Optimal. Leaf size=25 \[ -\frac{\sinh ^5(x)}{5}-\frac{\sinh ^3(x)}{3}+\frac{\cosh ^5(x)}{5} \]
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Rubi [A] time = 0.17683, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {3518, 3108, 3107, 2565, 30, 2564, 14} \[ -\frac{\sinh ^5(x)}{5}-\frac{\sinh ^3(x)}{3}+\frac{\cosh ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 3518
Rule 3108
Rule 3107
Rule 2565
Rule 30
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{1+\coth (x)} \, dx &=-\left (i \int \frac{\cosh ^3(x) \sinh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \cosh ^3(x) \sinh (x) (-\cosh (x)+\sinh (x)) \, dx\\ &=i \int \left (-i \cosh ^4(x) \sinh (x)+i \cosh ^3(x) \sinh ^2(x)\right ) \, dx\\ &=\int \cosh ^4(x) \sinh (x) \, dx-\int \cosh ^3(x) \sinh ^2(x) \, dx\\ &=-\left (i \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (x)\right )\right )+\operatorname{Subst}\left (\int x^4 \, dx,x,\cosh (x)\right )\\ &=\frac{\cosh ^5(x)}{5}-i \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,i \sinh (x)\right )\\ &=\frac{\cosh ^5(x)}{5}-\frac{\sinh ^3(x)}{3}-\frac{\sinh ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0579108, size = 34, normalized size = 1.36 \[ \frac{1}{120} (\cosh (x)-\sinh (x)) (10 \sinh (2 x)+\sinh (4 x)+20 \cosh (2 x)+4 \cosh (4 x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 82, normalized size = 3.3 \begin{align*} - \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}+{\frac{2}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+{\frac{4}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}- \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 = 1.03129, size = 36, normalized size = 1.44 \begin{align*} \frac{1}{48} \,{\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} + \frac{1}{24} \, 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 = 2.53558, size = 200, normalized size = 8. \begin{align*} \frac{\cosh \left (x\right )^{4} + \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{2} + 5 \, \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )}{30 \,{\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{\cosh ^{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 = 1.18732, size = 34, normalized size = 1.36 \begin{align*} \frac{1}{240} \,{\left (10 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{48} \, e^{\left (3 \, x\right )} + \frac{1}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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