Optimal. Leaf size=26 \[ \frac {3 e^{-1+3 \left (-x+\log \left (\log \left (-x+x^2\right )\right )\right )} \log (x)}{x^3} \]
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Rubi [F] time = 5.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{-x^4+x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-1-3 x} \log ^2\left (-x+x^2\right ) \left ((-9+18 x) \log (x)+\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{(-1+x) x^4} \, dx\\ &=\int \frac {e^{-1-3 x} \log ^2((-1+x) x) \left (-((-9+18 x) \log (x))-\left (-3+3 x+\left (9-9 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )\right )}{(1-x) x^4} \, dx\\ &=\int \left (\frac {9 e^{-1-3 x} (-1+2 x) \log (x) \log ^2((-1+x) x)}{(-1+x) x^4}-\frac {3 e^{-1-3 x} (-1+3 \log (x)+3 x \log (x)) \log ^3((-1+x) x)}{x^4}\right ) \, dx\\ &=-\left (3 \int \frac {e^{-1-3 x} (-1+3 \log (x)+3 x \log (x)) \log ^3((-1+x) x)}{x^4} \, dx\right )+9 \int \frac {e^{-1-3 x} (-1+2 x) \log (x) \log ^2((-1+x) x)}{(-1+x) x^4} \, dx\\ &=-\left (3 \int \left (-\frac {e^{-1-3 x} \log ^3((-1+x) x)}{x^4}+\frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^4}+\frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3}\right ) \, dx\right )+9 \int \left (\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{-1+x}+\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^4}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^3}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^2}-\frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x}\right ) \, dx\\ &=3 \int \frac {e^{-1-3 x} \log ^3((-1+x) x)}{x^4} \, dx+9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{-1+x} \, dx+9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^4} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^3} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x^2} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^2((-1+x) x)}{x} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^4} \, dx-9 \int \frac {e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.40, size = 22, normalized size = 0.85 \begin {gather*} \frac {3 e^{-1-3 x} \log (x) \log ^3((-1+x) x)}{x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 24, normalized size = 0.92 \begin {gather*} \frac {3 \, e^{\left (-3 \, x + 3 \, \log \left (\log \left (x^{2} - x\right )\right ) - 1\right )} \log \relax (x)}{x^{3}} \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 {3 \, {\left ({\left (3 \, {\left (x^{2} - 1\right )} \log \relax (x) - x + 1\right )} \log \left (x^{2} - x\right ) - 3 \, {\left (2 \, x - 1\right )} \log \relax (x)\right )} e^{\left (-3 \, x + 3 \, \log \left (\log \left (x^{2} - x\right )\right ) - 1\right )}}{{\left (x^{5} - x^{4}\right )} \log \left (x^{2} - x\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.59, size = 1414, normalized size = 54.38
method | result | size |
risch | \(-\frac {3 i \left (\ln \left (x -1\right )+\ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x \left (x -1\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x -1\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x -1\right )\right )+\mathrm {csgn}\left (i \left (x -1\right )\right )\right )}{2}\right )^{3} \left (8 \ln \relax (x )^{3}+24 \ln \relax (x )^{2} \ln \left (x -1\right )+24 \ln \relax (x ) \ln \left (x -1\right )^{2}+3 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{7}-3 i \pi ^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{8}-i \pi ^{3} \mathrm {csgn}\left (i \left (x -1\right )\right )^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6}-6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \relax (x )-6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \left (x -1\right )+12 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5} \ln \relax (x )+12 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5} \ln \left (x -1\right )-6 \pi ^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \relax (x )-6 \pi ^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \left (x -1\right )+12 \pi ^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5} \ln \relax (x )+12 \pi ^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5} \ln \left (x -1\right )+3 i \pi ^{3} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{7}-3 i \pi ^{3} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{8}-12 i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \relax (x )^{2}-12 i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \left (x -1\right )^{2}-i \pi ^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6}-3 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4}+9 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5}-9 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6}+3 i \pi ^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5}-9 i \pi ^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6}+9 i \pi ^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{7}+12 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x )^{2}+12 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \left (x -1\right )^{2}+12 i \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x )^{2}+12 i \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \left (x -1\right )^{2}-24 i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \relax (x ) \ln \left (x -1\right )-6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x )-6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \left (x -1\right )+12 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \relax (x )+12 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \left (x -1\right )+12 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \relax (x )+12 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3} \ln \left (x -1\right )-24 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \relax (x )-24 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4} \ln \left (x -1\right )-24 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right ) \ln \relax (x ) \ln \left (x -1\right )-6 \pi ^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6} \ln \relax (x )-6 \pi ^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{6} \ln \left (x -1\right )+i \pi ^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{9}-12 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right ) \ln \relax (x )^{2}-12 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right ) \ln \left (x -1\right )^{2}+24 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x ) \ln \left (x -1\right )+24 i \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \ln \relax (x ) \ln \left (x -1\right )+i \pi ^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i \left (x -1\right )\right )^{3} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}-3 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \left (x -1\right )\right )^{4}+3 i \pi ^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{5}+8 \ln \left (x -1\right )^{3}\right ) \ln \relax (x ) {\mathrm e}^{-3 x -1}}{\left (2 i \ln \relax (x )+2 i \ln \left (x -1\right )+\pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}\right )^{3} x^{3}}\) | \(1414\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 60, normalized size = 2.31 \begin {gather*} \frac {3 \, {\left (e^{\left (-3 \, x\right )} \log \left (x - 1\right )^{3} \log \relax (x) + 3 \, e^{\left (-3 \, x\right )} \log \left (x - 1\right )^{2} \log \relax (x)^{2} + 3 \, e^{\left (-3 \, x\right )} \log \left (x - 1\right ) \log \relax (x)^{3} + e^{\left (-3 \, x\right )} \log \relax (x)^{4}\right )} e^{\left (-1\right )}}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-1}\,{\mathrm {e}}^{3\,\ln \left (\ln \left (x^2-x\right )\right )-3\,x}\,\left (\ln \relax (x)\,\left (18\,x-9\right )-\ln \left (x^2-x\right )\,\left (\ln \relax (x)\,\left (9\,x^2-9\right )-3\,x+3\right )\right )}{\ln \left (x^2-x\right )\,\left (x^4-x^5\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 24, normalized size = 0.92 \begin {gather*} \frac {3 e^{- 3 x} \log {\relax (x )} \log {\left (x^{2} - x \right )}^{3}}{e x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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