Optimal. Leaf size=17 \[ a x-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.0101213, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3770} \[ a x-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int (a+b \text{csch}(c+d x)) \, dx &=a x+b \int \text{csch}(c+d x) \, dx\\ &=a x-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d}\\ \end{align*}
Mathematica [B] time = 0.0139126, size = 43, normalized size = 2.53 \[ a x+\frac{b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 1.2 \begin{align*} ax+{\frac{b}{d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982444, size = 26, normalized size = 1.53 \begin{align*} a x + \frac{b \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53912, size = 131, normalized size = 7.71 \begin{align*} \frac{a d x - b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d} \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 \left (a + b \operatorname{csch}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15953, size = 49, normalized size = 2.88 \begin{align*} a x - b{\left (\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{d} - \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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