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 = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 203
Rule 206
Rule 298
Rule 457
Rule 2282
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*}
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Mathematica [A] time = 0.06, 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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fricas [B] time = 0.53, size = 202, normalized size = 5.94 \[ -\frac {4 \, \cosh \relax (x)^{3} + 12 \, \cosh \relax (x)^{2} \sinh \relax (x) + 12 \, \cosh \relax (x) \sinh \relax (x)^{2} + 4 \, \sinh \relax (x)^{3} - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 1.03 \[ -\frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 48, normalized size = 1.41 \[ -\frac {{\mathrm e}^{3 x}}{{\mathrm e}^{4 x}-1}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 34, normalized size = 1.00 \[ -\frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 38, normalized size = 1.12 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )}{4}-\frac {\mathrm {atan}\left ({\mathrm {e}}^{-x}\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{4}-\frac {{\mathrm {e}}^{3\,x}}{{\mathrm {e}}^{4\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x} \coth {\left (2 x \right )} \operatorname {csch}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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