Optimal. Leaf size=92 \[ \frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0635706, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2282, 12, 297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int e^x \text{sech}(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2}{1+x^4} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,e^x\right )\\ &=-\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,e^x\right )+\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,e^x\right )}{2 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,e^x\right )}{2 \sqrt{2}}\\ &=\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} e^x\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} e^x\right )}{\sqrt{2}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (1+\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0102933, size = 24, normalized size = 0.26 \[ \frac{2}{3} e^{3 x} \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-e^{4 x}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.041, size = 25, normalized size = 0.3 \begin{align*} 2\,\sum _{{\it \_R}={\it RootOf} \left ( 256\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ( 64\,{{\it \_R}}^{3}+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56204, size = 103, normalized size = 1.12 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + 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 = 1.96356, size = 354, normalized size = 3.85 \begin{align*} -\sqrt{2} \arctan \left (-\sqrt{2} e^{x} + \sqrt{2} \sqrt{\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) - \sqrt{2} \arctan \left (-\sqrt{2} e^{x} + \frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\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 (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16802, size = 103, normalized size = 1.12 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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