# How to solve an integral involving hyperbolic sine/cosine functions

I should solve the following tricky integral:

$$\int_{0}^{+\infty}~\text{e}^{-ak^2}\left(\frac{b+ck^2}{\sqrt{(k^2-\alpha)(k^2-\beta)}}\sinh(d\sqrt{(k^2-\alpha)(k^2-\beta)})+\cosh(d\sqrt{(k^2-\alpha)(k^2-\beta)})\right)dk$$

with $a,\,b,\,c,\,d>0$ and $\alpha,\,\beta\in\mathbb{C}$.

Thanks

Since $a,d$ depends on time, if I compute the Laplace transform of the integrand, I obtain (making all the constants more explicit):

$$\mathcal{L^{-1}}\int_0^{+\infty} \exp(-4\pi^2\omega^2k^2)\,\frac{s+\lambda+\gamma+k^2(D_1+D_2)}{(s+\lambda+k^2D_1)(s+\gamma+k^2D_2)-\lambda\gamma}dk$$

where $D_1,\,D_2,\,\lambda,\,\gamma,\,\omega>0$.

That i'm not quite sure if it might help or not!

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Oh, dear! What a horror...I'm going to have nightmares tonight! –  DonAntonio May 27 '13 at 12:07
$\sinh x + \cosh x = e^x$ might help. –  fgp May 27 '13 at 12:10
@fgp: not here it won't. –  Ron Gordon May 27 '13 at 12:10
@RonGordon Oh, duh, yeah, there are no braces around that sum... Nevermind, then. –  fgp May 27 '13 at 12:11
@RonGordon I edited the question. Please have a look to it. –  JFNJr May 27 '13 at 12:23