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// Giac integration test file: "0 Independent test suites\Apostol Problems.txt"
// "nock" means we do not check the antiderivative
I:=i;
lst:=[
// Joel Moses - Symbolic Integration Ph.D. Thesis (1967)
// Chapter 2 - How SIN differs from SAINT
[cot(x)^4,x,3,x+cot(x)-1/3*cot(x)^3],
[1/(x^4*(1+x^2)),x,3,(-1/3)/x^3+1/x+arctan(x)],
[(x+x^2)/sqrt(x),x,2,2/3*x^(3/2)+2/5*x^(5/2)],
[cos(x),x,1,sin(x)],
[exp(x^2)*x,x,1,1/2*exp(x^2)],
[sec(x)^2*tan(x),x,2,1/2*sec(x)^2],
[x*sqrt(1+x^2),x,1,1/3*(1+x^2)^(3/2)],
[exp(x)*sin(x),x,1,-1/2*exp(x)*cos(x)+1/2*exp(x)*sin(x)],
// Chapter 3 - SCHATCHEN - A Matching Program for Algebraic Expressions
[cos(x)*csc(x)^2/sin(x)^2,x,2,-1/3*csc(x)^3],
// Chapter 4 - The First Stage of Sin
[sin(exp(x)),x,2,Si(exp(x))],
[sin(y)/y,y,1,Si(y)],
[exp(x)+sin(x),x,3,exp(x)-cos(x)],
[exp(x^2)+2*exp(x^2)*x^2,x,4,exp(x^2)*x],
[(exp(x)+x)^2,x,5,-2*exp(x)+1/2*exp(2*x)+2*exp(x)*x+1/3*x^3],
[2*exp(x)+exp(2*x)+x^2,x,3,2*exp(x)+1/2*exp(2*x)+1/3*x^3],
[cos(x)*sin(x),x,2,1/2*sin(x)^2],
[exp(x^2)*x,x,1,1/2*exp(x^2)],
[x*sqrt(1+x^2),x,1,1/3*(1+x^2)^(3/2)],
[exp(x)/(1+exp(x)),x,2,log(1+exp(x))],
[x^(3/2),x,1,2/5*x^(5/2)],
[cos(3+2*x),x,1,1/2*sin(3+2*x)],
[2*exp(2*x)*y*z,x,2,exp(2*x)*y*z],
[exp(x)*cos(exp(x))^2*sin(exp(x)),x,3,-1/3*cos(exp(x))^3],
// Chapter 4 - The Second Stage of Sin
[x*sqrt(1+x),x,2,-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)],
[1/(-1+x^4),x,3,-1/2*arctan(x)-1/2*arctanh(x)],
// Method 1) Elementary function of exponentials
[exp(x)/(2+3*exp(2*x)),x,2,arctan(exp(x)*sqrt(3/2))/sqrt(6)],
[exp(2*x)/(A+B*exp(4*x)),x,2,1/2*arctan(exp(2*x)*sqrt(B)/sqrt(A))/(sqrt(A)*sqrt(B))],
[exp(1+x)/(1+exp(x)),x,3,E*log(1+exp(x))],
[(10*E)^x,x,1,(10*E)^x/(1+log(10))],
// Method 2) Substitution for an integral power
[x^3*sin(x^2),x,3,-1/2*x^2*cos(x^2)+1/2*sin(x^2)],
[x^7/(1+x^12),x,7,-1/12*log(1+x^4)+1/24*log(1-x^4+x^8)-1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[x^(3*a)*sin(x^(2*a)),x,3,1/4*I*x^(1+3*a)*GAMMA(1/2*(3+1/a),-I*x^(2*a))/(a*(-I*x^(2*a))^(1/2*(1+3*a)/a))-1/4*I*x^(1+3*a)*GAMMA(1/2*(3+1/a),I*x^(2*a))/(a*(I*x^(2*a))^(1/2*(1+3*a)/a))],
// Method 3) Substitution for a rational root of a linear function of x
[cos(sqrt(x)),x,3,2*cos(sqrt(x))+2*sin(sqrt(x))*sqrt(x)],
[x*sqrt(1+x),x,2,-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)],
