"f(exp(a*x),1/a*exp(a*x))",
"f(exp(a*x+b),1/a*exp(a*x+b))",
"f(x*exp(a*x^2),exp(a*x^2)/(2*a))",
"f(x*exp(a*x^2+b),exp(a*x^2+b)/(2*a))",
# 14
"f(log(a*x),x*log(a*x)-x)",
# 15
"f(a^x,a^x/log(a),or(not(number(a)),a>0))",
# 16
"f(1/(a+x^2),1/sqrt(a)*arctan(x/sqrt(a)),or(not(number(a)),a>0))",
# 17
"f(1/(a-x^2),1/sqrt(a)*arctanh(x/sqrt(a)))",
# 19
"f(1/sqrt(a-x^2),arcsin(x/(sqrt(a))))",
# 20
"f(1/sqrt(a+x^2),log(x+sqrt(a+x^2)))",
# 27
"f(1/(a+b*x),1/b*log(a+b*x))",
# 28
"f(1/(a+b*x)^2,-1/(b*(a+b*x)))",
# 29
"f(1/(a+b*x)^3,-1/(2*b)*1/(a+b*x)^2)",
# 30
"f(x/(a+b*x),x/b-a*log(a+b*x)/b/b)",
# 31
"f(x/(a+b*x)^2,1/b^2*(log(a+b*x)+a/(a+b*x)))",
# 33
"f(x^2/(a+b*x),1/b^2*(1/2*(a+b*x)^2-2*a*(a+b*x)+a^2*log(a+b*x)))",
# 34
"f(x^2/(a+b*x)^2,1/b^3*(a+b*x-2*a*log(a+b*x)-a^2/(a+b*x)))",
# 35
"f(x^2/(a+b*x)^3,1/b^3*(log(a+b*x)+2*a/(a+b*x)-1/2*a^2/(a+b*x)^2))",
# 37
"f(1/x*1/(a+b*x),-1/a*log((a+b*x)/x))",
# 38
"f(1/x*1/(a+b*x)^2,1/a*1/(a+b*x)-1/a^2*log((a+b*x)/x))",
# 39
"f(1/x*1/(a+b*x)^3,1/a^3*(1/2*((2*a+b*x)/(a+b*x))^2+log(x/(a+b*x))))",
# 40
"f(1/x^2*1/(a+b*x),-1/(a*x)+b/a^2*log((a+b*x)/x))",
# 41
"f(1/x^3*1/(a+b*x),(2*b*x-a)/(2*a^2*x^2)+b^2/a^3*log(x/(a+b*x)))",
# 42
"f(1/x^2*1/(a+b*x)^2,-(a+2*b*x)/(a^2*x*(a+b*x))+2*b/a^3*log((a+b*x)/x))",
# 60
"f(1/(a+b*x^2),1/sqrt(a*b)*arctan(x*sqrt(a*b)/a),or(not(number(a*b)),a*b>0))",
# 61
"f(1/(a+b*x^2),1/(2*sqrt(-a*b))*log((a+x*sqrt(-a*b))/(a-x*sqrt(-a*b))),or(not(number(a*b)),a*b<0))",
# 62 is the same as 60
# 63
"f(x/(a+b*x^2),1/2*1/b*log(a+b*x^2))",
#64
"f(x^2/(a+b*x^2),x/b-a/b*integral(1/(a+b*x^2),x))",
#65
"f(1/(a+b*x^2)^2,x/(2*a*(a+b*x^2))+1/2*1/a*integral(1/(a+b*x^2),x))",
#66 is covered by 61
#70
"f(1/x*1/(a+b*x^2),1/2*1/a*log(x^2/(a+b*x^2)))",
#71
"f(1/x^2*1/(a+b*x^2),-1/(a*x)-b/a*integral(1/(a+b*x^2),x))",
#74
"f(1/(a+b*x^3),1/3*1/a*(a/b)^(1/3)*(1/2*log(((a/b)^(1/3)+x)^3/(a+b*x^3))+sqrt(3)*arctan((2*x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3))))",
#76
"f(x^2/(a+b*x^3),1/3*1/b*log(a+b*x^3))",
# float(defint(1/(2+3*X^4),X,0,pi)) gave wrong result.
# Also, the tests related to the indefinite integral
# fail since we rationalise expressions "better", so I'm thinking
# to take this out completely as it seemed to give the
# wrong results in the first place.
#77
#"f(1/(a+b*x^4),1/2*1/a*(a/b/4)^(1/4)*(1/2*log((x^2+2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2))/(x^2-2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2)))+arctan(2*(a/b/4)^(1/4)*x/(2*(a/b/4)^(1/2)-x^2))),or(not(number(a*b)),a*b>0))",
#78
#"f(1/(a+b*x^4),1/2*(-a/b)^(1/4)/a*(1/2*log((x+(-a/b)^(1/4))/(x-(-a/b)^(1/4)))+arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
#79
"f(x/(a+b*x^4),1/2*sqrt(b/a)/b*arctan(x^2*sqrt(b/a)),or(not(number(a*b)),a*b>0))",
#80
"f(x/(a+b*x^4),1/4*sqrt(-b/a)/b*log((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b))),or(not(number(a*b)),a*b<0))",
