Numerical integration using Gauss-Laguerre quadrature I have to evaluate the following integral for our numerical methods test:
$$\int_0^\infty e^{-x}\ln(1+\sin^2x)\, \mathrm dx$$
I managed to evaluate it numerically, but for the second point I have the following requierment:
a) What is the true value of the integral. 
I have tried to bring it to complex but I didn't get anywhere satisfactory and I also tried to use Feynman's method. I got this:
$$ I(a) = \int_0^\infty e^{-ax}\ln(1+\sin^2x)\, \mathrm dx$$
After I derivate and integrate I get this:
$$I(a) = Ce^{-a^2/2}$$
But I don't seem to be able to find any values for $a$ to nicely calculate $C$.
Also, I tried to play around in Mathematica, but I got nothing.
EDIT: My professor just wanted us to plug the equation into Mathematica and take the value it gives as "true" value. Thanks for the answers.
 A: I don't see a closed-form evaluation. Still, for $a>0$ and $|r|<1$ one has
$$I(a,r):=\int_{0}^{\infty} e^{-ax}\ln(1-2r\cos x+r^2)\,dx=-\sum_{n=1}^{\infty}\frac{r^n}{n}\frac{2a}{n^2+a^2}.$$
This is obtained from
\begin{gather}\ln(1-2r\cos x+r^2)=\ln(1-re^{ix})(1-re^{-ix})\\=-\sum_{n=1}^{\infty}\frac{r^n}{n}(e^{inx}+e^{-inx})=-2\sum_{n=1}^{\infty}\frac{r^n}{n}\cos nx\end{gather}
and $\displaystyle\int_{0}^{\infty}e^{-ax}\cos bx\,dx=\frac{a}{a^2+b^2}$ (termwise integration is clearly valid here).
The original integral is
$$\int_{0}^{\infty}e^{-x}\ln(1+\sin^2 x)\,dx=\int_{0}^{\infty}e^{-x}\ln\frac{3-\cos 2x}{2}\,dx\\=\left.\frac{1}{2}\int_{0}^{\infty}e^{-x/2}\ln\frac{1-2r\cos x+r^2}{4r}\,dx\right|_{r=3-2\sqrt{2}}\\=2\ln\frac{1+\sqrt{2}}{2}+\frac{1}{2}I\Big(\frac{1}2{},3-2\sqrt{2}\Big)$$
with the numerical value of $0.30599373695284849278809525044503\cdots$
A: This not an answer.
With CAS help:
$$\int_0^\infty e^{-x}\ln(1+\sin^2x)\,dx=-i\sum_{n=1}^\infty \frac{ \Gamma \left(\frac{i}{2}-n\right)\, \Gamma (2
   n)}{4^n\,\Gamma \left(n+1+\frac{i}{2}\right)}=\frac{2}{5} \, _3F_2\left(1,1,\frac{3}{2};2-\frac{i}{2},2+\frac{i}{2};-1\right)$$
where: $_3F_2$ is hypergeometric function.
