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 = 0.146851, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, 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 2282
Rule 390
Rule 288
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
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*}
Mathematica [C] time = 1.71621, size = 113, normalized size = 1.05 \[ \frac{36 e^{7 x} \left (e^{6 x}+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{7}{6},2,2,2\right \},\left \{1,1,\frac{25}{6}\right \},e^{6 x}\right )}{1729}+\frac{e^{-11 x} \left (7 \left (3708 e^{6 x}+538 e^{12 x}-684 e^{18 x}+e^{24 x}+2197\right ) \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};e^{6 x}\right )-28153 e^{6 x}-5633 e^{12 x}+3109 e^{18 x}-15379\right )}{3024} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.1, size = 150, normalized size = 1.4 \begin{align*}{{\rm e}^{x}}-{\frac{2\,{{\rm e}^{x}}}{3\,{{\rm e}^{6\,x}}-3}}+{\frac{1}{18}\ln \left ({{\rm e}^{x}}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }+{\frac{i}{18}}\ln \left ({{\rm e}^{x}}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}+{\frac{1}{18}\ln \left ({{\rm e}^{x}}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }-{\frac{i}{18}}\ln \left ({{\rm e}^{x}}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{9}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{9}}-{\frac{1}{18}\ln \left ({{\rm e}^{x}}+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }+{\frac{i}{18}}\ln \left ({{\rm e}^{x}}+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}-{\frac{1}{18}\ln \left ({{\rm e}^{x}}+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }-{\frac{i}{18}}\ln \left ({{\rm e}^{x}}+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64295, size = 117, normalized size = 1.08 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16464, size = 2221, normalized size = 20.56 \begin{align*} \text{result too large to display} \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 ^{2}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28956, size = 119, normalized size = 1.1 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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