Primitive of Square of Hyperbolic Tangent of a x

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Theorem

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


Proof

\(\ds \int \tanh^2 x \rd x\) \(=\) \(\ds x - \tanh x + C\) Primitive of $\tanh^2 x$
\(\ds \leadsto \ \ \) \(\ds \int \tanh^2 a x \rd x\) \(=\) \(\ds \frac 1 a \paren {a x - \tanh a x} + C\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds x - \frac {\tanh a x} a + C\) simplifying

$\blacksquare$


Also see


Sources