Optimal. Leaf size=24 \[ \frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]
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Rubi [A] time = 0.0500271, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3220, 3767, 8, 2638} \[ \frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3767
Rule 8
Rule 2638
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\int \left (-a \text{csch}^2(c+d x)-b \sinh (c+d x)\right ) \, dx\\ &=a \int \text{csch}^2(c+d x) \, dx+b \int \sinh (c+d x) \, dx\\ &=\frac{b \cosh (c+d x)}{d}-\frac{(i a) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac{b \cosh (c+d x)}{d}-\frac{a \coth (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0291625, size = 35, normalized size = 1.46 \[ -\frac{a \coth (c+d x)}{d}+\frac{b \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 23, normalized size = 1. \begin{align*}{\frac{-{\rm coth} \left (dx+c\right )a+b\cosh \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13484, size = 63, normalized size = 2.62 \begin{align*} \frac{1}{2} \, b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{2 \, a}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95052, size = 103, normalized size = 4.29 \begin{align*} -\frac{a \cosh \left (d x + c\right ) -{\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\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 [B] time = 1.20409, size = 84, normalized size = 3.5 \begin{align*} \frac{b e^{\left (d x + c\right )}}{2 \, d} + \frac{b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )} - b}{2 \, d{\left (e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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