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