98
# float(defint(X^2/(2+3*X^4),X,0,pi)) gave wrong result.
# Also, the tests related to the indefinite integral
# fail since we rationalise expressions "better", so I'm thinking
# to take this out completely as it seemed to give the
# wrong results in the first place.
#81
#"f(x^2/(a+b*x^4),1/4*1/b*(a/b/4)^(-1/4)*(1/2*log((x^2-2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4))/(x^2+2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4)))+arctan(2*(a/b/4)^(1/4)*x/(2*sqrt(a/b/4)-x^2))),or(not(number(a*b)),a*b>0))",
#82
#"f(x^2/(a+b*x^4),1/4*1/b*(-a/b)^(-1/4)*(log((x-(-a/b)^(1/4))/(x+(-a/b)^(1/4)))+2*arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
#83
"f(x^3/(a+b*x^4),1/4*1/b*log(a+b*x^4))",
#124
"f(sqrt(a+b*x),2/3*1/b*sqrt((a+b*x)^3))",
#125
"f(x*sqrt(a+b*x),-2*(2*a-3*b*x)*sqrt((a+b*x)^3)/15/b^2)",
#126
"f(x^2*sqrt(a+b*x),2*(8*a^2-12*a*b*x+15*b^2*x^2)*sqrt((a+b*x)^3)/105/b^3)",
#128
"f(sqrt(a+b*x)/x,2*sqrt(a+b*x)+a*integral(1/x*1/sqrt(a+b*x),x))",
#129
"f(sqrt(a+b*x)/x^2,-sqrt(a+b*x)/x+b/2*integral(1/x*1/sqrt(a+b*x),x))",
#131
"f(1/sqrt(a+b*x),2*sqrt(a+b*x)/b)",
#132
"f(x/sqrt(a+b*x),-2/3*(2*a-b*x)*sqrt(a+b*x)/b^2)",
#133
"f(x^2/sqrt(a+b*x),2/15*(8*a^2-4*a*b*x+3*b^2*x^2)*sqrt(a+b*x)/b^3)",
#135
"f(1/x*1/sqrt(a+b*x),1/sqrt(a)*log((sqrt(a+b*x)-sqrt(a))/(sqrt(a+b*x)+sqrt(a))),or(not(number(a)),a>0))",
#136
"f(1/x*1/sqrt(a+b*x),2/sqrt(-a)*arctan(sqrt(-(a+b*x)/a)),or(not(number(a)),a<0))",
#137
"f(1/x^2*1/sqrt(a+b*x),-sqrt(a+b*x)/a/x-1/2*b/a*integral(1/x*1/sqrt(a+b*x),x))",
#156
"f(sqrt(x^2+a),1/2*(x*sqrt(x^2+a)+a*log(x+sqrt(x^2+a))))",
#157
"f(1/sqrt(x^2+a),log(x+sqrt(x^2+a)))",
#158
"f(1/x*1/sqrt(x^2+a),arcsec(x/sqrt(-a))/sqrt(-a),or(not(number(a)),a<0))",
#159
"f(1/x*1/sqrt(x^2+a),-1/sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
#160
"f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
#161
"f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a)),or(not(number(a)),a<0))",
#162
"f(x/sqrt(x^2+a),sqrt(x^2+a))",
#163
"f(x*sqrt(x^2+a),1/3*sqrt((x^2+a)^3))",
#164 need an unexpanded version?
"f(sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2+a^(1/3))^3)+3/2*a^(1/3)*x*sqrt(x^2+a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2+a^(1/3)))))",
# match doesn't work for the following
"f(sqrt(-a+x^6-3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2-a^(1/3))^3)-3/2*a^(1/3)*x*sqrt(x^2-a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2-a^(1/3)))))",
#165
"f(1/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),x/a^(1/3)/sqrt(x^2+a^(1/3)))",
#166
"f(x/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),-1/sqrt(x^2+a^(1/3)))",
#167
"f(x*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/5*sqrt((x^2+a^(1/3))^5))",
#168
"f(x^2*sqrt(x^2+a),1/4*x*sqrt((x^2+a)^3)-1/8*a*x*sqrt(x^2+a)-1/8*a^2*log(x+sqrt(x^2+a)))",
#169
"f(x^3*sqrt(x^2+a),(1/5*x^2-2/15*a)*sqrt((x^2+a)^3),and(number(a),a>0))",
#170
"f(x^3*sqrt(x^2+a),sqrt((x^2+a)^5)/5-a*sqrt((x^2+a)^3)/3,and(number(a),a<0))",
#171
"f(x^2/sqrt(x^2+a),1/2*x*sqrt(x^2+a)-1/2*a*log(x+sqrt(x^2+a)))",
#172
"f(x^3/sqrt(x^2+a),1/3*sqrt((x^2+a)^3)-a*sqrt(x^2+a))",
#173
"f(1/x^2*1/sqrt(x^2+a),-sqrt(x^2+a)/a/x)",
#174
"f(1/x^3*1/sqrt(x^2+a),-1/2*sqrt(x^2+a)/a/x^2+1/2*log((sqrt(a)+sqrt(x^2+a))/x)/a^(3/2),or(not(number(a)),a>0))",
#175
"f(1/x^3*1/sqrt(x^2-a),1/2*sqrt(x^2-a)/a/x^2+1/2*1/(a^(3/2))*arcsec(x/(a^(1/2))),or(not(number(a)),a>0))",
#176+
"f(x^2*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/6*x*sqrt((x^2+a^(1/3))^5)-1/24*a^(1/3)*x*sqrt((x^2+a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2+a^(1/3))-1/16*a*log(x+sqrt(x^2+a^(1/3))),or(not(number(a)),a>0))",
#176-
