Optimal. Leaf size=108 \[ e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {1}{18} \log \left (-e^x+e^{2 x}+1\right )-\frac {1}{18} \log \left (e^x+e^{2 x}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 e^x+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2282, 390, 288, 210, 634, 618, 204, 628, 206} \[ e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {1}{18} \log \left (-e^x+e^{2 x}+1\right )-\frac {1}{18} \log \left (e^x+e^{2 x}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 e^x+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 288
Rule 390
Rule 618
Rule 628
Rule 634
Rule 2282
Rubi steps
\begin {align*} \int e^x \coth ^2(3 x) \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^2}{\left (1-x^6\right )^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {4 x^6}{\left (1-x^6\right )^2}\right ) \, dx,x,e^x\right )\\ &=e^x+4 \operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^6\right )^2} \, dx,x,e^x\right )\\ &=e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^6} \, dx,x,e^x\right )\\ &=e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,e^x\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,e^x\right )\\ &=e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{18} \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}{1-x+x^2} \, dx,x,e^x\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,e^x\right )\\ &=e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 e^x\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 e^x\right )\\ &=e^x+\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {\tan ^{-1}\left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \tanh ^{-1}\left (e^x\right )+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right )\\ \end {align*}
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Mathematica [C] time = 1.90, size = 113, normalized size = 1.05 \[ \frac {36 e^{7 x} \left (e^{6 x}+1\right )^2 \, _4F_3\left (\frac {7}{6},2,2,2;1,1,\frac {25}{6};e^{6 x}\right )}{1729}+\frac {e^{-11 x} \left (7 \left (3708 e^{6 x}+538 e^{12 x}-684 e^{18 x}+e^{24 x}+2197\right ) \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};e^{6 x}\right )-28153 e^{6 x}-5633 e^{12 x}+3109 e^{18 x}-15379\right )}{3024} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.56, size = 628, normalized size = 5.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 88, normalized size = 0.81 \[ -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} + e^{x} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 150, normalized size = 1.39 \[ {\mathrm e}^{x}-\frac {2 \,{\mathrm e}^{x}}{3 \left ({\mathrm e}^{6 x}-1\right )}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{18}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 87, normalized size = 0.81 \[ -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} + e^{x} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 93, normalized size = 0.86 \[ \frac {\ln \left (\frac {2}{3}-\frac {2\,{\mathrm {e}}^x}{3}\right )}{9}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x}{3}-\frac {2}{3}\right )}{9}+\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}-\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}+{\mathrm {e}}^x-\frac {2\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{6\,x}-1\right )}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )\right )}{9}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )\right )}{9} \]
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 \[ \int e^{x} \coth ^{2}{\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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