Optimal. Leaf size=22 \[ \frac {5 \left (x+\log \left (x^2\right )\right )^2}{10 x-\frac {16}{\log ^2(4)}} \]
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Rubi [B] time = 0.85, antiderivative size = 87, normalized size of antiderivative = 3.95, number of steps used = 19, number of rules used = 13, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.141, Rules used = {1594, 27, 12, 6688, 6742, 698, 2357, 2301, 2314, 31, 2317, 2391, 2318} \begin {gather*} -\frac {5}{16} \log ^2(4) \log ^2\left (x^2\right )-\frac {5 x \log ^2(4) \log \left (x^2\right )}{8-5 x \log ^2(4)}-\frac {25 x \log ^4(4) \log ^2\left (x^2\right )}{16 \left (8-5 x \log ^2(4)\right )}+\frac {x}{2}-\frac {32}{5 \log ^2(4) \left (8-5 x \log ^2(4)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 698
Rule 1594
Rule 2301
Rule 2314
Rule 2317
Rule 2318
Rule 2357
Rule 2391
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-160 x-80 x^2\right ) \log ^2(4)+\left (100 x^2+25 x^3\right ) \log ^4(4)+\left ((-160-80 x) \log ^2(4)+100 x \log ^4(4)\right ) \log \left (x^2\right )-25 x \log ^4(4) \log ^2\left (x^2\right )}{x \left (128-160 x \log ^2(4)+50 x^2 \log ^4(4)\right )} \, dx\\ &=\int \frac {\left (-160 x-80 x^2\right ) \log ^2(4)+\left (100 x^2+25 x^3\right ) \log ^4(4)+\left ((-160-80 x) \log ^2(4)+100 x \log ^4(4)\right ) \log \left (x^2\right )-25 x \log ^4(4) \log ^2\left (x^2\right )}{2 x \left (-8+5 x \log ^2(4)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {\left (-160 x-80 x^2\right ) \log ^2(4)+\left (100 x^2+25 x^3\right ) \log ^4(4)+\left ((-160-80 x) \log ^2(4)+100 x \log ^4(4)\right ) \log \left (x^2\right )-25 x \log ^4(4) \log ^2\left (x^2\right )}{x \left (-8+5 x \log ^2(4)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {5 \log ^2(4) \left (x+\log \left (x^2\right )\right ) \left (-32+5 x^2 \log ^2(4)+4 x \left (-4+5 \log ^2(4)\right )-5 x \log ^2(4) \log \left (x^2\right )\right )}{x \left (8-5 x \log ^2(4)\right )^2} \, dx\\ &=\frac {1}{2} \left (5 \log ^2(4)\right ) \int \frac {\left (x+\log \left (x^2\right )\right ) \left (-32+5 x^2 \log ^2(4)+4 x \left (-4+5 \log ^2(4)\right )-5 x \log ^2(4) \log \left (x^2\right )\right )}{x \left (8-5 x \log ^2(4)\right )^2} \, dx\\ &=\frac {1}{2} \left (5 \log ^2(4)\right ) \int \left (\frac {-32+5 x^2 \log ^2(4)-4 x \left (4-5 \log ^2(4)\right )}{\left (8-5 x \log ^2(4)\right )^2}+\frac {4 \left (-8-x \left (4-5 \log ^2(4)\right )\right ) \log \left (x^2\right )}{x \left (8-5 x \log ^2(4)\right )^2}-\frac {5 \log ^2(4) \log ^2\left (x^2\right )}{\left (-8+5 x \log ^2(4)\right )^2}\right ) \, dx\\ &=\frac {1}{2} \left (5 \log ^2(4)\right ) \int \frac {-32+5 x^2 \log ^2(4)-4 x \left (4-5 \log ^2(4)\right )}{\left (8-5 x \log ^2(4)\right )^2} \, dx+\left (10 \log ^2(4)\right ) \int \frac {\left (-8-x \left (4-5 \log ^2(4)\right )\right ) \log \left (x^2\right )}{x \left (8-5 x \log ^2(4)\right )^2} \, dx-\frac {1}{2} \left (25 \log ^4(4)\right ) \int \frac {\log ^2\left (x^2\right )}{\left (-8+5 x \log ^2(4)\right )^2} \, dx\\ &=-\frac {25 x \log ^4(4) \log ^2\left (x^2\right )}{16 \left (8-5 x \log ^2(4)\right )}+\frac {1}{2} \left (5 \log ^2(4)\right ) \int \left (\frac {1}{5 \log ^2(4)}-\frac {64}{5 \log ^2(4) \left (-8+5 x \log ^2(4)\right )^2}+\frac {4}{-8+5 x \log ^2(4)}\right ) \, dx+\left (10 \log ^2(4)\right ) \int \left (-\frac {\log \left (x^2\right )}{8 x}-\frac {4 \log \left (x^2\right )}{\left (-8+5 x \log ^2(4)\right )^2}+\frac {5 \log ^2(4) \log \left (x^2\right )}{8 \left (-8+5 x \log ^2(4)\right )}\right ) \, dx-\frac {1}{4} \left (25 \log ^4(4)\right ) \int \frac {\log \left (x^2\right )}{-8+5 x \log ^2(4)} \, dx\\ &=\frac {x}{2}-\frac {32}{5 \log ^2(4) \left (8-5 x \log ^2(4)\right )}-\frac {25 x \log ^4(4) \log ^2\left (x^2\right )}{16 \left (8-5 x \log ^2(4)\right )}+2 \log \left (8-5 x \log ^2(4)\right )-\frac {5}{4} \log ^2(4) \log \left (x^2\right ) \log \left (1-\frac {5}{8} x \log ^2(4)\right )-\frac {1}{4} \left (5 \log ^2(4)\right ) \int \frac {\log \left (x^2\right )}{x} \, dx+\frac {1}{2} \left (5 \log ^2(4)\right ) \int \frac {\log \left (1-\frac {5}{8} x \log ^2(4)\right )}{x} \, dx-\left (40 \log ^2(4)\right ) \int \frac {\log \left (x^2\right )}{\left (-8+5 x \log ^2(4)\right )^2} \, dx+\frac {1}{4} \left (25 \log ^4(4)\right ) \int \frac {\log \left (x^2\right )}{-8+5 x \log ^2(4)} \, dx\\ &=\frac {x}{2}-\frac {32}{5 \log ^2(4) \left (8-5 x \log ^2(4)\right )}-\frac {5 x \log ^2(4) \log \left (x^2\right )}{8-5 x \log ^2(4)}-\frac {5}{16} \log ^2(4) \log ^2\left (x^2\right )-\frac {25 x \log ^4(4) \log ^2\left (x^2\right )}{16 \left (8-5 x \log ^2(4)\right )}+2 \log \left (8-5 x \log ^2(4)\right )-\frac {5}{2} \log ^2(4) \text {Li}_2\left (\frac {5}{8} x \log ^2(4)\right )-\frac {1}{2} \left (5 \log ^2(4)\right ) \int \frac {\log \left (1-\frac {5}{8} x \log ^2(4)\right )}{x} \, dx-\left (10 \log ^2(4)\right ) \int \frac {1}{-8+5 x \log ^2(4)} \, dx\\ &=\frac {x}{2}-\frac {32}{5 \log ^2(4) \left (8-5 x \log ^2(4)\right )}-\frac {5 x \log ^2(4) \log \left (x^2\right )}{8-5 x \log ^2(4)}-\frac {5}{16} \log ^2(4) \log ^2\left (x^2\right )-\frac {25 x \log ^4(4) \log ^2\left (x^2\right )}{16 \left (8-5 x \log ^2(4)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.13, size = 60, normalized size = 2.73 \begin {gather*} \frac {64-40 x \log ^2(4)+25 x^2 \log ^4(4)+50 x \log ^4(4) \log \left (x^2\right )+25 \log ^4(4) \log ^2\left (x^2\right )}{10 \log ^2(4) \left (-8+5 x \log ^2(4)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 59, normalized size = 2.68 \begin {gather*} \frac {25 \, x^{2} \log \relax (2)^{4} + 50 \, x \log \relax (2)^{4} \log \left (x^{2}\right ) + 25 \, \log \relax (2)^{4} \log \left (x^{2}\right )^{2} - 10 \, x \log \relax (2)^{2} + 4}{10 \, {\left (5 \, x \log \relax (2)^{4} - 2 \, \log \relax (2)^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 