Optimal. Leaf size=47 \[ \frac{1}{12} \log \left (-2 e^{2 x}+3 e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 e^{2 x}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]
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Rubi [A] time = 0.0591705, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2282, 634, 618, 204, 628} \[ \frac{1}{12} \log \left (-2 e^{2 x}+3 e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 e^{2 x}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{e^{4 x}}{1-2 e^{2 x}+3 e^{4 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{1-2 x+3 x^2} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{-2+6 x}{1-2 x+3 x^2} \, dx,x,e^{2 x}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-2 x+3 x^2} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{12} \log \left (1-2 e^{2 x}+3 e^{4 x}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,-2+6 e^{2 x}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1-3 e^{2 x}}{\sqrt{2}}\right )}{6 \sqrt{2}}+\frac{1}{12} \log \left (1-2 e^{2 x}+3 e^{4 x}\right )\\ \end{align*}
Mathematica [A] time = 0.020503, size = 44, normalized size = 0.94 \[ \frac{1}{12} \left (\log \left (-2 e^{2 x}+3 e^{4 x}+1\right )+\sqrt{2} \tan ^{-1}\left (\frac{3 e^{2 x}-1}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 38, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 1-2\, \left ({{\rm e}^{x}} \right ) ^{2}+3\, \left ({{\rm e}^{x}} \right ) ^{4} \right ) }{12}}+{\frac{\sqrt{2}}{12}\arctan \left ({\frac{ \left ( 6\, \left ({{\rm e}^{x}} \right ) ^{2}-2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41344, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, e^{\left (4 \, x\right )} - 2 \, 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 = 1.78919, size = 127, normalized size = 2.7 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{3}{2} \, \sqrt{2} e^{\left (2 \, x\right )} - \frac{1}{2} \, \sqrt{2}\right ) + \frac{1}{12} \, \log \left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.134202, size = 22, normalized size = 0.47 \begin{align*} \operatorname{RootSum}{\left (96 z^{2} - 16 z + 1, \left ( i \mapsto i \log{\left (8 i + e^{2 x} - 1 \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13426, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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