Optimal. Leaf size=97 \[ e^x-\frac {2 \text {ArcTan}\left (e^x\right )}{3}+\frac {1}{3} \text {ArcTan}\left (\sqrt {3}-2 e^x\right )-\frac {1}{3} \text {ArcTan}\left (\sqrt {3}+2 e^x\right )+\frac {\log \left (1-\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.14, antiderivative size = 97, 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 = {2320, 396, 215,
648, 632, 210, 642, 209} \begin {gather*} -\frac {2}{3} \text {ArcTan}\left (e^x\right )+\frac {1}{3} \text {ArcTan}\left (\sqrt {3}-2 e^x\right )-\frac {1}{3} \text {ArcTan}\left (2 e^x+\sqrt {3}\right )+e^x+\frac {\log \left (-\sqrt {3} e^x+e^{2 x}+1\right )}{2 \sqrt {3}}-\frac {\log \left (\sqrt {3} e^x+e^{2 x}+1\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 215
Rule 396
Rule 632
Rule 642
Rule 648
Rule 2320
Rubi steps
\begin {align*} \int e^x \tanh (3 x) \, dx &=\text {Subst}\left (\int \frac {-1+x^6}{1+x^6} \, dx,x,e^x\right )\\ &=e^x-2 \text {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,e^x\right )\\ &=e^x-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1-\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,e^x\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1+\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,e^x\right )\\ &=e^x-\frac {2}{3} \tan ^{-1}\left (e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,e^x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,e^x\right )+\frac {\text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,e^x\right )}{2 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,e^x\right )}{2 \sqrt {3}}\\ &=e^x-\frac {2}{3} \tan ^{-1}\left (e^x\right )+\frac {\log \left (1-\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 e^x\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 e^x\right )\\ &=e^x-\frac {2}{3} \tan ^{-1}\left (e^x\right )+\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-2 e^x\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {3}+2 e^x\right )+\frac {\log \left (1-\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} e^x+e^{2 x}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 24, normalized size = 0.25 \begin {gather*} e^x-2 e^x \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-e^{6 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.91, size = 47, normalized size = 0.48
method | result | size |
risch | \({\mathrm e}^{x}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{3}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{3}+\left (\munderset {\textit {\_R} =\RootOf \left (81 \textit {\_Z}^{4}-9 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-3 \textit {\_R} \right )\right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 69, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, e^{x}\right ) - \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, e^{x}\right ) - \frac {2}{3} \, \arctan \left (e^{x}\right ) + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 107, normalized size = 1.10 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \log \left (4 \, \sqrt {3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-4 \, \sqrt {3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac {2}{3} \, \arctan \left (\sqrt {3} + \sqrt {-4 \, \sqrt {3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} - 2 \, e^{x}\right ) + \frac {2}{3} \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1} - 2 \, e^{x}\right ) - \frac {2}{3} \, \arctan \left (e^{x}\right ) + e^{x} \end {gather*}
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 \begin {gather*} \int e^{x} \tanh {\left (3 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 69, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \arctan \left (\sqrt {3} + 2 \, e^{x}\right ) - \frac {1}{3} \, \arctan \left (-\sqrt {3} + 2 \, e^{x}\right ) - \frac {2}{3} \, \arctan \left (e^{x}\right ) + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 70, normalized size = 0.72 \begin {gather*} {\mathrm {e}}^x-\frac {\mathrm {atan}\left (2\,{\mathrm {e}}^x+\sqrt {3}\right )}{3}-\frac {\mathrm {atan}\left (2\,{\mathrm {e}}^x-\sqrt {3}\right )}{3}-\frac {2\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{3}+\frac {\sqrt {3}\,\ln \left ({\left (2\,{\mathrm {e}}^x-\sqrt {3}\right )}^2+1\right )}{6}-\frac {\sqrt {3}\,\ln \left ({\left (2\,{\mathrm {e}}^x+\sqrt {3}\right )}^2+1\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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