Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]
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Rubi [A] time = 0.0347668, 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, Rules used = {2249, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]
Antiderivative was successfully verified.
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Rule 2249
Rule 205
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{A+B e^{4 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{A+B x^2} \, dx,x,e^{2 x}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}}\\ \end{align*}
Mathematica [A] time = 0.008076, 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.
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Maple [A] time = 0.007, size = 20, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arctan \left ({B \left ({{\rm e}^{x}} \right ) ^{2}{\frac{1}{\sqrt{AB}}}} \right ){\frac{1}{\sqrt{AB}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93193, size = 186, normalized size = 6. \begin{align*} \left [-\frac{\sqrt{-A B} \log \left (\frac{B e^{\left (4 \, x\right )} - 2 \, \sqrt{-A B} e^{\left (2 \, x\right )} - A}{B e^{\left (4 \, x\right )} + A}\right )}{4 \, A B}, -\frac{\sqrt{A B} \arctan \left (\frac{\sqrt{A B} e^{\left (-2 \, x\right )}}{B}\right )}{2 \, A B}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.150645, size = 22, normalized size = 0.71 \begin{align*} \operatorname{RootSum}{\left (16 z^{2} A B + 1, \left ( i \mapsto i \log{\left (4 i A + e^{2 x} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08468, size = 26, normalized size = 0.84 \begin{align*} \frac{\arctan \left (\frac{B e^{\left (2 \, x\right )}}{\sqrt{A B}}\right )}{2 \, \sqrt{A B}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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