Hyperbolic Function - Standard Integrals

Standard Integrals

For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions.

\begin{align} \int \sinh (ax)\,dx &= a^{-1} \cosh (ax) + C \\ \int \cosh (ax)\,dx &= a^{-1} \sinh (ax) + C \\ \int \tanh (ax)\,dx &= a^{-1} \ln (\cosh (ax)) + C \\ \int \coth (ax)\,dx &= a^{-1} \ln (\sinh (ax)) + C \\ \int \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\ \int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left( \tanh \left( \frac{ax}{2} \right) \right) + C \end{align}
\begin{align} \int {\frac{du}{\sqrt{a^2 + u^2}}} & = \sinh ^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{\sqrt{u^2 - a^2}}} &= \cosh ^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{a^2 - u^2}} & = a^{-1}\tanh ^{-1}\left( \frac{u}{a} \right) + C; u^2 < a^2 \\ \int {\frac{du}{a^2 - u^2}} & = a^{-1}\coth ^{-1}\left( \frac{u}{a} \right) + C; u^2 > a^2 \\ \int {\frac{du}{u\sqrt{a^2 - u^2}}} & = -a^{-1}\operatorname{sech}^{-1}\left( \frac{u}{a} \right) + C \\ \int {\frac{du}{u\sqrt{a^2 + u^2}}} & = -a^{-1}\operatorname{csch}^{-1}\left| \frac{u}{a} \right| + C \end{align}

, where C is the constant of integration.

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