Optimal. Leaf size=40 \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2} \]
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Rubi [A] time = 0.0559113, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3220, 3770, 2635, 8} \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b x}{2} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3770
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \text{csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=i \int \left (-i a \text{csch}(c+d x)-i b \sinh ^2(c+d x)\right ) \, dx\\ &=a \int \text{csch}(c+d x) \, dx+b \int \sinh ^2(c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{1}{2} b \int 1 \, dx\\ &=-\frac{b x}{2}-\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0542062, size = 72, normalized size = 1.8 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b (-c-d x)}{2 d}+\frac{b \sinh (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 40, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -2\,a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +b \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\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 = 1.1445, size = 68, normalized size = 1.7 \begin{align*} -\frac{1}{8} \, b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\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 = 1.99806, size = 721, normalized size = 18.02 \begin{align*} -\frac{4 \, b d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, b d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \,{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - 8 \,{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 4 \,{\left (2 \, b d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \,{\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13918, size = 97, normalized size = 2.42 \begin{align*} -\frac{{\left (d x + c\right )} b}{2 \, d} + \frac{b e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac{b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac{a \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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