Integrand size = 21, antiderivative size = 14 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3 \log ^2\left (1-22 e^{-x}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.93, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {12, 2320, 2516, 2512, 266, 2463, 2437, 2338, 2441, 2352, 2438} \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=-6 \operatorname {PolyLog}\left (2,22 e^{-x}\right )+6 \operatorname {PolyLog}\left (2,1-\frac {e^x}{22}\right )-3 \log ^2\left (e^x-22\right )+6 \log \left (\frac {e^x}{22}\right ) \log \left (e^x-22\right )+6 \log \left (1-22 e^{-x}\right ) \log \left (e^x-22\right ) \]
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Rule 12
Rule 266
Rule 2320
Rule 2338
Rule 2352
Rule 2437
Rule 2438
Rule 2441
Rule 2463
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = 132 \int \frac {\log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx \\ & = 132 \text {Subst}\left (\int \frac {\log \left (1-\frac {22}{x}\right )}{(-22+x) x} \, dx,x,e^x\right ) \\ & = 132 \text {Subst}\left (\int \left (\frac {\log \left (1-\frac {22}{x}\right )}{22 (-22+x)}-\frac {\log \left (1-\frac {22}{x}\right )}{22 x}\right ) \, dx,x,e^x\right ) \\ & = 6 \text {Subst}\left (\int \frac {\log \left (1-\frac {22}{x}\right )}{-22+x} \, dx,x,e^x\right )-6 \text {Subst}\left (\int \frac {\log \left (1-\frac {22}{x}\right )}{x} \, dx,x,e^x\right ) \\ & = 6 \log \left (1-22 e^{-x}\right ) \log \left (-22+e^x\right )-6 \text {Li}_2\left (22 e^{-x}\right )-132 \text {Subst}\left (\int \frac {\log (-22+x)}{\left (1-\frac {22}{x}\right ) x^2} \, dx,x,e^x\right ) \\ & = 6 \log \left (1-22 e^{-x}\right ) \log \left (-22+e^x\right )-6 \text {Li}_2\left (22 e^{-x}\right )-132 \text {Subst}\left (\int \left (\frac {\log (-22+x)}{22 (-22+x)}-\frac {\log (-22+x)}{22 x}\right ) \, dx,x,e^x\right ) \\ & = 6 \log \left (1-22 e^{-x}\right ) \log \left (-22+e^x\right )-6 \text {Li}_2\left (22 e^{-x}\right )-6 \text {Subst}\left (\int \frac {\log (-22+x)}{-22+x} \, dx,x,e^x\right )+6 \text {Subst}\left (\int \frac {\log (-22+x)}{x} \, dx,x,e^x\right ) \\ & = 6 \log \left (\frac {e^x}{22}\right ) \log \left (-22+e^x\right )+6 \log \left (1-22 e^{-x}\right ) \log \left (-22+e^x\right )-6 \text {Li}_2\left (22 e^{-x}\right )-6 \text {Subst}\left (\int \frac {\log \left (\frac {x}{22}\right )}{-22+x} \, dx,x,e^x\right )-6 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-22+e^x\right ) \\ & = 6 \log \left (\frac {e^x}{22}\right ) \log \left (-22+e^x\right )+6 \log \left (1-22 e^{-x}\right ) \log \left (-22+e^x\right )-3 \log ^2\left (-22+e^x\right )-6 \text {Li}_2\left (22 e^{-x}\right )+6 \text {Li}_2\left (1-\frac {e^x}{22}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3 \log ^2\left (1-22 e^{-x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(3 \ln \left ({\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}\) | \(15\) |
default | \(3 \ln \left ({\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}\) | \(15\) |
norman | \(3 \ln \left ({\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}\) | \(15\) |
risch | \(132 \left (\frac {x}{22}-\frac {\ln \left ({\mathrm e}^{x}-22\right )}{22}\right ) \ln \left ({\mathrm e}^{x}\right )+3 \ln \left ({\mathrm e}^{x}-22\right )^{2}-3 x^{2}+3 i \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )-3 i \pi x \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}-3 i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}+3 i \pi x \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{3}-3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )+3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}+3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}-3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{3}\) | \(260\) |
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Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3 \, \log \left ({\left (e^{x} - 22\right )} e^{\left (-x\right )}\right )^{2} \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3 \log {\left (\left (e^{x} - 22\right ) e^{- x} \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (13) = 26\).
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 3.14 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=-3 \, x^{2} - 6 \, {\left (x - \log \left (e^{x} - 22\right )\right )} \log \left ({\left (e^{x} - 22\right )} e^{\left (-x\right )}\right ) + 6 \, x \log \left (e^{x} - 22\right ) - 3 \, \log \left (e^{x} - 22\right )^{2} \]
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3 \, \log \left ({\left (e^{x} - 22\right )} e^{\left (-x\right )}\right )^{2} \]
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {132 \log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx=3\,{\ln \left (1-22\,{\mathrm {e}}^{-x}\right )}^2 \]
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