"f(x^2*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/6*x*sqrt((x^2-a^(1/3))^5)+1/24*a^(1/3)*x*sqrt((x^2-a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2-a^(1/3))+1/16*a*log(x+sqrt(x^2-a^(1/3))),or(not(number(a)),a>0))",
#177+
"f(x^3*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/7*sqrt((x^2+a^(1/3))^7)-1/5*a^(1/3)*sqrt((x^2+a^(1/3))^5),or(not(number(a)),a>0))",
#177-
"f(x^3*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/7*sqrt((x^2-a^(1/3))^7)+1/5*a^(1/3)*sqrt((x^2-a^(1/3))^5),or(not(number(a)),a>0))",
#196
"f(1/(x-a)/sqrt(x^2-a^2),-sqrt(x^2-a^2)/a/(x-a))",
#197
"f(1/(x+a)/sqrt(x^2-a^2),sqrt(x^2-a^2)/a/(x+a))",
#200+
"f(sqrt(a-x^2),1/2*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(abs(a)))))",
#201		(seems to be handled somewhere else)
#202
"f(1/x*1/sqrt(a-x^2),-1/sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
#203
"f(sqrt(a-x^2)/x,sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
#204
"f(x/sqrt(a-x^2),-sqrt(a-x^2))",
#205
"f(x*sqrt(a-x^2),-1/3*sqrt((a-x^2)^3))",
#210
"f(x^2*sqrt(a-x^2),-x/4*sqrt((a-x^2)^3)+1/8*a*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(a))),or(not(number(a)),a>0))",
#211
"f(x^3*sqrt(a-x^2),(-1/5*x^2-2/15*a)*sqrt((a-x^2)^3),or(not(number(a)),a>0))",
#214
"f(x^2/sqrt(a-x^2),-x/2*sqrt(a-x^2)+a/2*arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
#215
"f(1/x^2*1/sqrt(a-x^2),-sqrt(a-x^2)/a/x,or(not(number(a)),a>0))",
#216
"f(sqrt(a-x^2)/x^2,-sqrt(a-x^2)/x-arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
#217
"f(sqrt(a-x^2)/x^3,-1/2*sqrt(a-x^2)/x^2+1/2*log((sqrt(a)+sqrt(a-x^2))/x)/sqrt(a),or(not(number(a)),a>0))",
#218
"f(sqrt(a-x^2)/x^4,-1/3*sqrt((a-x^2)^3)/a/x^3,or(not(number(a)),a>0))",
# 273
"f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*log(x*sqrt(a)+sqrt(a*x^2+b))/2/sqrt(a),and(number(a),a>0))",
# 274
"f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*arcsin(x*sqrt(-a/b))/2/sqrt(-a),and(number(a),a<0))",
# 290
"f(sin(a*x),-cos(a*x)/a)",
# 291
"f(cos(a*x),sin(a*x)/a)",
# 292
"f(tan(a*x),-log(cos(a*x))/a)",
# 293
"f(1/tan(a*x),log(sin(a*x))/a)",
# 294
"f(1/cos(a*x),log(tan(pi/4+a*x/2))/a)",
# 295
"f(1/sin(a*x),log(tan(a*x/2))/a)",
# 296
"f(sin(a*x)^2,x/2-sin(2*a*x)/(4*a))",
# 297
"f(sin(a*x)^3,-cos(a*x)*(sin(a*x)^2+2)/(3*a))",
# 298
"f(sin(a*x)^4,3/8*x-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
# 302
"f(cos(a*x)^2,x/2+sin(2*a*x)/(4*a))",
# 303
"f(cos(a*x)^3,sin(a*x)*(cos(a*x)^2+2)/(3*a))",
# 304
"f(cos(a*x)^4,3/8*x+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
# 308
"f(1/sin(a*x)^2,-1/(a*tan(a*x)))",
# 312
"f(1/cos(a*x)^2,tan(a*x)/a)",
# 318
"f(sin(a*x)*cos(a*x),sin(a*x)^2/(2*a))",
# 320
"f(sin(a*x)^2*cos(a*x)^2,-sin(4*a*x)/(32*a)+x/8)",
# 326
"f(sin(a*x)/cos(a*x)^2,1/(a*cos(a*x)))",
# 327
"f(sin(a*x)^2/cos(a*x),(log(tan(pi/4+a*x/2))-sin(a*x))/a)",
# 328
"f(cos(a*x)/sin(a*x)^2,-1/(a*sin(a*x)))",
# 329
"f(1/(sin(a*x)*cos(a*x)),log(tan(a*x))/a)",
# 330
"f(1/(sin(a*x)*cos(a*x)^2),(1/cos(a*x)+log(tan(a*x/2)))/a)",
# 331
"f(1/(sin(a*x)^2*cos(a*x)),(log(tan(pi/4+a*x/2))-1/sin(a*x))/a)",
# 333
"f(1/(sin(a*x)^2*cos(a*x)^2),-2/(a*tan(2*a*x)))",
# 335
"f(sin(a+b*x),-cos(a+b*x)/b)",
# 336
"f(cos(a+b*x),sin(a+b*x)/b)",
# 337+ (with the addition of b)
"f(1/(b+b*sin(a*x)),-tan(pi/4-a*x/2)/a/b)",
# 337- (with the addition of b)
"f(1/(b-b*sin(a*x)),tan(pi/4+a*x/2)/a/b)",
# 338 (with the addition of b)
"f(1/(b+b*cos(a*x)),tan(a*x/2)/a/b)",
# 339 (with the addition of b)
"f(1/(b-b*cos(a*x)),-1/tan(a*x/2)/a/b)",
# 340
"f(1/(a+b*sin(x)),1/sqrt(b^2-a^2)*log((a*tan(x/2)+b-sqrt(b^2-a^2))/(a*tan(x/2)+b+sqrt(b^2-a^2))),b^2-a^2)", 	# check that b^2-a^2 is not zero
# 341
"f(1/(a+b*cos(x)),1/sqrt(b^2-a^2)*log((sqrt(b^2-a^2)*tan(x/2)+a+b)/(sqrt(b^2-a^2)*tan(x/2)-a-b)),b^2-a^2)", 	# check that b^2-a^2 is not zero
