Optimal. Leaf size=95 \[ e^x+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} e^x+1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2282, 388, 211, 1165, 628, 1162, 617, 204} \[ e^x+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} e^x+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 388
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2282
Rubi steps
\begin {align*} \int e^x \tanh (2 x) \, dx &=\operatorname {Subst}\left (\int \frac {-1+x^4}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-\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 )\\ &=e^x-\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}}\\ &=e^x+\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}}\\ &=e^x+\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*}
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Mathematica [A] time = 0.03, size = 95, normalized size = 1.00 \[ e^x+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} e^x+1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 113, normalized size = 1.19 \[ \sqrt {2} \arctan \left (-\sqrt {2} e^{x} + \sqrt {2} \sqrt {\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + \sqrt {2} \arctan \left (-\sqrt {2} e^{x} + \frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-4 \, \sqrt {2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 78, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 24, normalized size = 0.25 \[ {\mathrm e}^{x}+\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-2 \textit {\_R} \right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 78, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 81, normalized size = 0.85 \[ {\mathrm {e}}^x-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (2\,{\mathrm {e}}^x-\sqrt {2}\right )}{2}\right )}{2}+\frac {\sqrt {2}\,\ln \left ({\left (2\,{\mathrm {e}}^x-\sqrt {2}\right )}^2+2\right )}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (2\,{\mathrm {e}}^x+\sqrt {2}\right )}{2}\right )}{2}-\frac {\sqrt {2}\,\ln \left ({\left (2\,{\mathrm {e}}^x+\sqrt {2}\right )}^2+2\right )}{4} \]
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 \[ \int e^{x} \tanh {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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