Optimal. Leaf size=20 \[ \frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
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Rubi [A] time = 0.0183473, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 12, 264} \[ \frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 264
Rubi steps
\begin{align*} \int e^{2 x} \text{csch}^4(x) \, dx &=\operatorname{Subst}\left (\int \frac{16 x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \operatorname{Subst}\left (\int \frac{x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right )\\ &=\frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0145626, size = 20, normalized size = 1. \[ \frac{8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 20, normalized size = 1. \begin{align*} -{\frac{1}{3\, \left ( \tanh \left ( x \right ) \right ) ^{3}}}- \left ( \tanh \left ( x \right ) \right ) ^{-2}- \left ( \tanh \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938714, size = 30, normalized size = 1.5 \begin{align*} \frac{8}{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 = 2.14174, size = 254, normalized size = 12.7 \begin{align*} -\frac{8 \,{\left (4 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \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{e^{2 x}}{\sinh ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14871, size = 32, normalized size = 1.6 \begin{align*} -\frac{8 \,{\left (3 \, e^{\left (4 \, x\right )} - 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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