Optimal. Leaf size=14 \[ 3 \log ^2\left (1-22 e^{-x}\right ) \]
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Rubi [C] time = 0.17, antiderivative size = 69, normalized size of antiderivative = 4.93, number of steps used = 12, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {12, 2282, 2466, 2462, 260, 2416, 2390, 2301, 2394, 2315, 2391} \begin {gather*} -6 \text {Li}_2\left (22 e^{-x}\right )+6 \text {Li}_2\left (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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2282
Rule 2301
Rule 2315
Rule 2390
Rule 2391
Rule 2394
Rule 2416
Rule 2462
Rule 2466
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=132 \int \frac {\log \left (e^{-x} \left (-22+e^x\right )\right )}{-22+e^x} \, dx\\ &=132 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {22}{x}\right )}{(-22+x) x} \, dx,x,e^x\right )\\ &=132 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {22}{x}\right )}{-22+x} \, dx,x,e^x\right )-6 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log (-22+x)}{-22+x} \, dx,x,e^x\right )+6 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log \left (\frac {x}{22}\right )}{-22+x} \, dx,x,e^x\right )-6 \operatorname {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 {aligned} \end {gather*}
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Mathematica [C] time = 0.03, size = 122, normalized size = 8.71 \begin {gather*} 132 \left (\frac {1}{22} \log \left (22 e^{-x}\right ) \log \left (-e^{-x} \left (22-e^x\right )\right )+\frac {1}{22} \log \left (\frac {e^x}{22}\right ) \log \left (-22+e^x\right )+\frac {1}{22} \log \left (-e^{-x} \left (22-e^x\right )\right ) \log \left (-22+e^x\right )-\frac {1}{44} \log ^2\left (-22+e^x\right )+\frac {1}{22} \text {Li}_2\left (\frac {1}{22} \left (22-e^x\right )\right )+\frac {1}{22} \text {Li}_2\left (-e^{-x} \left (22-e^x\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 14, normalized size = 1.00 \begin {gather*} 3 \, \log \left ({\left (e^{x} - 22\right )} e^{\left (-x\right )}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {132 \, \log \left ({\left (e^{x} - 22\right )} e^{\left (-x\right )}\right )}{e^{x} - 22}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 14, normalized size = 1.00
method | result | size |
derivativedivides | \(3 \ln \left (1-22 \,{\mathrm e}^{-x}\right )^{2}\) | \(14\) |
default | \(3 \ln \left (1-22 \,{\mathrm e}^{-x}\right )^{2}\) | \(14\) |
norman | \(3 \ln \left ({\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}\) | \(15\) |
risch | \(132 \left (\frac {x}{22}-\frac {\ln \left ({\mathrm e}^{x}-22\right )}{22}\right ) \ln \left ({\mathrm e}^{x}\right )+3 \ln \left ({\mathrm e}^{x}-22\right )^{2}-3 x^{2}-3 i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}+3 i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+3 i \pi x \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{3}-3 i \pi x \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2}-3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{3}+3 i \pi \ln \left ({\mathrm e}^{x}-22\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-22\right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )\) | \(260\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 44, normalized size = 3.14 \begin {gather*} -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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 13, normalized size = 0.93 \begin {gather*} 3\,{\ln \left (1-22\,{\mathrm {e}}^{-x}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.86 \begin {gather*} 3 \log {\left (\left (e^{x} - 22\right ) e^{- x} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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