Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x

Theorem

$\ds \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$

$\ds \int \sinh a x \sinh p x \rd x$

$=$

$\ds \int \paren {\frac {\map \cosh {a x + p x} - \map \cosh {a x - p x} } 2} \rd x$

$\ds $

$=$

$\ds \frac 1 2 \int \map \cosh {a + p} x \rd x - \frac 1 2 \int \map \cosh {a - p} x \rd x$

$\ds $

$=$

$\ds \frac 1 2 \frac {\map \sinh {a + p} x} {a + p} - \frac 1 2 \frac {\map \sinh {a - p} x} {a - p} + C$

$\ds $

$=$

$\ds \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$

simplifying

$\blacksquare$