Optimal. Leaf size=24 \[ 1+\left ((1-2 x)^2-x\right )^2 \log ^{1-x}(2) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
time = 0.31, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps
used = 29, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2227, 2225,
2207} \begin {gather*} 16 x^4 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-10 x \log ^{1-x}(2)+\log ^{1-x}(2) \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-10 \log ^{1-x}(2)+66 x \log ^{1-x}(2)-120 x^2 \log ^{1-x}(2)+64 x^3 \log ^{1-x}(2)-\left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \log (\log (2))\right ) \, dx\\ &=-\left (10 \int \log ^{1-x}(2) \, dx\right )+64 \int x^3 \log ^{1-x}(2) \, dx+66 \int x \log ^{1-x}(2) \, dx-120 \int x^2 \log ^{1-x}(2) \, dx-\log (\log (2)) \int \left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \, dx\\ &=\frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}+\frac {66 \int \log ^{1-x}(2) \, dx}{\log (\log (2))}+\frac {192 \int x^2 \log ^{1-x}(2) \, dx}{\log (\log (2))}-\frac {240 \int x \log ^{1-x}(2) \, dx}{\log (\log (2))}-\log (\log (2)) \int \left (\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)\right ) \, dx\\ &=-\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}-\frac {240 \int \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}+\frac {384 \int x \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}-\log (\log (2)) \int \log ^{1-x}(2) \, dx+(10 \log (\log (2))) \int x \log ^{1-x}(2) \, dx-(16 \log (\log (2))) \int x^4 \log ^{1-x}(2) \, dx-(33 \log (\log (2))) \int x^2 \log ^{1-x}(2) \, dx+(40 \log (\log (2))) \int x^3 \log ^{1-x}(2) \, dx\\ &=\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {10 \log ^{1-x}(2)}{\log (\log (2))}-\frac {66 x \log ^{1-x}(2)}{\log (\log (2))}+\frac {120 x^2 \log ^{1-x}(2)}{\log (\log (2))}-\frac {64 x^3 \log ^{1-x}(2)}{\log (\log (2))}+10 \int \log ^{1-x}(2) \, dx-64 \int x^3 \log ^{1-x}(2) \, dx-66 \int x \log ^{1-x}(2) \, dx+120 \int x^2 \log ^{1-x}(2) \, dx+\frac {384 \int \log ^{1-x}(2) \, dx}{\log ^3(\log (2))}\\ &=\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {66 \log ^{1-x}(2)}{\log ^2(\log (2))}+\frac {240 x \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {192 x^2 \log ^{1-x}(2)}{\log ^2(\log (2))}-\frac {66 \int \log ^{1-x}(2) \, dx}{\log (\log (2))}-\frac {192 \int x^2 \log ^{1-x}(2) \, dx}{\log (\log (2))}+\frac {240 \int x \log ^{1-x}(2) \, dx}{\log (\log (2))}\\ &=\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}+\frac {240 \log ^{1-x}(2)}{\log ^3(\log (2))}-\frac {384 x \log ^{1-x}(2)}{\log ^3(\log (2))}+\frac {240 \int \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}-\frac {384 \int x \log ^{1-x}(2) \, dx}{\log ^2(\log (2))}\\ &=\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)-\frac {384 \log ^{1-x}(2)}{\log ^4(\log (2))}-\frac {384 \int \log ^{1-x}(2) \, dx}{\log ^3(\log (2))}\\ &=\log ^{1-x}(2)-10 x \log ^{1-x}(2)+33 x^2 \log ^{1-x}(2)-40 x^3 \log ^{1-x}(2)+16 x^4 \log ^{1-x}(2)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 21, normalized size = 0.88 \begin {gather*} \left (1-5 x+4 x^2\right )^2 \log ^{1-x}(2) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs.
\(2(30)=60\).
