Integrand size = 15, antiderivative size = 27 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2} \]
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Time = 0.02 (sec) , antiderivative size = 27, 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} \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,e^x\right ) \\ & = \frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {\frac {a}{a+b e^x}+\log \left (b \left (a+b e^x\right )\right )}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {a}{b^{2} \left (a +b \,{\mathrm e}^{x}\right )}+\frac {\ln \left (a +b \,{\mathrm e}^{x}\right )}{b^{2}}\) | \(26\) |
norman | \(\frac {a}{b^{2} \left (a +b \,{\mathrm e}^{x}\right )}+\frac {\ln \left (a +b \,{\mathrm e}^{x}\right )}{b^{2}}\) | \(26\) |
risch | \(\frac {a}{b^{2} \left (a +b \,{\mathrm e}^{x}\right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a}{b}\right )}{b^{2}}\) | \(28\) |
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none
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b^{3} e^{x} + a b^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {a}{a b^{2} + b^{3} e^{x}} + \frac {\log {\left (\frac {a}{b} + e^{x} \right )}}{b^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {a}{b^{3} e^{x} + a b^{2}} + \frac {\log \left (b e^{x} + a\right )}{b^{2}} \]
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none
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {\log \left ({\left | b e^{x} + a \right |}\right )}{b^{2}} + \frac {a}{{\left (b e^{x} + a\right )} b^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx=\frac {\ln \left (a+b\,{\mathrm {e}}^x\right )}{b^2}-\frac {{\mathrm {e}}^x}{b\,\left (a+b\,{\mathrm {e}}^x\right )} \]
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