Optimal. Leaf size=26 \[ x-\frac{2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac{2 \sinh (x)}{\cosh (x)+1} \]
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Rubi [A] time = 0.0848153, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2680, 8} \[ x-\frac{2 \sinh ^3(x)}{3 (\cosh (x)+1)^3}-\frac{2 \sinh (x)}{\cosh (x)+1} \]
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 [A] time = 0.0211706, size = 30, normalized size = 1.15 \[ x-\frac{8}{3} \tanh \left (\frac{x}{2}\right )+\frac{2}{3} \tanh \left (\frac{x}{2}\right ) \text{sech}^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 32, normalized size = 1.2 \begin{align*} -{\frac{2}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tanh \left ( x/2 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +\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 = 1.28264, size = 51, normalized size = 1.96 \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 = 1.97454, size = 234, normalized size = 9. \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 = 1.11941, size = 30, normalized size = 1.15 \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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