Optimal. Leaf size=30 \[ x+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac{2 \sinh (x)}{1-\cosh (x)} \]
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Rubi [A] time = 0.113769, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4392, 2670, 2680, 8} \[ x+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\frac{2 \sinh (x)}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2670
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int (\coth (x)+\text{csch}(x))^4 \, dx &=\int (i+i \cosh (x))^4 \text{csch}^4(x) \, dx\\ &=\int \frac{\sinh ^4(x)}{(i-i \cosh (x))^4} \, dx\\ &=\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}-\int \frac{\sinh ^2(x)}{(i-i \cosh (x))^2} \, dx\\ &=\frac{2 \sinh (x)}{1-\cosh (x)}+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}+\int 1 \, dx\\ &=x+\frac{2 \sinh (x)}{1-\cosh (x)}+\frac{2 \sinh ^3(x)}{3 (1-\cosh (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0479179, size = 30, normalized size = 1. \[ x-\frac{8}{3} \coth \left (\frac{x}{2}\right )-\frac{2}{3} \coth \left (\frac{x}{2}\right ) \text{csch}^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 57, normalized size = 1.9 \begin{align*} x-{\rm coth} \left (x\right )-{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{3}}{3}}-{\frac{8\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( x \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{3\,\sinh \left ( x \right ) }}+{\frac{4\,\sinh \left ( x \right ) }{3}}-3\,{\frac{\cosh \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{3}}}-2\, \left ( 2/3-1/3\, \left ({\rm csch} \left (x\right ) \right ) ^{2} \right ){\rm coth} \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18459, size = 247, normalized size = 8.23 \begin{align*} -2 \, \coth \left (x\right )^{3} + x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{4 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac{16 \, e^{\left (-3 \, x\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-5 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} - \frac{4}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{32}{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99215, size = 234, normalized size = 7.8 \begin{align*} \frac{3 \, x \cosh \left (x\right )^{2} + 3 \, x \sinh \left (x\right )^{2} - 4 \,{\left (3 \, x + 10\right )} \cosh \left (x\right ) + 2 \,{\left (3 \, x \cosh \left (x\right ) - 3 \, x - 4\right )} \sinh \left (x\right ) + 9 \, x + 24}{3 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) + 3\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 \left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11199, size = 30, normalized size = 1. \begin{align*} x - \frac{8 \,{\left (3 \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 2\right )}}{3 \,{\left (e^{x} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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