Optimal. Leaf size=29 \[ \frac{4 \sinh ^3(x)}{15}+\frac{4 \sinh (x)}{5}-\frac{\cosh ^3(x)}{5 (\tanh (x)+1)} \]
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Rubi [A] time = 0.0420266, 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} \[ \frac{4 \sinh ^3(x)}{15}+\frac{4 \sinh (x)}{5}-\frac{\cosh ^3(x)}{5 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cosh ^3(x)}{1+\tanh (x)} \, dx &=-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}+\frac{4}{5} \int \cosh ^3(x) \, dx\\ &=-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}+\frac{4}{5} i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )\\ &=\frac{4 \sinh (x)}{5}+\frac{4 \sinh ^3(x)}{15}-\frac{\cosh ^3(x)}{5 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0465145, size = 36, normalized size = 1.24 \[ \frac{\text{sech}(x) (40 \sinh (2 x)+4 \sinh (4 x)+20 \cosh (2 x)+\cosh (4 x)-45)}{120 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, 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{5}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{3}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{11}{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{5}{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 = 1.06333, 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 = 2.06148, 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 [B] time = 1.59062, size = 134, normalized size = 4.62 \begin{align*} - \frac{8 \sinh ^{3}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{2 \sinh ^{3}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{6 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{6 \sinh ^{2}{\left (x \right )} \cosh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{6 \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{9 \sinh{\left (x \right )} \cosh ^{2}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} + \frac{3 \cosh ^{3}{\left (x \right )} \tanh{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} - \frac{3 \cosh ^{3}{\left (x \right )}}{15 \tanh{\left (x \right )} + 15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17814, 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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