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 = 0.0830563, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4392, 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 2680
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(-\coth (x)+\text{csch}(x))^4} \, 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 [C] time = 0.0076706, size = 28, normalized size = 0.93 \[ -\frac{2}{3} \coth ^3\left (\frac{x}{2}\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\tanh ^2\left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.068, size = 34, normalized size = 1.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}-\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30636, size = 51, normalized size = 1.7 \begin{align*} x - \frac{8 \,{\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} - 2\right )}}{3 \,{\left (3 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.38497, 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 \frac{1}{\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 = 1.16775, 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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