# 389
"f(x*sin(a*x),sin(a*x)/a^2-x*cos(a*x)/a)",
# 390
"f(x^2*sin(a*x),2*x*sin(a*x)/a^2-(a^2*x^2-2)*cos(a*x)/a^3)",
# 393
"f(x*cos(a*x),cos(a*x)/a^2+x*sin(a*x)/a)",
# 394
"f(x^2*cos(a*x),2*x*cos(a*x)/a^2+(a^2*x^2-2)*sin(a*x)/a^3)",
# 441
"f(arcsin(a*x),x*arcsin(a*x)+sqrt(1-a^2*x^2)/a)",
# 442
"f(arccos(a*x),x*arccos(a*x)-sqrt(1-a^2*x^2)/a)",
# 443
"f(arctan(a*x),x*arctan(a*x)-1/2*log(1+a^2*x^2)/a)",
# 485 (with addition of a)
"f(log(a*x),x*log(a*x)-x)",
# 486 (with addition of a)
"f(x*log(a*x),x^2*log(a*x)/2-x^2/4)",
# 487 (with addition of a)
"f(x^2*log(a*x),x^3*log(a*x)/3-1/9*x^3)",
# 489
"f(log(x)^2,x*log(x)^2-2*x*log(x)+2*x)",
# 493 (with addition of a)
"f(1/x*1/(a+log(x)),log(a+log(x)))",
# 499
"f(log(a*x+b),(a*x+b)*log(a*x+b)/a-x)",
# 500
"f(log(a*x+b)/x^2,a/b*log(x)-(a*x+b)*log(a*x+b)/b/x)",
# 554
"f(sinh(x),cosh(x))",
# 555
"f(cosh(x),sinh(x))",
# 556
"f(tanh(x),log(cosh(x)))",
# 560
"f(x*sinh(x),x*cosh(x)-sinh(x))",
# 562
"f(x*cosh(x),x*sinh(x)-cosh(x))",
# 566
"f(sinh(x)^2,sinh(2*x)/4-x/2)",
# 569
"f(tanh(x)^2,x-tanh(x))",
# 572
"f(cosh(x)^2,sinh(2*x)/4+x/2)",
# ?
"f(x^3*exp(a*x^2),exp(a*x^2)*(x^2/a-1/(a^2))/2)",
# ?
"f(x^3*exp(a*x^2+b),exp(a*x^2)*exp(b)*(x^2/a-1/(a^2))/2)",
# ?
"f(exp(a*x^2),-i*sqrt(pi)*erf(i*sqrt(a)*x)/sqrt(a)/2)",
# ?
"f(erf(a*x),x*erf(a*x)+exp(-a^2*x^2)/a/sqrt(pi))",
329
# these are needed for the surface integral in the manual
331
"f(x^2*(1-x^2)^(3/2),(x*sqrt(1-x^2)*(-8*x^4+14*x^2-3)+3*arcsin(x))/48)",
"f(x^2*(1-x^2)^(5/2),(x*sqrt(1-x^2)*(48*x^6-136*x^4+118*x^2-15)+15*arcsin(x))/384)",
"f(x^4*(1-x^2)^(3/2),(-x*sqrt(1-x^2)*(16*x^6-24*x^4+2*x^2+3)+3*arcsin(x))/128)",
335
"f(x*exp(a*x),exp(a*x)*(a*x-1)/(a^2))",
"f(x*exp(a*x+b),exp(a*x+b)*(a*x-1)/(a^2))",
338
"f(x^2*exp(a*x),exp(a*x)*(a^2*x^2-2*a*x+2)/(a^3))",
"f(x^2*exp(a*x+b),exp(a*x+b)*(a^2*x^2-2*a*x+2)/(a^3))",
341
"f(x^3*exp(a*x),exp(a*x)*x^3/a-3/a*integral(x^2*exp(a*x),x))",
"f(x^3*exp(a*x+b),exp(a*x+b)*x^3/a-3/a*integral(x^2*exp(a*x+b),x))",
0
345
346]
347
348
349#define F p3
350#define X p4
351#define N p5
352
353Eval_integral = ->
i = 0
n = 0
356
# evaluate 1st arg to get function F
358
p1 = cdr(p1)
push(car(p1))
Eval()
362
# evaluate 2nd arg and then...
364
# example		result of 2nd arg	what to do
#
# integral(f)		nil			guess X, N = nil
# integral(f,2)	2			guess X, N = 2
# integral(f,x)	x			X = x, N = nil
# integral(f,x,2)	x			X = x, N = 2
# integral(f,x,y)	x			X = x, N = y
372
p1 = cdr(p1)
push(car(p1))
Eval()
376
p2 = pop()
if (p2 == symbol(NIL))
guess()
push(symbol(NIL))
else if (isnum(p2))
guess()
push(p2)
else
push(p2)
p1 = cdr(p1)
push(car(p1))
Eval()
389
p5 = pop()
p4 = pop()
p3 = pop()
393
while (1)
395
# N might be a symbol instead of a number
397
if (isnum(p5))
	push(p5)
	n = pop_integer()
	if (isNaN(n))
		stop("nth integral: check n")
else
	n = 1
405
push(p3)
407
if (n >= 0)
	for i in [0...n]
		push(p4)
		integral()
else
	n = -n
	for i in [0...n]
		push(p4)
		derivative()
417
p3 = pop()
419
# if N is nil then arglist is exhausted
421
if (p5 == symbol(NIL))
	break
424
# otherwise...
426
# N		arg1		what to do
#
# number	nil		break
# number	number		N = arg1, continue
# number	symbol		X = arg1, N = arg2, continue
#
# symbol	nil		X = N, N = nil, continue
# symbol	number		X = N, N = arg1, continue
# symbol	symbol		X = N, N = arg1, continue
436
if (isnum(p5))
	p1 = cdr(p1)
	push(car(p1))
	Eval()
	p5 = pop()
	if (p5 == symbol(NIL))
		break;		# arglist exhausted
	if (isnum(p5))
		doNothing = 1		# N = arg1
	else
		p4 = p5;		# X = arg1
		p1 = cdr(p1)
		push(car(p1))
		Eval()
		p5 = pop();	# N = arg2
else
	p4 = p5;			# X = N
	p1 = cdr(p1)
	push(car(p1))
	Eval()
	p5 = pop();		# N = arg1
458
push(p3);	# final result
460
461integral = ->
save()
p2 = pop()
p1 = pop()
if (car(p1) == symbol(ADD))
integral_of_sum()
else if (car(p1) == symbol(MULTIPLY))
integral_of_product()
else
integral_of_form()
p1 = pop()
if (Find(p1, symbol(INTEGRAL)))
stop("integral: sorry, could not find a solution")
push(p1)
simplify();	# polish the result
Eval();		# normalize the result
restore()
478
479integral_of_sum = ->
p1 = cdr(p1)
push(car(p1))
push(p2)
integral()
p1 = cdr(p1)
while (iscons(p1))
push(car(p1))
push(p2)
integral()
add()
p1 = cdr(p1)
491
492integral_of_product = ->
push(p1)
push(p2)
partition()
p1 = pop();			# pop variable part