66, normalized size = 3.00 \begin {gather*} \frac {5 \, \log \relax (2)^{2} \log \left (x^{2}\right )^{2}}{2 \, {\left (5 \, x \log \relax (2)^{2} - 2\right )}} + \frac {1}{2} \, x + \frac {2 \, \log \left (x^{2}\right )}{5 \, x \log \relax (2)^{2} - 2} + \frac {2}{5 \, {\left (5 \, x \log \relax (2)^{4} - 2 \, \log \relax (2)^{2}\right )}} + 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 46, normalized size = 2.09
method | result | size |
norman | \(\frac {5 \ln \relax (2)^{2} \ln \left (x^{2}\right ) x +\frac {5 x^{2} \ln \relax (2)^{2}}{2}+\frac {5 \ln \relax (2)^{2} \ln \left (x^{2}\right )^{2}}{2}}{5 x \ln \relax (2)^{2}-2}\) | \(46\) |
risch | \(\frac {5 \ln \relax (2)^{2} \ln \left (x^{2}\right )^{2}}{2 \left (5 x \ln \relax (2)^{2}-2\right )}+\frac {2 \ln \left (x^{2}\right )}{5 x \ln \relax (2)^{2}-2}+\frac {100 \ln \relax (2)^{4} \ln \left (-x \right ) x +25 x^{2} \ln \relax (2)^{4}-40 \ln \left (-x \right ) \ln \relax (2)^{2}-10 x \ln \relax (2)^{2}+4}{10 \ln \relax (2)^{2} \left (5 x \ln \relax (2)^{2}-2\right )}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 197, normalized size = 8.95 \begin {gather*} -\frac {1}{10} \, {\left (\frac {4}{5 \, x \log \relax (2)^{8} - 2 \, \log \relax (2)^{6}} - \frac {5 \, x}{\log \relax (2)^{4}} - \frac {4 \, \log \left (5 \, x \log \relax (2)^{2} - 2\right )}{\log \relax (2)^{6}}\right )} \log \relax (2)^{4} - 2 \, {\left (\frac {2}{5 \, x \log \relax (2)^{6} - 2 \, \log \relax (2)^{4}} - \frac {\log \left (5 \, x \log \relax (2)^{2} - 2\right )}{\log \relax (2)^{4}}\right )} \log \relax (2)^{4} + \frac {2}{5} \, {\left (\frac {2}{5 \, x \log \relax (2)^{6} - 2 \, \log \relax (2)^{4}} - \frac {\log \left (5 \, x \log \relax (2)^{2} - 2\right )}{\log \relax (2)^{4}}\right )} \log \relax (2)^{2} + \frac {4 \, \log \relax (2)^{2}}{5 \, x \log \relax (2)^{4} - 2 \, \log \relax (2)^{2}} + \frac {2 \, {\left (5 \, \log \relax (2)^{2} \log \relax (x)^{2} + 2 \, \log \relax (x)\right )}}{5 \, x \log \relax (2)^{2} - 2} - 2 \, \log \left (5 \, x \log \relax (2)^{2} - 2\right ) + 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.78, size = 65, normalized size = 2.95 \begin {gather*} \frac {x}{2}+2\,\ln \relax (x)+\frac {2\,\ln \left (x^2\right )}{5\,x\,{\ln \relax (2)}^2-2}+\frac {2}{5\,{\ln \relax (2)}^2\,\left (5\,x\,{\ln \relax (2)}^2-2\right )}+\frac {5\,{\ln \left (x^2\right )}^2\,{\ln \relax (2)}^2}{2\,\left (5\,x\,{\ln \relax (2)}^2-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 65, normalized size = 2.95 \begin {gather*} \frac {x}{2} + 2 \log {\relax (x )} + \frac {2}{25 x \log {\relax (2 )}^{4} - 10 \log {\relax (2 )}^{2}} + \frac {5 \log {\relax (2 )}^{2} \log {\left (x^{2} \right )}^{2}}{10 x \log {\relax (2 )}^{2} - 4} + \frac {2 \log {\left (x^{2} \right )}}{5 x \log {\relax (2 )}^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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