I am stuck on taking the integral of this:
$$\int \sin(9\sinh(2x))\,{\rm d}x$$
I tried using using a double $u$ substitution, but i got confused. Nothing seems to cancel out.
2026-03-27 08:46:58.1774601218
Problem with integral of hyperbolic function
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In general, $~\displaystyle\int_0^\infty\sin(a~\sinh t)~dt~=~\dfrac\pi2~\Big(I_0(a)-L_0(a)\Big),~$ where I and L represent the Bessel and Struve functions, respectively. Letting $t=2x$ and $a=9$ should therefore yield a closed form expression for the definite integral in terms of these two special functions. However, evaluating the indefinite integral would require the existence of “incomplete” Bessel and Struve functions, which, unfortunately, do not formally exist $($as a conventional object of study within the larger realm of mathematics$)$.