integral_of_form()
multiply();			# multiply constant part
499
500integral_of_form = ->
push(p1) # free variable
push(p2) # input expression
transform(itab, false)
p3 = pop()
if (p3 == symbol(NIL))
push_symbol(INTEGRAL)
push(p1)
push(p2)
list(3)
else
push(p3)
512
513

1	`###`
2	`Table of integrals`
3
4	`The symbol f is just a dummy symbol for creating a list f(A,B,C,C,...) where`
5
6	`A is the template expression`
7
8	`B is the result expression`
9
10	`C is an optional list of conditional expressions`
11	`###`
12
13	`itab = [`
14	`# 1`
15	`"f(a,a*x)",`
16	`# 9 (need a caveat for 7 so we can put 9 after 7)`
17	`"f(1/x,log(x))",`
18	`# 7`
19	`"f(x^a,x^(a+1)/(a+1))",`
20	`# 12`
21	`"f(exp(ax),1/aexp(a*x))",`
22	`"f(exp(ax+b),1/aexp(a*x+b))",`
23	`"f(xexp(ax^2),exp(ax^2)/(2a))",`
24	`"f(xexp(ax^2+b),exp(ax^2+b)/(2a))",`
25	`# 14`
26	`"f(log(ax),xlog(a*x)-x)",`
27	`# 15`
28	`"f(a^x,a^x/log(a),or(not(number(a)),a>0))",`
29	`# 16`
30	`"f(1/(a+x^2),1/sqrt(a)*arctan(x/sqrt(a)),or(not(number(a)),a>0))",`
31	`# 17`
32	`"f(1/(a-x^2),1/sqrt(a)*arctanh(x/sqrt(a)))",`
33	`# 19`
34	`"f(1/sqrt(a-x^2),arcsin(x/(sqrt(a))))",`
35	`# 20`
36	`"f(1/sqrt(a+x^2),log(x+sqrt(a+x^2)))",`
37	`# 27`
38	`"f(1/(a+bx),1/blog(a+b*x))",`
39	`# 28`
40	`"f(1/(a+bx)^2,-1/(b(a+b*x)))",`
41	`# 29`
42	`"f(1/(a+bx)^3,-1/(2b)1/(a+bx)^2)",`
43	`# 30`
44	`"f(x/(a+bx),x/b-alog(a+b*x)/b/b)",`
45	`# 31`
46	`"f(x/(a+bx)^2,1/b^2(log(a+bx)+a/(a+bx)))",`
47	`# 33`
48	`"f(x^2/(a+bx),1/b^2(1/2(a+bx)^2-2a(a+bx)+a^2log(a+b*x)))",`
49	`# 34`
50	`"f(x^2/(a+bx)^2,1/b^3(a+bx-2alog(a+bx)-a^2/(a+b*x)))",`
51	`# 35`
52	`"f(x^2/(a+bx)^3,1/b^3(log(a+bx)+2a/(a+bx)-1/2a^2/(a+b*x)^2))",`
53	`# 37`
54	`"f(1/x1/(a+bx),-1/alog((a+bx)/x))",`
55	`# 38`
56	`"f(1/x1/(a+bx)^2,1/a1/(a+bx)-1/a^2log((a+bx)/x))",`
57	`# 39`
58	`"f(1/x1/(a+bx)^3,1/a^3(1/2((2a+bx)/(a+bx))^2+log(x/(a+bx))))",`
59	`# 40`
60	`"f(1/x^21/(a+bx),-1/(ax)+b/a^2log((a+b*x)/x))",`
61	`# 41`
62	`"f(1/x^31/(a+bx),(2bx-a)/(2a^2x^2)+b^2/a^3log(x/(a+bx)))",`
63	`# 42`
64	`"f(1/x^21/(a+bx)^2,-(a+2bx)/(a^2x(a+bx))+2b/a^3log((a+bx)/x))",`
65	`# 60`
66	`"f(1/(a+bx^2),1/sqrt(ab)arctan(xsqrt(ab)/a),or(not(number(ab)),a*b>0))",`
67	`# 61`
68	`"f(1/(a+bx^2),1/(2sqrt(-ab))log((a+xsqrt(-ab))/(a-xsqrt(-ab))),or(not(number(ab)),ab<0))",`
69	`# 62 is the same as 60`
70	`# 63`
71	`"f(x/(a+bx^2),1/21/blog(a+bx^2))",`
72	`#64`
73	`"f(x^2/(a+bx^2),x/b-a/bintegral(1/(a+b*x^2),x))",`
74	`#65`
75	`"f(1/(a+bx^2)^2,x/(2a(a+bx^2))+1/21/aintegral(1/(a+b*x^2),x))",`
76	`#66 is covered by 61`
77	`#70`
78	`"f(1/x1/(a+bx^2),1/21/alog(x^2/(a+b*x^2)))",`
79	`#71`
80	`"f(1/x^21/(a+bx^2),-1/(ax)-b/aintegral(1/(a+b*x^2),x))",`
81	`#74`
82	`"f(1/(a+bx^3),1/31/a(a/b)^(1/3)(1/2log(((a/b)^(1/3)+x)^3/(a+bx^3))+sqrt(3)arctan((2x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3))))",`
83	`#76`
84	`"f(x^2/(a+bx^3),1/31/blog(a+bx^3))",`
85	`# float(defint(1/(2+3*X^4),X,0,pi)) gave wrong result.`
86	`# Also, the tests related to the indefinite integral`
87	`# fail since we rationalise expressions "better", so I'm thinking`
88	`# to take this out completely as it seemed to give the`
89	`# wrong results in the first place.`
90	`#77`
91	`#"f(1/(a+bx^4),1/21/a(a/b/4)^(1/4)(1/2log((x^2+2(a/b/4)^(1/4)x+2(a/b/4)^(1/2))/(x^2-2(a/b/4)^(1/4)x+2(a/b/4)^(1/2)))+arctan(2(a/b/4)^(1/4)x/(2(a/b/4)^(1/2)-x^2))),or(not(number(ab)),ab>0))",`
92	`#78`
93	`#"f(1/(a+bx^4),1/2(-a/b)^(1/4)/a(1/2log((x+(-a/b)^(1/4))/(x-(-a/b)^(1/4)))+arctan(x(-a/b)^(-1/4))),or(not(number(ab)),a*b<0))",`
94	`#79`
95	`"f(x/(a+bx^4),1/2sqrt(b/a)/barctan(x^2sqrt(b/a)),or(not(number(ab)),ab>0))",`
96	`#80`
97	`"f(x/(a+bx^4),1/4sqrt(-b/a)/blog((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b))),or(not(number(ab)),a*b<0))",`
98
99	`# float(defint(X^2/(2+3*X^4),X,0,pi)) gave wrong result.`
100	`# Also, the tests related to the indefinite integral`
101	`# fail since we rationalise expressions "better", so I'm thinking`
102	`# to take this out completely as it seemed to give the`
103	`# wrong results in the first place.`
104	`#81`
105	`#"f(x^2/(a+bx^4),1/41/b(a/b/4)^(-1/4)(1/2log((x^2-2(a/b/4)^(1/4)x+2sqrt(a/b/4))/(x^2+2(a/b/4)^(1/4)x+2sqrt(a/b/4)))+arctan(2(a/b/4)^(1/4)x/(2sqrt(a/b/4)-x^2))),or(not(number(ab)),ab>0))",`
106	`#82`
