Integrand size = 17, antiderivative size = 38 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=\frac {a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac {1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2280, 45} \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=\frac {a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac {1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^4} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{2} \text {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*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=\frac {-a-3 b e^{2 x}}{12 b^2 \left (a+b e^{2 x}\right )^3} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {3 b \,{\mathrm e}^{2 x}+a}{12 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{3}}\) | \(25\) |
norman | \(\frac {-\frac {a}{12 b^{2}}-\frac {{\mathrm e}^{2 x}}{4 b}}{\left (a +b \,{\mathrm e}^{2 x}\right )^{3}}\) | \(28\) |
default | \(\frac {a}{6 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{3}}-\frac {1}{4 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{2}}\) | \(33\) |
parallelrisch | \(\frac {b^{4} {\mathrm e}^{4 x} {\mathrm e}^{2 x}+3 \,{\mathrm e}^{4 x} b^{3} a}{12 b^{3} a^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{3}}\) | \(42\) |
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none
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=-\frac {3 \, b e^{\left (2 \, x\right )} + a}{12 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=\frac {- a - 3 b e^{2 x}}{12 a^{3} b^{2} + 36 a^{2} b^{3} e^{2 x} + 36 a b^{4} e^{4 x} + 12 b^{5} e^{6 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (32) = 64\).
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.39 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=-\frac {b e^{\left (2 \, x\right )}}{4 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} - \frac {a}{12 \, {\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=-\frac {3 \, b e^{\left (2 \, x\right )} + a}{12 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{3} b^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx=\frac {\frac {{\mathrm {e}}^{4\,x}}{4\,a}+\frac {b\,{\mathrm {e}}^{6\,x}}{12\,a^2}}{a^3+3\,{\mathrm {e}}^{2\,x}\,a^2\,b+3\,{\mathrm {e}}^{4\,x}\,a\,b^2+{\mathrm {e}}^{6\,x}\,b^3} \]
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