Optimal. Leaf size=17 \[ \frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3} \]
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Rubi [A] time = 0.044374, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3516, 848, 43} \[ \frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 848
Rule 43
Rubi steps
\begin{align*} \int \frac{\text{csch}^4(x)}{1+\tanh (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 (1+x)} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{-1+x}{x^4} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\tanh (x)\right )\\ &=\frac{\coth ^2(x)}{2}-\frac{\coth ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0477782, size = 20, normalized size = 1.18 \[ -\frac{1}{6} \text{csch}(x) (2 \cosh (x)+(2 \coth (x)-3) \text{csch}(x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 48, normalized size = 2.8 \begin{align*} -{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16533, size = 101, normalized size = 5.94 \begin{align*} -\frac{2 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{4 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1} + \frac{2}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28356, size = 286, normalized size = 16.82 \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{3 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} +{\left (10 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{3} +{\left (10 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right ) + 2 \, \cosh \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{\operatorname{csch}^{4}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22717, size = 24, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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