107	`#"f(x^2/(a+bx^4),1/41/b(-a/b)^(-1/4)(log((x-(-a/b)^(1/4))/(x+(-a/b)^(1/4)))+2arctan(x(-a/b)^(-1/4))),or(not(number(ab)),ab<0))",`
108	`#83`
109	`"f(x^3/(a+bx^4),1/41/blog(a+bx^4))",`
110	`#124`
111	`"f(sqrt(a+bx),2/31/bsqrt((a+bx)^3))",`
112	`#125`
113	`"f(xsqrt(a+bx),-2(2a-3bx)sqrt((a+bx)^3)/15/b^2)",`
114	`#126`
115	`"f(x^2sqrt(a+bx),2(8a^2-12abx+15b^2x^2)sqrt((a+b*x)^3)/105/b^3)",`
116	`#128`
117	`"f(sqrt(a+bx)/x,2sqrt(a+bx)+aintegral(1/x1/sqrt(a+bx),x))",`
118	`#129`
119	`"f(sqrt(a+bx)/x^2,-sqrt(a+bx)/x+b/2integral(1/x1/sqrt(a+b*x),x))",`
120	`#131`
121	`"f(1/sqrt(a+bx),2sqrt(a+b*x)/b)",`
122	`#132`
123	`"f(x/sqrt(a+bx),-2/3(2a-bx)sqrt(a+bx)/b^2)",`
124	`#133`
125	`"f(x^2/sqrt(a+bx),2/15(8a^2-4abx+3b^2x^2)sqrt(a+bx)/b^3)",`
126	`#135`
127	`"f(1/x1/sqrt(a+bx),1/sqrt(a)log((sqrt(a+bx)-sqrt(a))/(sqrt(a+b*x)+sqrt(a))),or(not(number(a)),a>0))",`
128	`#136`
129	`"f(1/x1/sqrt(a+bx),2/sqrt(-a)arctan(sqrt(-(a+bx)/a)),or(not(number(a)),a<0))",`
130	`#137`
131	`"f(1/x^21/sqrt(a+bx),-sqrt(a+bx)/a/x-1/2b/aintegral(1/x1/sqrt(a+b*x),x))",`
132	`#156`
133	`"f(sqrt(x^2+a),1/2(xsqrt(x^2+a)+a*log(x+sqrt(x^2+a))))",`
134	`#157`
135	`"f(1/sqrt(x^2+a),log(x+sqrt(x^2+a)))",`
136	`#158`
137	`"f(1/x*1/sqrt(x^2+a),arcsec(x/sqrt(-a))/sqrt(-a),or(not(number(a)),a<0))",`
138	`#159`
139	`"f(1/x1/sqrt(x^2+a),-1/sqrt(a)log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",`
140	`#160`
141	`"f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",`
142	`#161`
143	`"f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a)),or(not(number(a)),a<0))",`
144	`#162`
145	`"f(x/sqrt(x^2+a),sqrt(x^2+a))",`
146	`#163`
147	`"f(xsqrt(x^2+a),1/3sqrt((x^2+a)^3))",`
148	`#164 need an unexpanded version?`
149	`"f(sqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),1/4(xsqrt((x^2+a^(1/3))^3)+3/2a^(1/3)xsqrt(x^2+a^(1/3))+3/2a^(2/3)*log(x+sqrt(x^2+a^(1/3)))))",`
150	`# match doesn't work for the following`
151	`"f(sqrt(-a+x^6-3a^(1/3)x^4+3a^(2/3)x^2),1/4(xsqrt((x^2-a^(1/3))^3)-3/2a^(1/3)xsqrt(x^2-a^(1/3))+3/2a^(2/3)*log(x+sqrt(x^2-a^(1/3)))))",`
152	`#165`
153	`"f(1/sqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),x/a^(1/3)/sqrt(x^2+a^(1/3)))",`
154	`#166`
155	`"f(x/sqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),-1/sqrt(x^2+a^(1/3)))",`
156	`#167`
157	`"f(xsqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),1/5sqrt((x^2+a^(1/3))^5))",`
158	`#168`
159	`"f(x^2sqrt(x^2+a),1/4xsqrt((x^2+a)^3)-1/8axsqrt(x^2+a)-1/8a^2log(x+sqrt(x^2+a)))",`
160	`#169`
161	`"f(x^3sqrt(x^2+a),(1/5x^2-2/15a)sqrt((x^2+a)^3),and(number(a),a>0))",`
162	`#170`
163	`"f(x^3sqrt(x^2+a),sqrt((x^2+a)^5)/5-asqrt((x^2+a)^3)/3,and(number(a),a<0))",`
164	`#171`
165	`"f(x^2/sqrt(x^2+a),1/2xsqrt(x^2+a)-1/2alog(x+sqrt(x^2+a)))",`
166	`#172`
167	`"f(x^3/sqrt(x^2+a),1/3sqrt((x^2+a)^3)-asqrt(x^2+a))",`
168	`#173`
169	`"f(1/x^2*1/sqrt(x^2+a),-sqrt(x^2+a)/a/x)",`
170	`#174`
171	`"f(1/x^31/sqrt(x^2+a),-1/2sqrt(x^2+a)/a/x^2+1/2*log((sqrt(a)+sqrt(x^2+a))/x)/a^(3/2),or(not(number(a)),a>0))",`
172	`#175`
173	`"f(1/x^31/sqrt(x^2-a),1/2sqrt(x^2-a)/a/x^2+1/21/(a^(3/2))arcsec(x/(a^(1/2))),or(not(number(a)),a>0))",`
174	`#176+`
175	`"f(x^2sqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),1/6xsqrt((x^2+a^(1/3))^5)-1/24a^(1/3)xsqrt((x^2+a^(1/3))^3)-1/16a^(2/3)xsqrt(x^2+a^(1/3))-1/16a*log(x+sqrt(x^2+a^(1/3))),or(not(number(a)),a>0))",`
176	`#176-`
177	`"f(x^2sqrt(-a-3a^(1/3)x^4+3a^(2/3)x^2+x^6),1/6xsqrt((x^2-a^(1/3))^5)+1/24a^(1/3)xsqrt((x^2-a^(1/3))^3)-1/16a^(2/3)xsqrt(x^2-a^(1/3))+1/16a*log(x+sqrt(x^2-a^(1/3))),or(not(number(a)),a>0))",`
178	`#177+`
179	`"f(x^3sqrt(a+x^6+3a^(1/3)x^4+3a^(2/3)x^2),1/7sqrt((x^2+a^(1/3))^7)-1/5a^(1/3)sqrt((x^2+a^(1/3))^5),or(not(number(a)),a>0))",`
180	`#177-`
181	`"f(x^3sqrt(-a-3a^(1/3)x^4+3a^(2/3)x^2+x^6),1/7sqrt((x^2-a^(1/3))^7)+1/5a^(1/3)sqrt((x^2-a^(1/3))^5),or(not(number(a)),a>0))",`
182	`#196`
183	`"f(1/(x-a)/sqrt(x^2-a^2),-sqrt(x^2-a^2)/a/(x-a))",`
184	`#197`
185	`"f(1/(x+a)/sqrt(x^2-a^2),sqrt(x^2-a^2)/a/(x+a))",`
186	`#200+`
187	`"f(sqrt(a-x^2),1/2(xsqrt(a-x^2)+a*arcsin(x/sqrt(abs(a)))))",`
188	`#201 (seems to be handled somewhere else)`
189	`#202`
190	`"f(1/x1/sqrt(a-x^2),-1/sqrt(a)log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",`
191	`#203`
192	`"f(sqrt(a-x^2)/x,sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",`
193	`#204`
194	`"f(x/sqrt(a-x^2),-sqrt(a-x^2))",`
195	`#205`
