Primitive of Cube of Hyperbolic Tangent of a x

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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 1

\(\ds \int \tanh^3 a x \rd x\) \(=\) \(\ds \int \tanh a x \tanh^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \int \tanh a x \paren {1 - \sech^2 a x} \rd x\) Sum of Squares of Hyperbolic Secant and Tangent
\(\ds \) \(=\) \(\ds \int \tanh a x \rd x - \int \tanh a x \sech^2 a x \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {\ln \size {\cosh a x} } a - \int \tanh a x \sech^2 a x \rd x\) Primitive of $\tanh a x$
\(\ds \) \(=\) \(\ds \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C\) Primitive of $\tanh^n a x \sech^2 a x$: $n = 1$

$\blacksquare$


Proof 2

\(\ds \int \tanh^3 a x \rd x\) \(=\) \(\ds -\frac {\tanh^2 a x} {2 a} + \int \tanh a x \rd x\) 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 $\tanh a x$

$\blacksquare$


Also see


Sources

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