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 = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.43, size = 51, normalized size = 1.34 \[ -\frac {3 \, b e^{\left (2 \, x\right )} + a}{12 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 24, normalized size = 0.63 \[ -\frac {3 \, b e^{\left (2 \, x\right )} + a}{12 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 33, normalized size = 0.87 \[ \frac {a}{6 \left (b \,{\mathrm e}^{2 x}+a \right )^{3} b^{2}}-\frac {1}{4 \left (b \,{\mathrm e}^{2 x}+a \right )^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 91, normalized size = 2.39 \[ -\frac {b e^{\left (2 \, x\right )}}{4 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} - \frac {a}{12 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 55, normalized size = 1.45 \[ \frac {\frac {{\mathrm {e}}^{4\,x}}{4\,a}+\frac {b\,{\mathrm {e}}^{6\,x}}{12\,a^2}}{a^3+3\,{\mathrm {e}}^{2\,x}\,a^2\,b+3\,{\mathrm {e}}^{4\,x}\,a\,b^2+{\mathrm {e}}^{6\,x}\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 54, normalized size = 1.42 \[ \frac {- a - 3 b e^{2 x}}{12 a^{3} b^{2} + 36 a^{2} b^{3} e^{2 x} + 36 a b^{4} e^{4 x} + 12 b^{5} e^{6 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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