[1/(x^(1/3)+x^(1/2)),x,4,6*x^(1/6)-3*x^(1/3)-6*log(1+x^(1/6))+2*sqrt(x)],
[sqrt((1+x)/(3+2*x)),x,4,-1/2*arcsinh(sqrt(2)*sqrt(1+x))/sqrt(2)+1/2*sqrt(1+x)*sqrt(3+2*x)],
// Method 4) Binomial - Chebyschev
[x^4/(1-x^2)^(5/2),x,3,1/3*x^3/(1-x^2)^(3/2)+arcsin(x)-x/sqrt(1-x^2)],
[x^(1/2)*(1+x)^(5/2),x,6,5/24*x^(3/2)*(1+x)^(3/2)+1/4*x^(3/2)*(1+x)^(5/2)-5/64*arcsinh(sqrt(x))+5/32*x^(3/2)*sqrt(1+x)+5/64*sqrt(x)*sqrt(1+x)],
// Method 5) Arctrigonometric substitutions
[x^4/(1-x^2)^(5/2),x,3,1/3*x^3/(1-x^2)^(3/2)+arcsin(x)-x/sqrt(1-x^2)],
[sqrt(A^2+B^2-B^2*y^2)/(1-y^2),y,5,B*arctan(B*y/sqrt(A^2+B^2-B^2*y^2))+A*arctanh(A*y/sqrt(A^2+B^2-B^2*y^2))],
// Method 6) Elementary function of trigonometric functions
[sin(x)^2,x,2,1/2*x-1/2*cos(x)*sin(x)],
[sqrt(A^2+B^2*sin(x)^2)/sin(x),x,6,-B*arctan(B*cos(x)/sqrt(A^2+B^2*sin(x)^2))-A*arctanh(A*cos(x)/sqrt(A^2+B^2*sin(x)^2)),-B*arctan(B*cos(x)/sqrt(A^2+B^2-B^2*cos(x)^2))-A*arctanh(A*cos(x)/sqrt(A^2+B^2-B^2*cos(x)^2))],
[1/(1+cos(x)),x,1,sin(x)/(1+cos(x))],
// Method 7) Rational function times an exponential
[exp(x)*x,x,2,-exp(x)+exp(x)*x],
[exp(x)*x/(1+x)^2,x,1,exp(x)/(1+x)],
[exp(x^2)*(1+2*x^2),x,5,exp(x^2)*x],
[exp(x^2),x,1,1/2*erfi(x)*sqrt(Pi)],
[exp(x)/x,x,1,Ei(x)],
// Method 8) Rational functions
[x/(1+x^3),x,6,-1/3*log(1+x)+1/6*log(1-x+x^2)-arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(-1+x^6),x,10,-1/3*arctanh(x)-1/6*arctanh(x/(1+x^2))-1/2*arctan(x*sqrt(3)/(1-x^2))/sqrt(3),-1/3*arctanh(x)+1/12*log(1-x+x^2)-1/12*log(1+x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[1/(A^4-A^2*B^2+(-A^2+B^2)*x^2),x,1,arctanh(x/A)/(A*(A^2-B^2))],
// Method 9) Rational function times a log or arctrigonometric function
[x*log(x),x,1,-1/4*x^2+1/2*x^2*log(x)],
[x^2*arcsin(x),x,4,-1/9*(1-x^2)^(3/2)+1/3*x^3*arcsin(x)+1/3*sqrt(1-x^2)],
[1/(1+2*x+x^2),x,2,(-1)/(1+x)],
// Method 10) Rational function times an elementary function of log(a+b x)
[log(x)/(1+log(x))^2,x,7,x/(1+log(x))],
[1/(x*(1+log(x)^2)),x,2,arctan(log(x))],
[1/log(x),x,1,Li(x)],
// Method 11) Expansion of the integrand
[x*(cos(x)+sin(x)),x,6,cos(x)-x*cos(x)+sin(x)+x*sin(x)],
[(exp(x)+x)/exp(x),x,4,(-1)/exp(x)+x-x/exp(x)],
[(1+exp(x))^2*x,x,6,-2*exp(x)-1/4*exp(2*x)+2*exp(x)*x+1/2*exp(2*x)*x+1/2*x^2],
// Chapter 4 - The Third Stage of Sin
[x*cos(x),x,2,cos(x)+x*sin(x)],
[cos(sqrt(x)),x,3,2*cos(sqrt(x))+2*sin(sqrt(x))*sqrt(x)],
