Primitive of Cube of Hyperbolic Tangent of a x/Proof 2

Theorem

$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$

Proof

\(\ds \int \tanh^3 a x \rd x\)

\(=\)

\(\ds -\frac {\tanh^2 a x} {2 a} + \int \tanh a x \rd x\)

Primitive of Power of $\tanh^n a x$ with $n = 3$

\(\ds \)

\(=\)

\(\ds \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C\)

Primitive of $\tanh a x$

$\blacksquare$