Optimal. Leaf size=31 \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]
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Rubi [A] time = 0.0338813, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 x}}{a+b e^{2 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0155778, size = 30, normalized size = 0.97 \[ \frac{1}{2} \left (\frac{e^{2 x}}{b}-\frac{a \log \left (a+b e^{2 x}\right )}{b^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2\,b}}-{\frac{a\ln \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13591, size = 34, normalized size = 1.1 \begin{align*} \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49142, size = 59, normalized size = 1.9 \begin{align*} \frac{b e^{\left (2 \, x\right )} - a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.140633, size = 29, normalized size = 0.94 \begin{align*} - \frac{a \log{\left (\frac{a}{b} + e^{2 x} \right )}}{2 b^{2}} + \begin{cases} \frac{e^{2 x}}{2 b} & \text{for}\: 2 b \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29887, size = 35, normalized size = 1.13 \begin{align*} \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a \log \left ({\left | b e^{\left (2 \, x\right )} + a \right |}\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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