Optimal. Leaf size=34 \[ \frac{e^{3 x}}{1-e^{4 x}}+\frac{1}{2} \tan ^{-1}\left (e^x\right )-\frac{1}{2} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0283708, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2282, 12, 457, 298, 203, 206} \[ \frac{e^{3 x}}{1-e^{4 x}}+\frac{1}{2} \tan ^{-1}\left (e^x\right )-\frac{1}{2} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 457
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^x \coth (2 x) \text{csch}(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2 \left (1+x^4\right )}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2 \left (1+x^4\right )}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{1-e^{4 x}}-\operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{1-e^{4 x}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{1-e^{4 x}}+\frac{1}{2} \tan ^{-1}\left (e^x\right )-\frac{1}{2} \tanh ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0558284, size = 31, normalized size = 0.91 \[ \frac{1}{2} \left (-\frac{2 e^{3 x}}{e^{4 x}-1}+\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.061, size = 48, normalized size = 1.4 \begin{align*} -{\frac{{{\rm e}^{3\,x}}}{{{\rm e}^{4\,x}}-1}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{4}}+{\frac{i}{4}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{i}{4}}\ln \left ({{\rm e}^{x}}-i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55826, size = 46, normalized size = 1.35 \begin{align*} -\frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac{1}{2} \, \arctan \left (e^{x}\right ) - \frac{1}{4} \, \log \left (e^{x} + 1\right ) + \frac{1}{4} \, \log \left (e^{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84365, size = 745, normalized size = 21.91 \begin{align*} -\frac{4 \, \cosh \left (x\right )^{3} + 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sinh \left (x\right )^{3} - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\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 e^{x} \coth{\left (2 x \right )} \operatorname{csch}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16302, size = 47, normalized size = 1.38 \begin{align*} -\frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac{1}{2} \, \arctan \left (e^{x}\right ) - \frac{1}{4} \, \log \left (e^{x} + 1\right ) + \frac{1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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