Integrand size = 18, antiderivative size = 32 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=-\frac {a^2 e^{-n x}}{n}+\frac {b^2 e^{n x}}{n}+2 a b x \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2280, 45} \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=-\frac {a^2 e^{-n x}}{n}+2 a b x+\frac {b^2 e^{n x}}{n} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,e^{n x}\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,e^{n x}\right )}{n} \\ & = -\frac {a^2 e^{-n x}}{n}+\frac {b^2 e^{n x}}{n}+2 a b x \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=\frac {-a^2 e^{-n x}+b^2 e^{n x}-2 a b \log \left (e^{-n x}\right )}{n} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {a^{2} {\mathrm e}^{-n x}}{n}+\frac {b^{2} {\mathrm e}^{n x}}{n}+2 a b x\) | \(32\) |
parts | \(-\frac {a^{2} {\mathrm e}^{-n x}}{n}+\frac {b^{2} {\mathrm e}^{n x}}{n}+2 a b x\) | \(32\) |
derivativedivides | \(\frac {{\mathrm e}^{n x} b^{2}-a^{2} {\mathrm e}^{-n x}+2 a b \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(34\) |
default | \(\frac {{\mathrm e}^{n x} b^{2}-a^{2} {\mathrm e}^{-n x}+2 a b \ln \left ({\mathrm e}^{n x}\right )}{n}\) | \(34\) |
parallelrisch | \(\frac {\left (2 x \,{\mathrm e}^{n x} a b n +{\mathrm e}^{2 n x} b^{2}-a^{2}\right ) {\mathrm e}^{-n x}}{n}\) | \(37\) |
norman | \(\left (\frac {b^{2} {\mathrm e}^{2 n x}}{n}-\frac {a^{2}}{n}+2 a b x \,{\mathrm e}^{n x}\right ) {\mathrm e}^{-n x}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=\frac {{\left (2 \, a b n x e^{\left (n x\right )} + b^{2} e^{\left (2 \, n x\right )} - a^{2}\right )} e^{\left (-n x\right )}}{n} \]
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Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=2 a b x + \begin {cases} \frac {- a^{2} n e^{- n x} + b^{2} n e^{n x}}{n^{2}} & \text {for}\: n^{2} \neq 0 \\x \left (a^{2} + b^{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \]
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Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=2 \, a b x + \frac {b^{2} e^{\left (n x\right )}}{n} - \frac {a^{2} e^{\left (-n x\right )}}{n} \]
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Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int e^{-n x} \left (a+b e^{n x}\right )^2 \, dx=2\,a\,b\,x-\frac {{\mathrm {e}}^{-n\,x}\,\left (a^2-b^2\,{\mathrm {e}}^{2\,n\,x}\right )}{n} \]
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