Integrand size = 15, antiderivative size = 34 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=\frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2280, 45} \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=\frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{(a+b x)^4} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^4}+\frac {1}{b (a+b x)^3}\right ) \, dx,x,e^x\right ) \\ & = \frac {a}{3 b^2 \left (a+b e^x\right )^3}-\frac {1}{2 b^2 \left (a+b e^x\right )^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=\frac {-a-3 b e^x}{6 b^2 \left (a+b e^x\right )^3} \]
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Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {3 b \,{\mathrm e}^{x}+a}{6 b^{2} \left (a +b \,{\mathrm e}^{x}\right )^{3}}\) | \(21\) |
norman | \(\frac {-\frac {a}{6 b^{2}}-\frac {{\mathrm e}^{x}}{2 b}}{\left (a +b \,{\mathrm e}^{x}\right )^{3}}\) | \(24\) |
default | \(\frac {a}{3 b^{2} \left (a +b \,{\mathrm e}^{x}\right )^{3}}-\frac {1}{2 b^{2} \left (a +b \,{\mathrm e}^{x}\right )^{2}}\) | \(29\) |
parallelrisch | \(\frac {b^{4} {\mathrm e}^{2 x} {\mathrm e}^{x}+3 \,{\mathrm e}^{2 x} b^{3} a}{6 b^{3} a^{2} \left (a +b \,{\mathrm e}^{x}\right )^{3}}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=-\frac {3 \, b e^{x} + a}{6 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=\frac {- a - 3 b e^{x}}{6 a^{3} b^{2} + 18 a^{2} b^{3} e^{x} + 18 a b^{4} e^{2 x} + 6 b^{5} e^{3 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (28) = 56\).
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.50 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=-\frac {b e^{x}}{2 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} - \frac {a}{6 \, {\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=-\frac {3 \, b e^{x} + a}{6 \, {\left (b e^{x} + a\right )}^{3} b^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^4} \, dx=\frac {\frac {{\mathrm {e}}^{2\,x}}{2\,a}+\frac {b\,{\mathrm {e}}^{3\,x}}{6\,a^2}}{a^3+3\,{\mathrm {e}}^x\,a^2\,b+3\,{\mathrm {e}}^{2\,x}\,a\,b^2+{\mathrm {e}}^{3\,x}\,b^3} \]
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