Optimal. Leaf size=85 \[ e^x+\frac {1}{6} \log \left (-e^x+e^{2 x}+1\right )-\frac {1}{6} \log \left (e^x+e^{2 x}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 e^x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2282, 388, 210, 634, 618, 204, 628, 206} \[ e^x+\frac {1}{6} \log \left (-e^x+e^{2 x}+1\right )-\frac {1}{6} \log \left (e^x+e^{2 x}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 e^x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 388
Rule 618
Rule 628
Rule 634
Rule 2282
Rubi steps
\begin {align*} \int e^x \coth (3 x) \, dx &=\operatorname {Subst}\left (\int \frac {-1-x^6}{1-x^6} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname {Subst}\left (\int \frac {1}{1-x^6} \, dx,x,e^x\right )\\ &=e^x-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,e^x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,e^x\right )\\ &=e^x-\frac {2}{3} \tanh ^{-1}\left (e^x\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,e^x\right )\\ &=e^x-\frac {2}{3} \tanh ^{-1}\left (e^x\right )+\frac {1}{6} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{6} \log \left (1+e^x+e^{2 x}\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 e^x\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 e^x\right )\\ &=e^x-\frac {\tan ^{-1}\left (\frac {-1+2 e^x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 e^x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (e^x\right )+\frac {1}{6} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{6} \log \left (1+e^x+e^{2 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 22, normalized size = 0.26 \[ e^x-2 e^x \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};e^{6 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 113, normalized size = 1.33 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \cosh \relax (x) + \frac {2}{3} \, \sqrt {3} \sinh \relax (x) + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \cosh \relax (x) + \frac {2}{3} \, \sqrt {3} \sinh \relax (x) - \frac {1}{3} \, \sqrt {3}\right ) + \cosh \relax (x) - \frac {1}{6} \, \log \left (\frac {2 \, \cosh \relax (x) + 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) + \frac {1}{6} \, \log \left (\frac {2 \, \cosh \relax (x) - 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) - \frac {1}{3} \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{3} \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 76, normalized size = 0.89 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) + e^{x} - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{3} \, \log \left (e^{x} + 1\right ) + \frac {1}{3} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 138, normalized size = 1.62 \[ {\mathrm e}^{x}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{3}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}+\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}+\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 75, normalized size = 0.88 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) + e^{x} - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{3} \, \log \left (e^{x} + 1\right ) + \frac {1}{3} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 81, normalized size = 0.95 \[ \frac {\ln \left (2-2\,{\mathrm {e}}^x\right )}{3}-\frac {\ln \left (-2\,{\mathrm {e}}^x-2\right )}{3}+\frac {\ln \left ({\left (2\,{\mathrm {e}}^x-1\right )}^2+3\right )}{6}-\frac {\ln \left ({\left (2\,{\mathrm {e}}^x+1\right )}^2+3\right )}{6}+{\mathrm {e}}^x-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^x-1\right )}{3}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^x+1\right )}{3}\right )}{3} \]
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 \[ \int e^{x} \coth {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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