Optimal. Leaf size=37 \[ \frac{a}{2 b^2 \left (a+b e^{2 x}\right )}+\frac{\log \left (a+b e^{2 x}\right )}{2 b^2} \]
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Rubi [A] time = 0.037703, antiderivative size = 37, 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} \[ \frac{a}{2 b^2 \left (a+b e^{2 x}\right )}+\frac{\log \left (a+b e^{2 x}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 x}}{\left (a+b e^{2 x}\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{a}{2 b^2 \left (a+b e^{2 x}\right )}+\frac{\log \left (a+b e^{2 x}\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0255883, size = 31, normalized size = 0.84 \[ \frac{\frac{a}{a+b e^{2 x}}+\log \left (a+b e^{2 x}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2\,{b}^{2}}}+{\frac{a}{2\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04988, size = 46, normalized size = 1.24 \begin{align*} \frac{a}{2 \,{\left (b^{3} e^{\left (2 \, x\right )} + a b^{2}\right )}} + \frac{\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 = 1.48779, size = 92, normalized size = 2.49 \begin{align*} \frac{{\left (b e^{\left (2 \, x\right )} + a\right )} \log \left (b e^{\left (2 \, x\right )} + a\right ) + a}{2 \,{\left (b^{3} e^{\left (2 \, x\right )} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133682, size = 32, normalized size = 0.86 \begin{align*} \frac{a}{2 a b^{2} + 2 b^{3} e^{2 x}} + \frac{\log{\left (\frac{a}{b} + e^{2 x} \right )}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27111, size = 43, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | b e^{\left (2 \, x\right )} + a \right |}\right )}{2 \, b^{2}} + \frac{a}{2 \,{\left (b e^{\left (2 \, x\right )} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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