Optimal. Leaf size=110 \[ -\frac{2 e^x}{3 \left (e^{6 x}+1\right )}-\frac{\log \left (-\sqrt{3} e^x+e^{2 x}+1\right )}{6 \sqrt{3}}+\frac{\log \left (\sqrt{3} e^x+e^{2 x}+1\right )}{6 \sqrt{3}}+\frac{2}{9} \tan ^{-1}\left (e^x\right )-\frac{1}{9} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )+\frac{1}{9} \tan ^{-1}\left (2 e^x+\sqrt{3}\right ) \]
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Rubi [A] time = 0.205163, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {2282, 12, 288, 209, 634, 618, 204, 628, 203} \[ -\frac{2 e^x}{3 \left (e^{6 x}+1\right )}-\frac{\log \left (-\sqrt{3} e^x+e^{2 x}+1\right )}{6 \sqrt{3}}+\frac{\log \left (\sqrt{3} e^x+e^{2 x}+1\right )}{6 \sqrt{3}}+\frac{2}{9} \tan ^{-1}\left (e^x\right )-\frac{1}{9} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )+\frac{1}{9} \tan ^{-1}\left (2 e^x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 288
Rule 209
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int e^x \text{sech}^2(3 x) \, dx &=\operatorname{Subst}\left (\int \frac{4 x^6}{\left (1+x^6\right )^2} \, dx,x,e^x\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^6\right )^2} \, dx,x,e^x\right )\\ &=-\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 )\\ &=-\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{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )+\frac{2}{9} \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )\\ &=-\frac{2 e^x}{3 \left (1+e^{6 x}\right )}+\frac{2}{9} \tan ^{-1}\left (e^x\right )+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )}{6 \sqrt{3}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )}{6 \sqrt{3}}\\ &=-\frac{2 e^x}{3 \left (1+e^{6 x}\right )}+\frac{2}{9} \tan ^{-1}\left (e^x\right )-\frac{\log \left (1-\sqrt{3} e^x+e^{2 x}\right )}{6 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} e^x+e^{2 x}\right )}{6 \sqrt{3}}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 e^x\right )-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 e^x\right )\\ &=-\frac{2 e^x}{3 \left (1+e^{6 x}\right )}+\frac{2}{9} \tan ^{-1}\left (e^x\right )-\frac{1}{9} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )+\frac{1}{9} \tan ^{-1}\left (\sqrt{3}+2 e^x\right )-\frac{\log \left (1-\sqrt{3} e^x+e^{2 x}\right )}{6 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} e^x+e^{2 x}\right )}{6 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0200028, size = 34, normalized size = 0.31 \[ \frac{2}{3} e^x \left (\, _2F_1\left (\frac{1}{6},1;\frac{7}{6};-e^{6 x}\right )-\frac{1}{e^{6 x}+1}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 59, normalized size = 0.5 \begin{align*} -{\frac{2\,{{\rm e}^{x}}}{3+3\,{{\rm e}^{6\,x}}}}+{\frac{i}{9}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{i}{9}}\ln \left ({{\rm e}^{x}}-i \right ) +4\,\sum _{{\it \_R}={\it RootOf} \left ( 1679616\,{{\it \_Z}}^{4}-1296\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}+36\,{\it \_R} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59934, size = 107, normalized size = 0.97 \begin{align*} \frac{1}{18} \, \sqrt{3} \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{18} \, \sqrt{3} \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{2 \, e^{x}}{3 \,{\left (e^{\left (6 \, x\right )} + 1\right )}} + \frac{1}{9} \, \arctan \left (\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{9} \, \arctan \left (-\sqrt{3} + 2 \, e^{x}\right ) + \frac{2}{9} \, \arctan \left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08446, size = 474, normalized size = 4.31 \begin{align*} -\frac{4 \,{\left (e^{\left (6 \, x\right )} + 1\right )} \arctan \left (\sqrt{3} + \sqrt{-4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} - 2 \, e^{x}\right ) + 4 \,{\left (e^{\left (6 \, x\right )} + 1\right )} \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1} - 2 \, e^{x}\right ) - 4 \,{\left (e^{\left (6 \, x\right )} + 1\right )} \arctan \left (e^{x}\right ) -{\left (\sqrt{3} e^{\left (6 \, x\right )} + \sqrt{3}\right )} \log \left (4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) +{\left (\sqrt{3} e^{\left (6 \, x\right )} + \sqrt{3}\right )} \log \left (-4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + 12 \, e^{x}}{18 \,{\left (e^{\left (6 \, x\right )} + 1\right )}} \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} \operatorname{sech}^{2}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \operatorname{sech}\left (3 \, x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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