An outline to get the closed-form of $\int_0^\infty\frac{(\operatorname{gd}(x))^3}{(\cosh(x))^4}dx$

Question

With $\operatorname{gd}(x)$ we denote the so-called Gudermannian function, see for example this MathWorld.

I know the closed-form of the following integral for the more simple cases of integers $n,m\geq 1$ $$\int_0^\infty\frac{(\operatorname{gd}(x))^n}{(\cosh(x))^m}dx.\tag{1}$$

I don't know if such family was in the literature.

Question. Can you provide me a summary of the strategy to get the closed-form of an integral in $(1)$, for example with $n=3$ and $m=4$? Are not required all tedious calculations, nor the exact closed-form: what it is required is the strategy to get the result. Thanks in advance.

Answer 1

$$I(m,n)=\int_{0}^{+\infty}\frac{\left(\arctan\sinh x\right)^n}{\left(\cosh x\right)^m }\,dx=\int_{0}^{+\infty}\frac{\left(\arctan u\right)^n}{\left(1+u^2\right)^{m+1/2}}\,du$$ can be simply computed by repeated integration by parts or by Fourier series, since $$ I(m,n) = \int_{0}^{\pi/2}\theta^n \left(\cos\theta\right)^{2m-1}\,d\theta. $$ In particular $$ I(4,3)=\frac{44658302}{13505625}-\frac{206656}{128625}\,\pi +\frac{2}{35}\,\pi^3.$$