196	`"f(xsqrt(a-x^2),-1/3sqrt((a-x^2)^3))",`
197	`#210`
198	`"f(x^2sqrt(a-x^2),-x/4sqrt((a-x^2)^3)+1/8a(xsqrt(a-x^2)+aarcsin(x/sqrt(a))),or(not(number(a)),a>0))",`
199	`#211`
200	`"f(x^3sqrt(a-x^2),(-1/5x^2-2/15a)sqrt((a-x^2)^3),or(not(number(a)),a>0))",`
201	`#214`
202	`"f(x^2/sqrt(a-x^2),-x/2sqrt(a-x^2)+a/2arcsin(x/sqrt(a)),or(not(number(a)),a>0))",`
203	`#215`
204	`"f(1/x^2*1/sqrt(a-x^2),-sqrt(a-x^2)/a/x,or(not(number(a)),a>0))",`
205	`#216`
206	`"f(sqrt(a-x^2)/x^2,-sqrt(a-x^2)/x-arcsin(x/sqrt(a)),or(not(number(a)),a>0))",`
207	`#217`
208	`"f(sqrt(a-x^2)/x^3,-1/2sqrt(a-x^2)/x^2+1/2log((sqrt(a)+sqrt(a-x^2))/x)/sqrt(a),or(not(number(a)),a>0))",`
209	`#218`
210	`"f(sqrt(a-x^2)/x^4,-1/3*sqrt((a-x^2)^3)/a/x^3,or(not(number(a)),a>0))",`
211	`# 273`
212	`"f(sqrt(ax^2+b),xsqrt(ax^2+b)/2+blog(xsqrt(a)+sqrt(ax^2+b))/2/sqrt(a),and(number(a),a>0))",`
213	`# 274`
214	`"f(sqrt(ax^2+b),xsqrt(ax^2+b)/2+barcsin(x*sqrt(-a/b))/2/sqrt(-a),and(number(a),a<0))",`
215	`# 290`
216	`"f(sin(ax),-cos(ax)/a)",`
217	`# 291`
218	`"f(cos(ax),sin(ax)/a)",`
219	`# 292`
220	`"f(tan(ax),-log(cos(ax))/a)",`
221	`# 293`
222	`"f(1/tan(ax),log(sin(ax))/a)",`
223	`# 294`
224	`"f(1/cos(ax),log(tan(pi/4+ax/2))/a)",`
225	`# 295`
226	`"f(1/sin(ax),log(tan(ax/2))/a)",`
227	`# 296`
228	`"f(sin(ax)^2,x/2-sin(2ax)/(4a))",`
229	`# 297`
230	`"f(sin(ax)^3,-cos(ax)(sin(ax)^2+2)/(3*a))",`
231	`# 298`
232	`"f(sin(ax)^4,3/8x-sin(2ax)/(4a)+sin(4ax)/(32a))",`
233	`# 302`
234	`"f(cos(ax)^2,x/2+sin(2ax)/(4a))",`
235	`# 303`
236	`"f(cos(ax)^3,sin(ax)(cos(ax)^2+2)/(3*a))",`
237	`# 304`
238	`"f(cos(ax)^4,3/8x+sin(2ax)/(4a)+sin(4ax)/(32a))",`
239	`# 308`
240	`"f(1/sin(ax)^2,-1/(atan(a*x)))",`
241	`# 312`
242	`"f(1/cos(ax)^2,tan(ax)/a)",`
243	`# 318`
244	`"f(sin(ax)cos(ax),sin(ax)^2/(2*a))",`
245	`# 320`
246	`"f(sin(ax)^2cos(ax)^2,-sin(4ax)/(32a)+x/8)",`
247	`# 326`
248	`"f(sin(ax)/cos(ax)^2,1/(acos(ax)))",`
249	`# 327`
250	`"f(sin(ax)^2/cos(ax),(log(tan(pi/4+ax/2))-sin(ax))/a)",`
251	`# 328`
252	`"f(cos(ax)/sin(ax)^2,-1/(asin(ax)))",`
253	`# 329`
254	`"f(1/(sin(ax)cos(ax)),log(tan(ax))/a)",`
255	`# 330`
256	`"f(1/(sin(ax)cos(ax)^2),(1/cos(ax)+log(tan(a*x/2)))/a)",`
257	`# 331`
258	`"f(1/(sin(ax)^2cos(ax)),(log(tan(pi/4+ax/2))-1/sin(a*x))/a)",`
259	`# 333`
260	`"f(1/(sin(ax)^2cos(ax)^2),-2/(atan(2ax)))",`
261	`# 335`
262	`"f(sin(a+bx),-cos(a+bx)/b)",`
263	`# 336`
264	`"f(cos(a+bx),sin(a+bx)/b)",`
265	`# 337+ (with the addition of b)`
266	`"f(1/(b+bsin(ax)),-tan(pi/4-a*x/2)/a/b)",`
267	`# 337- (with the addition of b)`
268	`"f(1/(b-bsin(ax)),tan(pi/4+a*x/2)/a/b)",`
269	`# 338 (with the addition of b)`
270	`"f(1/(b+bcos(ax)),tan(a*x/2)/a/b)",`
271	`# 339 (with the addition of b)`
272	`"f(1/(b-bcos(ax)),-1/tan(a*x/2)/a/b)",`
273	`# 340`
274	`"f(1/(a+bsin(x)),1/sqrt(b^2-a^2)log((atan(x/2)+b-sqrt(b^2-a^2))/(atan(x/2)+b+sqrt(b^2-a^2))),b^2-a^2)", # check that b^2-a^2 is not zero`
275	`# 341`
276	`"f(1/(a+bcos(x)),1/sqrt(b^2-a^2)log((sqrt(b^2-a^2)tan(x/2)+a+b)/(sqrt(b^2-a^2)tan(x/2)-a-b)),b^2-a^2)", # check that b^2-a^2 is not zero`
277	`# 389`
278	`"f(xsin(ax),sin(ax)/a^2-xcos(a*x)/a)",`
279	`# 390`
280	`"f(x^2sin(ax),2xsin(ax)/a^2-(a^2x^2-2)cos(ax)/a^3)",`
281	`# 393`
282	`"f(xcos(ax),cos(ax)/a^2+xsin(a*x)/a)",`
283	`# 394`
284	`"f(x^2cos(ax),2xcos(ax)/a^2+(a^2x^2-2)sin(ax)/a^3)",`
285	`# 441`
286	`"f(arcsin(ax),xarcsin(ax)+sqrt(1-a^2x^2)/a)",`
287	`# 442`
288	`"f(arccos(ax),xarccos(ax)-sqrt(1-a^2x^2)/a)",`
289	`# 443`
290	`"f(arctan(ax),xarctan(ax)-1/2log(1+a^2*x^2)/a)",`
291	`# 485 (with addition of a)`
292	`"f(log(ax),xlog(a*x)-x)",`
293	`# 486 (with addition of a)`
294	`"f(xlog(ax),x^2log(ax)/2-x^2/4)",`
295	`# 487 (with addition of a)`
296	`"f(x^2log(ax),x^3log(ax)/3-1/9*x^3)",`
297	`# 489`
298	`"f(log(x)^2,xlog(x)^2-2xlog(x)+2x)",`
299	`# 493 (with addition of a)`
300	`"f(1/x*1/(a+log(x)),log(a+log(x)))",`
301	`# 499`
302	`"f(log(ax+b),(ax+b)log(ax+b)/a-x)",`
303	`# 500`
304	`"f(log(ax+b)/x^2,a/blog(x)-(ax+b)log(a*x+b)/b/x)",`
305	`# 554`
306	`"f(sinh(x),cosh(x))",`
307	`# 555`
308	`"f(cosh(x),sinh(x))",`
309	`# 556`
310	`"f(tanh(x),log(cosh(x)))",`
311	`# 560`
312	`"f(xsinh(x),xcosh(x)-sinh(x))",`
313	`# 562`
314	`"f(xcosh(x),xsinh(x)-cosh(x))",`
315	`# 566`
316	`"f(sinh(x)^2,sinh(2*x)/4-x/2)",`
317	`# 569`
318	`"f(tanh(x)^2,x-tanh(x))",`
319	`# 572`
