Optimal. Leaf size=47 \[ -\frac {\tan ^{-1}\left (\frac {1-3 e^{2 x}}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {1}{12} \log \left (1-2 e^{2 x}+3 e^{4 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {2320, 648, 632,
210, 642} \begin {gather*} \frac {1}{12} \log \left (-2 e^{2 x}+3 e^{4 x}+1\right )-\frac {\text {ArcTan}\left (\frac {1-3 e^{2 x}}{\sqrt {2}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 2320
Rubi steps
\begin {align*} \int \frac {e^{4 x}}{1-2 e^{2 x}+3 e^{4 x}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{1-2 x+3 x^2} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{12} \text {Subst}\left (\int \frac {-2+6 x}{1-2 x+3 x^2} \, dx,x,e^{2 x}\right )+\frac {1}{6} \text {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} \text {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*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (-\sqrt {2} \tan ^{-1}\left (\frac {1-3 e^{2 x}}{\sqrt {2}}\right )+\log \left (1-2 e^{2 x}+3 e^{4 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 38, normalized size = 0.81
method | result | size |
default | \(\frac {\ln \left (1-2 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{4 x}\right )}{12}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 \,{\mathrm e}^{2 x}-2\right ) \sqrt {2}}{4}\right )}{12}\) | \(38\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}-\frac {1}{3}+\frac {i \sqrt {2}}{3}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 x}-\frac {1}{3}+\frac {i \sqrt {2}}{3}\right ) \sqrt {2}}{24}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {1}{3}-\frac {i \sqrt {2}}{3}\right )}{12}-\frac {i \ln \left ({\mathrm e}^{2 x}-\frac {1}{3}-\frac {i \sqrt {2}}{3}\right ) \sqrt {2}}{24}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.93, size = 37, normalized size = 0.79 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 39, normalized size = 0.83 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.47 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.27, size = 37, normalized size = 0.79 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 39, normalized size = 0.83 \begin {gather*} \frac {\ln \left (3\,{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}}{2}-\frac {3\,\sqrt {2}\,{\mathrm {e}}^{2\,x}}{2}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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