time = 0.23, size = 160, normalized size = 6.67
method | result | size |
risch | \(\frac {\left (16 x^{4}-40 x^{3}+33 x^{2}-10 x +1\right ) \ln \left (2\right )^{3-x}}{\ln \left (2\right )^{2}}\) | \(34\) |
gosper | \(\frac {{\mathrm e}^{-\left (x -3\right ) \ln \left (\ln \left (2\right )\right )} \left (4 x -1\right ) \left (4 x^{3}-9 x^{2}+6 x -1\right )}{\ln \left (2\right )^{2}}\) | \(35\) |
norman | \(\frac {\frac {{\mathrm e}^{\left (3-x \right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}-\frac {10 x \,{\mathrm e}^{\left (3-x \right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}+\frac {33 x^{2} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}-\frac {40 x^{3} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}+\frac {16 x^{4} {\mathrm e}^{\left (3-x \right ) \ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )}}{\ln \left (2\right )}\) | \(96\) |
derivativedivides | \(-\frac {836 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )-484 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \ln \left (\ln \left (2\right )\right )-\frac {16 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{4}}{\ln \left (\ln \left (2\right )\right )^{3}}+\frac {152 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{3}}{\ln \left (\ln \left (2\right )\right )^{2}}-\frac {537 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{2}}{\ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )}\) | \(160\) |
default | \(-\frac {836 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )-484 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \ln \left (\ln \left (2\right )\right )-\frac {16 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{4}}{\ln \left (\ln \left (2\right )\right )^{3}}+\frac {152 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{3}}{\ln \left (\ln \left (2\right )\right )^{2}}-\frac {537 \,{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )} \left (-x \ln \left (\ln \left (2\right )\right )+3 \ln \left (\ln \left (2\right )\right )\right )^{2}}{\ln \left (\ln \left (2\right )\right )}}{\ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )}\) | \(160\) |
meijerg | \(-\frac {16 \ln \left (2\right ) \left (24-\frac {\left (5 x^{4} \ln \left (\ln \left (2\right )\right )^{4}+20 x^{3} \ln \left (\ln \left (2\right )\right )^{3}+60 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+120 x \ln \left (\ln \left (2\right )\right )+120\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{5}\right )}{\ln \left (\ln \left (2\right )\right )^{4}}+\frac {\left (40 \ln \left (\ln \left (2\right )\right )+64\right ) \ln \left (2\right ) \left (6-\frac {\left (4 x^{3} \ln \left (\ln \left (2\right )\right )^{3}+12 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+24 x \ln \left (\ln \left (2\right )\right )+24\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{4}\right )}{\ln \left (\ln \left (2\right )\right )^{4}}+\frac {\left (-33 \ln \left (\ln \left (2\right )\right )-120\right ) \ln \left (2\right ) \left (2-\frac {\left (3 x^{2} \ln \left (\ln \left (2\right )\right )^{2}+6 x \ln \left (\ln \left (2\right )\right )+6\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{3}\right )}{\ln \left (\ln \left (2\right )\right )^{3}}+\frac {\left (10 \ln \left (\ln \left (2\right )\right )+66\right ) \ln \left (2\right ) \left (1-\frac {\left (2+2 x \ln \left (\ln \left (2\right )\right )\right ) {\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}}{2}\right )}{\ln \left (\ln \left (2\right )\right )^{2}}-\ln \left (2\right ) \left (1-{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}\right )-\frac {10 \ln \left (2\right ) \left (1-{\mathrm e}^{-x \ln \left (\ln \left (2\right )\right )}\right )}{\ln \left (\ln \left (2\right )\right )}\) | \(227\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs.
\(2 (28) = 56\).
time = 0.49, size = 303, normalized size = 12.62 \begin {gather*} \log \left (2\right )^{-x + 1} - \frac {10 \, {\left (x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )} + \frac {33 \, {\left (x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 2 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{2}} - \frac {66 \, {\left (x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{2}} + \frac {10 \, \log \left (2\right )^{-x + 1}}{\log \left (\log \left (2\right )\right )} - \frac {40 \, {\left (x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 3 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 6 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 6 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{3}} + \frac {120 \, {\left (x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 2 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{3}} + \frac {16 \, {\left (x^{4} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{4} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 12 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 24 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 24 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{4}} - \frac {64 \, {\left (x^{3} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{3} + 3 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2} + 6 \, x \log \left (2\right ) \log \left (\log \left (2\right )\right ) + 6 \, \log \left (2\right )\right )} \log \left (2\right )^{-x}}{\log \left (\log \left (2\right )\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 33, normalized size = 1.38 \begin {gather*} \frac {{\left (16 \, x^{4} - 40 \, x^{3} + 33 \, x^{2} - 10 \, x + 1\right )} \log \left (2\right )^{-x + 3}}{\log \left (2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 34, normalized size = 1.42 \begin {gather*} \frac {\left (16 x^{4} - 40 x^{3} + 33 x^{2} - 10 x + 1\right ) e^{\left (3 - x\right ) \log {\left (\log {\left (2 \right )} \right )}}}{\log {\left (2 \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs.
\(2 (28) = 56\).
time = 0.41, size = 67, normalized size = 2.79 \begin {gather*} \frac {{\left (16 \, x^{4} \log \left (\log \left (2\right )\right )^{5} - 40 \, x^{3} \log \left (\log \left (2\right )\right )^{5} + 33 \, x^{2} \log \left (\log \left (2\right )\right )^{5} - 10 \, x \log \left (\log \left (2\right )\right )^{5} + \log \left (\log \left (2\right )\right )^{5}\right )} e^{\left (-x \log \left (\log \left (2\right )\right ) + 3 \, \log \left (\log \left (2\right )\right )\right )}}{\log \left (2\right )^{2} \log \left (\log \left (2\right )\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 21, normalized size = 0.88 \begin {gather*} {\ln \left (2\right )}^{1-x}\,{\left (4\,x^2-5\,x+1\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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