3.27 \(\int \frac{e^{2 x}}{A+B e^{4 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[B])

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Rubi [A]  time = 0.0590941, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)/(A + B*E^(4*x)),x]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[B])

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Rubi in Sympy [A]  time = 4.47356, size = 27, normalized size = 0.87 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}} \right )}}{2 \sqrt{A} \sqrt{B}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)/(A+B*exp(4*x)),x)

[Out]

atan(sqrt(B)*exp(2*x)/sqrt(A))/(2*sqrt(A)*sqrt(B))

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Mathematica [A]  time = 0.0122646, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)/(A + B*E^(4*x)),x]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[B])

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Maple [A]  time = 0.01, size = 20, normalized size = 0.7 \[{\frac{1}{2}\arctan \left ({B \left ({{\rm e}^{x}} \right ) ^{2}{\frac{1}{\sqrt{AB}}}} \right ){\frac{1}{\sqrt{AB}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)/(A+B*exp(4*x)),x)

[Out]

1/2/(A*B)^(1/2)*arctan(B*exp(x)^2/(A*B)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(B*e^(4*x) + A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214702, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, A B e^{\left (2 \, x\right )} + \sqrt{-A B}{\left (B e^{\left (4 \, x\right )} - A\right )}}{B e^{\left (4 \, x\right )} + A}\right )}{4 \, \sqrt{-A B}}, -\frac{\arctan \left (\frac{A e^{\left (-2 \, x\right )}}{\sqrt{A B}}\right )}{2 \, \sqrt{A B}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(B*e^(4*x) + A),x, algorithm="fricas")

[Out]

[1/4*log((2*A*B*e^(2*x) + sqrt(-A*B)*(B*e^(4*x) - A))/(B*e^(4*x) + A))/sqrt(-A*B
), -1/2*arctan(A*e^(-2*x)/sqrt(A*B))/sqrt(A*B)]

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Sympy [A]  time = 0.170636, size = 22, normalized size = 0.71 \[ \operatorname{RootSum}{\left (16 z^{2} A B + 1, \left ( i \mapsto i \log{\left (4 i A + e^{2 x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)/(A+B*exp(4*x)),x)

[Out]

RootSum(16*_z**2*A*B + 1, Lambda(_i, _i*log(4*_i*A + exp(2*x))))

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GIAC/XCAS [A]  time = 0.202735, size = 26, normalized size = 0.84 \[ \frac{\arctan \left (\frac{B e^{\left (2 \, x\right )}}{\sqrt{A B}}\right )}{2 \, \sqrt{A B}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(B*e^(4*x) + A),x, algorithm="giac")

[Out]

1/2*arctan(B*e^(2*x)/sqrt(A*B))/sqrt(A*B)