#93 certain integrals cannot be calculated

[Anonymous]

This may not be a bug however, the following integral can be computed by Mathematica (giving pi * J_o(1) ) but not by Maxima.

integrate(cos(x)/sqrt(1-x^2),x,-1,1)

[Aleksas]

solving with Maxima 5.24.0:

(%i1) load(simplify_sum)$
(%i2) S:integrate(cos(x)/sqrt(1-x^2),x,-1,1)=2*integrate(cos(x)/sqrt(1-x^2),x,0,1);
(%o2) integrate(cos(x)/sqrt(1-x^2),x,-1,1)=2*integrate(cos(x)/sqrt(1-x^2),x,0,1)
(%i3) cos(x)=niceindices(powerseries(cos(x),x,0));
powerseries: first simplification returned
cos(x)
(%o3) cos(x)=sum(((-1)^i*x^(2*i))/(2*i)!,i,0,inf)
(%i4) T:intosum(%/sqrt(1-x^2));
(%o4) cos(x)/sqrt(1-x^2)=sum(((-1)^i*x^(2*i))/((2*i)!*sqrt(1-x^2)),i,0,inf)
(%i5) assume(i>=0)$
(%i6) integrate(lhs(T),x,0,1)=sum(integrate(first(rhs(T)),x,0,1),i,0,inf);
(%o6) integrate(cos(x)/sqrt(1-x^2),x,0,1)=sum((beta(1/2,(2*i+1)/2)*(-1)^i)/(2*i)!,i,0,inf)/2
(%i7) simplify_sum(%);
(%o7) integrate(cos(x)/sqrt(1-x^2),x,0,1)=(%pi*bessel_j(0,1))/2
(%i8) subst(%,S);
(%o8) integrate(cos(x)/sqrt(1-x^2),x,-1,1)=%pi*bessel_j(0,1)
(%i9) lhs(%)=float(rhs(%));
(%o9) integrate(cos(x)/sqrt(1-x^2),x,-1,1)=2.403939430634413

[Dieter Kaiser]

Maxima gives a noun form for the reported integral.

(%i3) integrate(cos(x)/sqrt(1-x^2),x,-1,1);
(%o3) 'integrate(cos(x)/sqrt(1-x^2),x,-1,1)

This is not a bug. Moving this report to the tracker "Feature Requests".
Dieter Kaiser