Integrand size = 15, antiderivative size = 21 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=\frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2280, 37} \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=\frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
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Rule 37
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{(a+b x)^3} \, dx,x,e^x\right ) \\ & = \frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=\frac {-a-2 b e^x}{2 b^2 \left (a+b e^x\right )^2} \]
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Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{2 x}}{2 a \left (a +b \,{\mathrm e}^{x}\right )^{2}}\) | \(18\) |
risch | \(-\frac {2 b \,{\mathrm e}^{x}+a}{2 b^{2} \left (a +b \,{\mathrm e}^{x}\right )^{2}}\) | \(21\) |
norman | \(\frac {-\frac {{\mathrm e}^{x}}{b}-\frac {a}{2 b^{2}}}{\left (a +b \,{\mathrm e}^{x}\right )^{2}}\) | \(24\) |
default | \(\frac {a}{2 b^{2} \left (a +b \,{\mathrm e}^{x}\right )^{2}}-\frac {1}{b^{2} \left (a +b \,{\mathrm e}^{x}\right )}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=-\frac {2 \, b e^{x} + a}{2 \, {\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=\frac {- a - 2 b e^{x}}{2 a^{2} b^{2} + 4 a b^{3} e^{x} + 2 b^{4} e^{2 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (17) = 34\).
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=-\frac {b e^{x}}{b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}} - \frac {a}{2 \, {\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=-\frac {2 \, b e^{x} + a}{2 \, {\left (b e^{x} + a\right )}^{2} b^{2}} \]
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Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx=\frac {{\mathrm {e}}^{2\,x}}{2\,a\,\left (a^2+2\,{\mathrm {e}}^x\,a\,b+{\mathrm {e}}^{2\,x}\,b^2\right )} \]
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