3.525 \(\int \frac{e^{4 x}}{1-2 e^{2 x}+3 e^{4 x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{12} \log \left (-2 e^{2 x}+3 e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 e^{2 x}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

[Out]

-ArcTan[(1 - 3*E^(2*x))/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*E^(2*x) + 3*E^(4*x)]/12

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Rubi [A]  time = 0.0997838, 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 \[ \frac{1}{12} \log \left (-2 e^{2 x}+3 e^{4 x}+1\right )-\frac{\tan ^{-1}\left (\frac{1-3 e^{2 x}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[E^(4*x)/(1 - 2*E^(2*x) + 3*E^(4*x)),x]

[Out]

-ArcTan[(1 - 3*E^(2*x))/Sqrt[2]]/(6*Sqrt[2]) + Log[1 - 2*E^(2*x) + 3*E^(4*x)]/12

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Rubi in Sympy [A]  time = 11.9864, size = 42, normalized size = 0.89 \[ \frac{\log{\left (3 e^{4 x} - 2 e^{2 x} + 1 \right )}}{12} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 e^{2 x}}{2} - \frac{1}{2}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(4*x)/(1-2*exp(2*x)+3*exp(4*x)),x)

[Out]

log(3*exp(4*x) - 2*exp(2*x) + 1)/12 + sqrt(2)*atan(sqrt(2)*(3*exp(2*x)/2 - 1/2))
/12

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Mathematica [A]  time = 0.0283617, 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.

[In]  Integrate[E^(4*x)/(1 - 2*E^(2*x) + 3*E^(4*x)),x]

[Out]

(Sqrt[2]*ArcTan[(-1 + 3*E^(2*x))/Sqrt[2]] + Log[1 - 2*E^(2*x) + 3*E^(4*x)])/12

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Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[{\frac{\ln \left ( 1-2\, \left ({{\rm e}^{x}} \right ) ^{2}+3\, \left ({{\rm e}^{x}} \right ) ^{4} \right ) }{12}}+{\frac{\sqrt{2}}{12}\arctan \left ({\frac{ \left ( 6\, \left ({{\rm e}^{x}} \right ) ^{2}-2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(4*x)/(1-2*exp(2*x)+3*exp(4*x)),x)

[Out]

1/12*ln(1-2*exp(x)^2+3*exp(x)^4)+1/12*2^(1/2)*arctan(1/4*(6*exp(x)^2-2)*2^(1/2))

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Maxima [A]  time = 1.53689, size = 50, normalized size = 1.06 \[ \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac{1}{12} \, \log \left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(3*e^(4*x) - 2*e^(2*x) + 1),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*arctan(1/2*sqrt(2)*(3*e^(2*x) - 1)) + 1/12*log(3*e^(4*x) - 2*e^(2*x
) + 1)

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Fricas [A]  time = 0.219026, size = 58, normalized size = 1.23 \[ \frac{1}{24} \, \sqrt{2}{\left (\sqrt{2} \log \left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right ) + 2 \, \arctan \left (\frac{3}{2} \, \sqrt{2} e^{\left (2 \, x\right )} - \frac{1}{2} \, \sqrt{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(3*e^(4*x) - 2*e^(2*x) + 1),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*(sqrt(2)*log(3*e^(4*x) - 2*e^(2*x) + 1) + 2*arctan(3/2*sqrt(2)*e^(2
*x) - 1/2*sqrt(2)))

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Sympy [A]  time = 0.152781, size = 22, normalized size = 0.47 \[ \operatorname{RootSum}{\left (96 z^{2} - 16 z + 1, \left ( i \mapsto i \log{\left (8 i + e^{2 x} - 1 \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(4*x)/(1-2*exp(2*x)+3*exp(4*x)),x)

[Out]

RootSum(96*_z**2 - 16*_z + 1, Lambda(_i, _i*log(8*_i + exp(2*x) - 1)))

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GIAC/XCAS [A]  time = 0.197918, size = 50, normalized size = 1.06 \[ \frac{1}{12} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(4*x)/(3*e^(4*x) - 2*e^(2*x) + 1),x, algorithm="giac")

[Out]

1/12*sqrt(2)*arctan(1/2*sqrt(2)*(3*e^(2*x) - 1)) + 1/12*ln(3*e^(4*x) - 2*e^(2*x)
+ 1)