// The Integration-by-Parts Methods
[x*cos(x),x,2,cos(x)+x*sin(x)],
[x*log(x)^2,x,2,1/4*x^2-1/2*x^2*log(x)+1/2*x^2*log(x)^2],
// The Derivative-divides Method
[cos(x)*(1+sin(x)^3),x,2,sin(x)+1/4*sin(x)^4],
[1/(x*(1+log(x)^2)),x,2,arctan(log(x))],
[1/((1+arcsin(x)^2)*sqrt(1-x^2)),x,2,arctan(arcsin(x))],
[sin(x)/(cos(x)+sin(x)),x,2,1/2*x-1/2*log(cos(x)+sin(x))],
// An Example of SIN's Performance
[-sqrt(A^2+B^2*(1-y^2))/(1-y^2),y,6,-B*arctan(B*y/sqrt(A^2+B^2-B^2*y^2))-A*arctanh(A*y/sqrt(A^2+B^2-B^2*y^2))],
[(-A^2-B^2)*cos(z)^2/(B*(1-(A^2+B^2)*sin(z)^2/B^2)),z,5,-B*z-A*arctanh(A*tan(z)/B)],
[(-A^2-B^2)/(B*(1+w^2)^2*(1-(A^2+B^2)*w^2/(B^2*(1+w^2)))),w,6,-B*arctan(w)-A*arctanh(A*w/B)],
[-B*(A^2+B^2)/((1+w^2)*(B^2-A^2*w^2)),w,4,-B*arctan(w)-A*arctanh(A*w/B)],
// SAINT and SIN solutions of the same problem
[x^4/(1-x^2)^(5/2),x,3,1/3*x^3/(1-x^2)^(3/2)+arcsin(x)-x/sqrt(1-x^2)],
[sin(y)^4/cos(y)^4,y,3,y-tan(y)+1/3*tan(y)^3],
[z^4/(1+z^2),z,3,-z+1/3*z^3+arctan(z)],
// Chapter 5 - The Edge Heuristic
[exp(x^2)*(1+2*x^2),x,5,exp(x^2)*x],
[exp(x^2)*(1+4*x^2+x^3+5*x^4+2*x^6)/(1+x^2)^2,x,10,exp(x^2)*x+1/2*exp(x^2)/(1+x^2)],
[exp(-1-x),x,1,-exp(-1-x)],
[(1/x+x)*log(x),x,5,-1/4*x^2+1/2*x^2*log(x)+1/2*log(x)^2],
[x/(1+x^4),x,2,1/2*arctan(x^2)],
[x^5/(1+x^4),x,3,1/2*x^2-1/2*arctan(x^2)],
[1/(1+tan(x)^2),x,3,1/2*x+1/2*cos(x)*sin(x)],
[x^4/(1-x^2)^(5/2),x,3,1/3*x^3/(1-x^2)^(3/2)+arcsin(x)-x/sqrt(1-x^2)],
[-x^2/(1-x^2)^(3/2),x,2,arcsin(x)-x/sqrt(1-x^2)],
[exp(x)*sin(x),x,1,-1/2*exp(x)*cos(x)+1/2*exp(x)*sin(x)],
// Appendix C - Slagle's Thesis Integration Problems
[1/x,x,1,log(x)],
[sec(2*t)/(1+sec(t)^2+3*tan(t)),t,4,-1/12*log(cos(t)-sin(t))-1/4*log(cos(t)+sin(t))+1/3*log(2*cos(t)+sin(t))+(-1/2)/(1+tan(t))],
[1/sec(x)^2,x,2,1/2*x+1/2*cos(x)*sin(x)],
[(1+x^2)/sqrt(x),x,2,2/5*x^(5/2)+2*sqrt(x)],
[x/sqrt(5+2*x+x^2),x,3,-arcsinh(1/2*(1+x))+sqrt(5+2*x+x^2)],
[cos(x)*sin(x)^2,x,2,1/3*sin(x)^3],
[exp(x)/(1+exp(x)),x,2,log(1+exp(x))],
[exp(2*x)/(1+exp(x)),x,3,exp(x)-log(1+exp(x))],
[1/(1-cos(x)),x,1,-sin(x)/(1-cos(x))],
[sec(x)^2*tan(x),x,2,1/2*sec(x)^2],
[x*log(x),x,1,-1/4*x^2+1/2*x^2*log(x)],
[cos(x)*sin(x),x,2,1/2*sin(x)^2],
[(1+x)/sqrt(2*x-x^2),x,3,-2*arcsin(1-x)-sqrt(2*x-x^2)],
[2*exp(x)/(2+3*exp(2*x)),x,3,arctan(exp(x)*sqrt(3/2))*sqrt(2/3)],
[x^4/(1-x^2)^(5/2),x,3,1/3*x^3/(1-x^2)^(3/2)+arcsin(x)-x/sqrt(1-x^2)],
