Is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})\ dx$?

Question

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp\left(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a}\right)dx $$where is $a\in (-1,3)$

I've tried with $\tan x=u$ and I got the result of sum in term of HurwitzLerchPhi but I failed.

Answer 1

$$\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a}=\tan^2(x)\;\Phi\left(\tan^2(x)\;,\;1\;,\;1+a \right)$$

where $\Phi$ is Hurwitz-Lerch transcendent function. $$I(a)=\int_0^{\pi/4}e^{-\tan^2(x)\;\Phi\left(\tan^2(x)\;,\;1\;,\;1+a\right)}dx $$ For a few particular values of $a$, the Hurwitz function reduces to a function of lower level. So, in particular cases, the integral is likely to have a closed form (for example in case of $a=0$).

But, in the general case, it is highly unlikely that such an integral has a closed form.