I'm trying to evaluate this cursed integral (finite for $ 0 < a < 1 $):
$$\int_0^1 \frac{x}{\log{\left(ax+(1-x)^2\right)}}{\rm d}x$$
using Sagemath 9.0 and 9.1:
sage: a = var('a')
sage: assume(a>0)
sage: assume(a<1)
sage: f = integrate(x*log(1/(a*x+(1-x)^2)), x, 0, 1)
leads to
RuntimeError: ECL says: C-STACK overflow at size 1048576. Stack can probably be resized. Proceed with caution.
Sympy gives a result even without the assumptions (which is odd by itself):
sage: f = integrate(x*log(1/(a*x+(1-x)*(1-x))), x, 0, 1, algorithm='sympy')
$$\frac{1}{4} {\left(a^{2} + \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 2\right)} \log\left(\frac{a^{2} + \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 4}{2 {\left(a - 2\right)}} + 1\right) - \frac{1}{4} {\left(a^{2} + \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 2\right)} \log\left(\frac{a^{2} + \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 4}{2 {\left(a - 2\right)}}\right) + \frac{1}{4} {\left(a^{2} - \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 2\right)} \log\left(\frac{a^{2} - \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 4}{2 {\left(a - 2\right)}} + 1\right) - \frac{1}{4} {\left(a^{2} - \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 2\right)} \log\left(\frac{a^{2} - \sqrt{{\left(a - 4\right)} a} {\left(a - 2\right)} - 4 a + 4}{2 {\left(a - 2\right)}}\right) - \frac{1}{2} a + \frac{1}{2} \log\left(\frac{1}{a}\right) + \frac{3}{2}$$
But then problems continue:
sage: f.simplify_full()
RuntimeError: ECL says: C-STACK overflow at size 1048576. Stack can probably be resized. Proceed with caution.
But:
sage: f.expand().simplify_full()
$$ -\frac{1}{2} {\left(2 \pi - {\left(\pi - \arctan\left(\frac{\sqrt{-a + 4}}{\sqrt{a}}\right) + \arctan\left(\frac{\sqrt{a} \sqrt{-a + 4}}{a - 2}\right)\right)} a - 2 \arctan\left(\frac{\sqrt{-a + 4}}{\sqrt{a}}\right) + 2 \arctan\left(\frac{\sqrt{a} \sqrt{-a + 4}}{a - 2}\right)\right)} \sqrt{a} \sqrt{-a + 4} + \frac{1}{4} {\left(a^{2} - 4 a\right)} \log\left(a\right) - \frac{1}{2} a + \frac{3}{2} $$ If I want to find the limit when $a\to 0$:
sage: f.limit(a=0)
RuntimeError: ECL says: C-STACK overflow at size 1048576. Stack can probably be resized. Proceed with caution.
and once again
sage: f.limit(a=0, algorithm="sympy")
$$-i \pi + \frac{3}{2}$$
Which is only true for $a<0$.
I previously found this strange bug a while ago (fixed upstream, still present in Sagemath 9.1), this makes me think the assumption assume(a>0) is the root of it all, unfortunately it cannot be skipped to evaluate the integral properly.
Do you know when the updated version of Maxima will be included with Sagemath?
Do you think all these issues are linked to this old ticket or should it be investigated independently?
