Optimal. Leaf size=113 \[ -\frac {e^{3 x}}{e^{4 x}+1}+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} e^x+1\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {2282, 12, 457, 297, 1162, 617, 204, 1165, 628} \[ -\frac {e^{3 x}}{e^{4 x}+1}+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} e^x+1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 297
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2282
Rubi steps
\begin {align*} \int e^x \text {sech}(2 x) \tanh (2 x) \, dx &=\operatorname {Subst}\left (\int \frac {2 x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=-\frac {e^{3 x}}{1+e^{4 x}}+\operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,e^x\right )\\ &=-\frac {e^{3 x}}{1+e^{4 x}}-\frac {1}{2} \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+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=-\frac {e^{3 x}}{1+e^{4 x}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,e^x\right )+\frac {1}{4} \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 )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}\\ &=-\frac {e^{3 x}}{1+e^{4 x}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} e^x\right )}{2 \sqrt {2}}\\ &=-\frac {e^{3 x}}{1+e^{4 x}}-\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 42, normalized size = 0.37 \[ \frac {2}{3} e^{3 x} \left (\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-e^{4 x}\right )-2 \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-e^{4 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 164, normalized size = 1.45 \[ -\frac {4 \, {\left (\sqrt {2} e^{\left (4 \, x\right )} + \sqrt {2}\right )} \arctan \left (-\sqrt {2} e^{x} + \sqrt {2} \sqrt {\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + 4 \, {\left (\sqrt {2} e^{\left (4 \, x\right )} + \sqrt {2}\right )} \arctan \left (-\sqrt {2} e^{x} + \frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + 1\right ) + {\left (\sqrt {2} e^{\left (4 \, x\right )} + \sqrt {2}\right )} \log \left (4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) - {\left (\sqrt {2} e^{\left (4 \, x\right )} + \sqrt {2}\right )} \log \left (-4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + 8 \, e^{\left (3 \, x\right )}}{8 \, {\left (e^{\left (4 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 90, normalized size = 0.80 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 40, normalized size = 0.35 \[ -\frac {{\mathrm e}^{3 x}}{1+{\mathrm e}^{4 x}}+2 \left (\munderset {\textit {\_R} =\RootOf \left (4096 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (512 \textit {\_R}^{3}+{\mathrm e}^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 90, normalized size = 0.80 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 91, normalized size = 0.81 \[ -\frac {{\mathrm {e}}^{3\,x}}{{\mathrm {e}}^{4\,x}+1}+\sqrt {2}\,\ln \left (1+\sqrt {2}\,{\mathrm {e}}^x\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (1+\sqrt {2}\,{\mathrm {e}}^x\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (1+\sqrt {2}\,{\mathrm {e}}^x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (1+\sqrt {2}\,{\mathrm {e}}^x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right ) \]
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 \[ \int e^{x} \tanh {\left (2 x \right )} \operatorname {sech}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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