[exp(6*x)/(1+exp(4*x)),x,3,1/2*exp(2*x)-1/2*arctan(exp(2*x))],
[log(2+3*x^2),x,3,-2*x+x*log(2+3*x^2)+2*arctan(x*sqrt(3/2))*sqrt(2/3)],
// Appendix D - MacIntosh Integration Problems
[1/(r*sqrt(-a^2+2*H*r^2)),x,1,x/(r*sqrt(-a^2+2*H*r^2))],
[1/(r*sqrt(-a^2-e^2+2*H*r^2)),x,1,x/(r*sqrt(-a^2-e^2+2*H*r^2))],
[1/(r*sqrt(-a^2+2*H*r^2-2*K*r^4)),x,1,x/(r*sqrt(-a^2+2*H*r^2-2*K*r^4))],
[1/(r*sqrt(-a^2-e^2+2*H*r^2-2*K*r^4)),x,1,x/(r*sqrt(-a^2-e^2+2*H*r^2-2*K*r^4))],
[1/(r*sqrt(-a^2-2*K*r+2*H*r^2)),x,1,x/(r*sqrt(-a^2-2*r*(K-H*r)))],
[1/(r*sqrt(-a^2-e^2-2*K*r+2*H*r^2)),x,1,x/(r*sqrt(-a^2-e^2-2*K*r+2*H*r^2))],
[r/sqrt(-a^2+2*E*r^2),x,1,r*x/sqrt(-a^2+2*E*r^2)],
[r/sqrt(-a^2-e^2+2*E*r^2),x,1,r*x/sqrt(-a^2-e^2+2*E*r^2)],
[r/sqrt(-a^2+2*E*r^2-2*K*r^4),x,1,r*x/sqrt(-a^2+2*E*r^2-2*K*r^4)],
[r/sqrt(-a^2-e^2+2*E*r^2-2*K*r^4),x,1,r*x/sqrt(-a^2-e^2+2*E*r^2-2*K*r^4)],
[r/sqrt(-a^2-e^2-2*K*r+2*H*r^2),x,1,r*x/sqrt(-a^2-e^2-2*K*r+2*H*r^2)]
];
res:=[]:;
S:=182; S:=size(lst);
failint:=[]; failsimp:=[]; nock:=[]; ass:=[];
print("Integration independent test suites, Moses "+S);
file:=fopen("intindm.tst");
T0:=time();
for j from 0 to S-1 do
l:=eval(lst,1)[j];
f:=l[0]; v:=l[1]; hyp:=l[2]; print(f);
purge(unquote(v));
if (type(hyp)==string) expr(hyp); // eval assumption
print(j+1,f,v,hyp,about(unquote(v)));
try { g:=integrate(f,unquote(v)); } catch(err){ g:="integrate(err)"; }
s:=""+eval(g,1);
h:=false;
fail:=size(s.find("integrate("))>0 || hyp==x;
if (fail) failint.append(j+1);
if (hyp=="nock") nock.append(j+1);
if (hyp.type==string && size(hyp.find("assume("))>0) ass.append(j+1);
if (hyp!="nock" && !fail) h:=simplify(diff(g,unquote(v))-f); else print("nock");
purge(unquote(assumptions));
fgh:=""+eval([j+1,f,g,h],1);
if (eval(h,1)!=0) failsimp.append(j+1);
print(fgh);
//res.append([f,g,h]); print(res[size(res)-1]);
fprint(file,"",fgh);
od:;
fprint(file,"","Time:",time()-T0,", tests: ",S,", integration failures: ",size(failint),failint,", simplification failures: ",size(failsimp),failsimp,", not cheked: ",size(nock),nock,", assumptions: ",size(ass),ass);
fclose(file);
print("Integration independent test suites, Moses ","tests: ",S,", integration failures: ",size(failint),failint,", simplification failures: ",size(failsimp),failsimp,", not cheked: ",size(nock),nock,", assumptions: ",size(ass),ass);
//res;
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