Optimal. Leaf size=82 \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b^2 x}{8} \]
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Rubi [A] time = 0.0979725, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3220, 3767, 8, 2638, 2635} \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b^2 x}{8} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3767
Rule 8
Rule 2638
Rule 2635
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\int \left (-a^2 \text{csch}^2(c+d x)-2 a b \sinh (c+d x)-b^2 \sinh ^4(c+d x)\right ) \, dx\\ &=a^2 \int \text{csch}^2(c+d x) \, dx+(2 a b) \int \sinh (c+d x) \, dx+b^2 \int \sinh ^4(c+d x) \, dx\\ &=\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{1}{4} \left (3 b^2\right ) \int \sinh ^2(c+d x) \, dx-\frac{\left (i a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac{2 a b \cosh (c+d x)}{d}-\frac{a^2 \coth (c+d x)}{d}-\frac{3 b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac{1}{8} \left (3 b^2\right ) \int 1 \, dx\\ &=\frac{3 b^2 x}{8}+\frac{2 a b \cosh (c+d x)}{d}-\frac{a^2 \coth (c+d x)}{d}-\frac{3 b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.284753, size = 92, normalized size = 1.12 \[ -\frac{a^2 \coth (c+d x)}{d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}+\frac{3 b^2 (c+d x)}{8 d}-\frac{b^2 \sinh (2 (c+d x))}{4 d}+\frac{b^2 \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 65, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{a}^{2}{\rm coth} \left (dx+c\right )+2\,ab\cosh \left ( dx+c \right ) +{b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07606, size = 153, normalized size = 1.87 \begin{align*} \frac{1}{64} \, b^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{2 \, a^{2}}{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 = 1.91237, size = 360, normalized size = 4.39 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{5} + 5 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 9 \, b^{2} \cosh \left (d x + c\right )^{3} +{\left (10 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 8 \,{\left (8 \, a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) + 8 \,{\left (3 \, b^{2} d x + 16 \, a b \cosh \left (d x + c\right ) + 8 \, a^{2}\right )} \sinh \left (d x + c\right )}{64 \, 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 = 1.22934, size = 224, normalized size = 2.73 \begin{align*} \frac{3 \,{\left (d x + c\right )} b^{2}}{8 \, d} + \frac{{\left (64 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 64 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2} - 8 \,{\left (16 \, a^{2} - b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d{\left (e^{\left (d x + c\right )} + 1\right )}{\left (e^{\left (d x + c\right )} - 1\right )}} + \frac{b^{2} d^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{2} d^{3} e^{\left (2 \, d x + 2 \, c\right )} + 64 \, a b d^{3} e^{\left (d x + c\right )}}{64 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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