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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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2.4 Hyperbolic Functions

2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x

2.411 sinhpxcoshqx dx=sinhp+1x coshq1xp+q+q1p+qsinhpx coshq2x dx=sinhp1x coshq+1xp+qp1p+qsinhp2x coshqx dx=sinhp1x coshq+1xq+1p1q+1sinhp2x coshq+2x dx=sinhp+1x coshq1xp+1q1p+1sinhp+2x coshq2x dx=sinhp+1x coshq+1xp+1p+q+2p+1sinhp+2x coshqx dx=sinhp+1x coshq+1xq+1+p+q+2q+1sinhpx coshq+2x dxsi724_e

2.412

1. 

sinhpxcosh2nxdx=sinhp+12n+p[cosh2n1x+k=1n1(2n1)(2n3)(2n2k+1)(2n+p2)(2n+p4)(2n+p2k)cosh2n2k1x]+(2n1)!!(2n+p)(2n+p2)(p2)sinhpxdx

This formula is applicable for arbitrary real p except for the following negative even integers: ...

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