Optimal. Leaf size=55 \[ -\frac{1}{3} \log \left (e^{2 x}+1\right )+\frac{1}{6} \log \left (-e^{2 x}+e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 e^{2 x}}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0584103, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.125, Rules used = {2282, 12, 275, 292, 31, 634, 618, 204, 628} \[ -\frac{1}{3} \log \left (e^{2 x}+1\right )+\frac{1}{6} \log \left (-e^{2 x}+e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 e^{2 x}}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 275
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int e^x \text{sech}(3 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^3}{1+x^6} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^3}{1+x^6} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \frac{x}{1+x^3} \, dx,x,e^{2 x}\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^{2 x}\right )\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1+x}{1-x+x^2} \, dx,x,e^{2 x}\right )\\ &=-\frac{1}{3} \log \left (1+e^{2 x}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,e^{2 x}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,e^{2 x}\right )\\ &=-\frac{1}{3} \log \left (1+e^{2 x}\right )+\frac{1}{6} \log \left (1-e^{2 x}+e^{4 x}\right )-\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 e^{2 x}\right )\\ &=\frac{\tan ^{-1}\left (\frac{-1+2 e^{2 x}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{3} \log \left (1+e^{2 x}\right )+\frac{1}{6} \log \left (1-e^{2 x}+e^{4 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0107056, size = 24, normalized size = 0.44 \[ \frac{1}{2} e^{4 x} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-e^{6 x}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.04, size = 79, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( 1+{{\rm e}^{2\,x}} \right ) }{3}}+{\frac{1}{6}\ln \left ({{\rm e}^{2\,x}}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }+{\frac{i}{6}}\ln \left ({{\rm e}^{2\,x}}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}+{\frac{1}{6}\ln \left ({{\rm e}^{2\,x}}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }-{\frac{i}{6}}\ln \left ({{\rm e}^{2\,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 = 1.56601, size = 96, normalized size = 1.75 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{3} \, \sqrt{3} \arctan \left (-\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{6} \, \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{6} \, \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06412, size = 288, normalized size = 5.24 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right ) + 3 \, \sqrt{3} \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + \frac{1}{6} \, \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac{1}{3} \, \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\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}{\left (3 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20861, size = 59, normalized size = 1.07 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac{1}{6} \, \log \left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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