Optimal. Leaf size=12 \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
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Rubi [A] time = 0.0062549, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3770} \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \text{csch}(a+b x) \, dx &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}\\ \end{align*}
Mathematica [B] time = 0.0176559, size = 38, normalized size = 3.17 \[ \frac{\log \left (\sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 1.3 \begin{align*}{\frac{1}{b}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00528, size = 19, normalized size = 1.58 \begin{align*} \frac{\log \left (\tanh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55341, size = 116, normalized size = 9.67 \begin{align*} -\frac{\log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14747, size = 39, normalized size = 3.25 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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