Optimal. Leaf size=29 \[ \frac{4 \cosh ^3(x)}{15}-\frac{4 \cosh (x)}{5}-\frac{\sinh ^3(x)}{5 (\coth (x)+1)} \]
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Rubi [A] time = 0.0474609, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3502, 2633} \[ \frac{4 \cosh ^3(x)}{15}-\frac{4 \cosh (x)}{5}-\frac{\sinh ^3(x)}{5 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 2633
Rubi steps
\begin{align*} \int \frac{\sinh ^3(x)}{1+\coth (x)} \, dx &=-\frac{\sinh ^3(x)}{5 (1+\coth (x))}+\frac{4}{5} \int \sinh ^3(x) \, dx\\ &=-\frac{\sinh ^3(x)}{5 (1+\coth (x))}-\frac{4}{5} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )\\ &=-\frac{4 \cosh (x)}{5}+\frac{4 \cosh ^3(x)}{15}-\frac{\sinh ^3(x)}{5 (1+\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.0763492, size = 36, normalized size = 1.24 \[ \frac{\text{csch}(x) (-40 \sinh (2 x)+4 \sinh (4 x)-20 \cosh (2 x)+\cosh (4 x)-45)}{120 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 80, normalized size = 2.8 \begin{align*} -{\frac{2}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{6} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05927, size = 45, normalized size = 1.55 \begin{align*} -\frac{1}{48} \,{\left (12 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac{3}{8} \, e^{\left (-x\right )} + \frac{1}{12} \, e^{\left (-3 \, x\right )} - \frac{1}{80} \, e^{\left (-5 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40185, size = 221, normalized size = 7.62 \begin{align*} \frac{\cosh \left (x\right )^{4} + 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 10\right )} \sinh \left (x\right )^{2} - 20 \, \cosh \left (x\right )^{2} + 16 \,{\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 45}{120 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \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 \frac{\sinh ^{3}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15439, size = 42, normalized size = 1.45 \begin{align*} -\frac{1}{240} \,{\left (90 \, e^{\left (4 \, x\right )} - 20 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{48} \, e^{\left (3 \, x\right )} - \frac{1}{4} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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