320	`"f(cosh(x)^2,sinh(2*x)/4+x/2)",`
321	`# ?`
322	`"f(x^3exp(ax^2),exp(ax^2)(x^2/a-1/(a^2))/2)",`
323	`# ?`
324	`"f(x^3exp(ax^2+b),exp(ax^2)exp(b)*(x^2/a-1/(a^2))/2)",`
325	`# ?`
326	`"f(exp(ax^2),-isqrt(pi)erf(isqrt(a)*x)/sqrt(a)/2)",`
327	`# ?`
328	`"f(erf(ax),xerf(ax)+exp(-a^2x^2)/a/sqrt(pi))",`
329
330	`# these are needed for the surface integral in the manual`
331
332	`"f(x^2(1-x^2)^(3/2),(xsqrt(1-x^2)(-8x^4+14x^2-3)+3arcsin(x))/48)",`
333	`"f(x^2(1-x^2)^(5/2),(xsqrt(1-x^2)(48x^6-136x^4+118x^2-15)+15*arcsin(x))/384)",`
334	`"f(x^4(1-x^2)^(3/2),(-xsqrt(1-x^2)(16x^6-24x^4+2x^2+3)+3*arcsin(x))/128)",`
335
336	`"f(xexp(ax),exp(ax)(a*x-1)/(a^2))",`
337	`"f(xexp(ax+b),exp(ax+b)(a*x-1)/(a^2))",`
338
339	`"f(x^2exp(ax),exp(ax)(a^2x^2-2a*x+2)/(a^3))",`
340	`"f(x^2exp(ax+b),exp(ax+b)(a^2x^2-2a*x+2)/(a^3))",`
341
342	`"f(x^3exp(ax),exp(ax)x^3/a-3/aintegral(x^2exp(a*x),x))",`
343	`"f(x^3exp(ax+b),exp(ax+b)x^3/a-3/aintegral(x^2exp(a*x+b),x))",`
344	`0`
345
346	`]`
347
348
349	`#define F p3`
350	`#define X p4`
351	`#define N p5`
352
353	`Eval_integral = ->`
354	`i = 0`
355	`n = 0`
356
357	`# evaluate 1st arg to get function F`
358
359	`p1 = cdr(p1)`
360	`push(car(p1))`
361	`Eval()`
362
363	`# evaluate 2nd arg and then...`
364
365	`# example result of 2nd arg what to do`
366	`#`
367	`# integral(f) nil guess X, N = nil`
368	`# integral(f,2) 2 guess X, N = 2`
369	`# integral(f,x) x X = x, N = nil`
370	`# integral(f,x,2) x X = x, N = 2`
371	`# integral(f,x,y) x X = x, N = y`
372
373	`p1 = cdr(p1)`
374	`push(car(p1))`
375	`Eval()`
376
377	`p2 = pop()`
378	`if (p2 == symbol(NIL))`
379	`guess()`
380	`push(symbol(NIL))`
381	`else if (isnum(p2))`
382	`guess()`
383	`push(p2)`
384	`else`
385	`push(p2)`
386	`p1 = cdr(p1)`
387	`push(car(p1))`
388	`Eval()`
389
390	`p5 = pop()`
391	`p4 = pop()`
392	`p3 = pop()`
393
394	`while (1)`
395
396	`# N might be a symbol instead of a number`
397
398	`if (isnum(p5))`
399	`push(p5)`
400	`n = pop_integer()`
401	`if (isNaN(n))`
402	`stop("nth integral: check n")`
403	`else`
404	`n = 1`
405
406	`push(p3)`
407
408	`if (n >= 0)`
409	`for i in [0...n]`
410	`push(p4)`
411	`integral()`
412	`else`
413	`n = -n`
414	`for i in [0...n]`
415	`push(p4)`
416	`derivative()`
417
418	`p3 = pop()`
419
420	`# if N is nil then arglist is exhausted`
421
422	`if (p5 == symbol(NIL))`
423	`break`
424
425	`# otherwise...`
426
427	`# N arg1 what to do`
428	`#`
429	`# number nil break`
430	`# number number N = arg1, continue`
431	`# number symbol X = arg1, N = arg2, continue`
432	`#`
433	`# symbol nil X = N, N = nil, continue`
434	`# symbol number X = N, N = arg1, continue`
435	`# symbol symbol X = N, N = arg1, continue`
436
437	`if (isnum(p5))`
438	`p1 = cdr(p1)`
439	`push(car(p1))`
440	`Eval()`
441	`p5 = pop()`
442	`if (p5 == symbol(NIL))`
443	`break; # arglist exhausted`
444	`if (isnum(p5))`
445	`doNothing = 1 # N = arg1`
446	`else`
447	`p4 = p5; # X = arg1`
448	`p1 = cdr(p1)`
449	`push(car(p1))`
450	`Eval()`
451	`p5 = pop(); # N = arg2`
452	`else`
453	`p4 = p5; # X = N`
454	`p1 = cdr(p1)`
455	`push(car(p1))`
456	`Eval()`
457	`p5 = pop(); # N = arg1`
458
459	`push(p3); # final result`
460
461	`integral = ->`
462	`save()`
463	`p2 = pop()`
464	`p1 = pop()`
465	`if (car(p1) == symbol(ADD))`
466	`integral_of_sum()`
467	`else if (car(p1) == symbol(MULTIPLY))`
468	`integral_of_product()`
469	`else`
470	`integral_of_form()`
471	`p1 = pop()`
472	`if (Find(p1, symbol(INTEGRAL)))`
473	`stop("integral: sorry, could not find a solution")`
474	`push(p1)`
475	`simplify(); # polish the result`
476	`Eval(); # normalize the result`
477	`restore()`
478
479	`integral_of_sum = ->`
480	`p1 = cdr(p1)`
481	`push(car(p1))`
482	`push(p2)`
483	`integral()`
484	`p1 = cdr(p1)`
485	`while (iscons(p1))`
486	`push(car(p1))`
487	`push(p2)`
488	`integral()`
489	`add()`
490	`p1 = cdr(p1)`
491
492	`integral_of_product = ->`
493	`push(p1)`
494	`push(p2)`
495	`partition()`
496	`p1 = pop(); # pop variable part`
497	`integral_of_form()`
498	`multiply(); # multiply constant part`
499
500	`integral_of_form = ->`
501	`push(p1) # free variable`
502	`push(p2) # input expression`
503	`transform(itab, false)`
504	`p3 = pop()`
505	`if (p3 == symbol(NIL))`
506	`push_symbol(INTEGRAL)`
507	`push(p1)`
508	`push(p2)`
509	`list(3)`
510	`else`
511	`push(p3)`
512
513