Optimal. Leaf size=49 \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.0743924, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3666, 3770, 2611} \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3666
Rule 3770
Rule 2611
Rubi steps
\begin{align*} \int \text{csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx &=i \int \left (-i a \text{csch}(c+d x)-i b \text{sech}(c+d x) \tanh ^2(c+d x)\right ) \, dx\\ &=a \int \text{csch}(c+d x) \, dx+b \int \text{sech}(c+d x) \tanh ^2(c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b \text{sech}(c+d x) \tanh (c+d x)}{2 d}+\frac{1}{2} b \int \text{sech}(c+d x) \, dx\\ &=\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b \text{sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0311553, size = 75, normalized size = 1.53 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 65, normalized size = 1.3 \begin{align*} -2\,{\frac{a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}-{\frac{b\sinh \left ( dx+c \right ) }{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51615, size = 112, normalized size = 2.29 \begin{align*} -b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac{a \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 = 2.63424, size = 1453, normalized size = 29.65 \begin{align*} \text{result too large to display} \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 \tanh ^{3}{\left (c + d x \right )}\right ) \operatorname{csch}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21699, size = 100, normalized size = 2.04 \begin{align*} \frac{b \arctan \left (e^{\left (d x + c\right )}\right ) - a \log \left (e^{\left (d x + c\right )} + 1\right ) + a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac{b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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