Optimal. Leaf size=38 \[ \frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]
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Rubi [A] time = 0.0366779, antiderivative size = 38, 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} \[ \frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]
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 )^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^4} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^4}+\frac{1}{b (a+b x)^3}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0181263, size = 28, normalized size = 0.74 \[ -\frac{a+3 b e^{2 x}}{12 b^2 \left (a+b e^{2 x}\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{2}}}+{\frac{a}{6\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1199, size = 123, normalized size = 3.24 \begin{align*} -\frac{b e^{\left (2 \, x\right )}}{4 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} - \frac{a}{12 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48252, size = 117, normalized size = 3.08 \begin{align*} -\frac{3 \, b e^{\left (2 \, x\right )} + a}{12 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.178177, size = 54, normalized size = 1.42 \begin{align*} \frac{- a - 3 b e^{2 x}}{12 a^{3} b^{2} + 36 a^{2} b^{3} e^{2 x} + 36 a b^{4} e^{4 x} + 12 b^{5} e^{6 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34235, size = 32, normalized size = 0.84 \begin{align*} -\frac{3 \, b e^{\left (2 \, x\right )} + a}{12 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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