Optimal. Leaf size=30 \[ 4 x^2 \left (x+\frac {x}{4 \left (25+x \log \left (\frac {4}{x}\right )\right ) \log (x)}\right )^2 \]
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Rubi [F] time = 4.13, 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 {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (32 x^3 \log ^3\left (\frac {4}{x}\right ) \log ^3(x)+4 x^2 \log ^2\left (\frac {4}{x}\right ) \log (x) \left (-1+3 \log (x)+600 \log ^2(x)\right )+(1+100 \log (x)) \left (-25+(50+x) \log (x)+5000 \log ^2(x)\right )+x \log \left (\frac {4}{x}\right ) \left (-1-199 \log (x)+4 (175+x) \log ^2(x)+60000 \log ^3(x)\right )\right )}{2 \left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{2} \int \frac {x^3 \left (32 x^3 \log ^3\left (\frac {4}{x}\right ) \log ^3(x)+4 x^2 \log ^2\left (\frac {4}{x}\right ) \log (x) \left (-1+3 \log (x)+600 \log ^2(x)\right )+(1+100 \log (x)) \left (-25+(50+x) \log (x)+5000 \log ^2(x)\right )+x \log \left (\frac {4}{x}\right ) \left (-1-199 \log (x)+4 (175+x) \log ^2(x)+60000 \log ^3(x)\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^3(x)} \, dx\\ &=\frac {1}{2} \int \left (\frac {500000 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {60000 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {2400 x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {32 x^6 \log ^3\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}-\frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)}-\frac {x^3 \left (2450-x+199 x \log \left (\frac {4}{x}\right )+4 x^2 \log ^2\left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {4 x^3 \left (100+x+3 x \log \left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx\right )-\frac {1}{2} \int \frac {x^3 \left (2450-x+199 x \log \left (\frac {4}{x}\right )+4 x^2 \log ^2\left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \int \frac {x^3 \left (100+x+3 x \log \left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx+16 \int \frac {x^6 \log ^3\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+1200 \int \frac {x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+30000 \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+250000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {2450 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}-\frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {199 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {4 x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}\right ) \, dx\right )-\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx+2 \int \left (\frac {100 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}+\frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}+\frac {3 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}\right ) \, dx+16 \int \left (x^3-\frac {15625 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {1875 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}-\frac {75 x^3}{25+x \log \left (\frac {4}{x}\right )}\right ) \, dx+1200 \int \left (\frac {625 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}-\frac {50 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}+\frac {x^3}{25+x \log \left (\frac {4}{x}\right )}\right ) \, dx+30000 \int \left (-\frac {25 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}\right ) \, dx+250000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx\\ &=4 x^4-\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx-2 \int \frac {x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \int \frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx+6 \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx-\frac {199}{2} \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+200 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx-1225 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \left (30000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2} \, dx\right )-60000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 41, normalized size = 1.37 \begin {gather*} \frac {x^4 \left (1+4 \left (25+x \log \left (\frac {4}{x}\right )\right ) \log (x)\right )^2}{4 \left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 218, normalized size = 7.27 \begin {gather*} \frac {16 \, x^{6} \log \left (\frac {4}{x}\right )^{4} + 40000 \, x^{4} \log \relax (2)^{2} + 400 \, x^{4} \log \relax (2) + x^{4} - 32 \, {\left (2 \, x^{6} \log \relax (2) - 25 \, x^{5}\right )} \log \left (\frac {4}{x}\right )^{3} + 8 \, {\left (8 \, x^{6} \log \relax (2)^{2} - 400 \, x^{5} \log \relax (2) - x^{5} + 1250 \, x^{4}\right )} \log \left (\frac {4}{x}\right )^{2} + 8 \, {\left (400 \, x^{5} \log \relax (2)^{2} - 25 \, x^{4} + 2 \, {\left (x^{5} - 2500 \, x^{4}\right )} \log \relax (2)\right )} \log \left (\frac {4}{x}\right )}{4 \, {\left (x^{2} \log \left (\frac {4}{x}\right )^{4} - 2 \, {\left (2 \, x^{2} \log \relax (2) - 25 \, x\right )} \log \left (\frac {4}{x}\right )^{3} + {\left (4 \, x^{2} \log \relax (2)^{2} - 200 \, x \log \relax (2) + 625\right )} \log \left (\frac {4}{x}\right )^{2} + 2500 \, \log \relax (2)^{2} + 100 \, {\left (2 \, x \log \relax (2)^{2} - 25 \, \log \relax (2)\right )} \log \left (\frac {4}{x}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 316, normalized size = 10.53 \begin {gather*} 4 \, x^{4} + \frac {64 \, x^{8} \log \relax (2)^{3} - 32 \, x^{8} \log \relax (2)^{2} \log \relax (x) + 2400 \, x^{7} \log \relax (2)^{2} - 800 \, x^{7} \log \relax (2) \log \relax (x) + 6 \, x^{7} \log \relax (2) - 2 \, x^{7} \log \relax (x) + 30000 \, x^{6} \log \relax (2) - 5000 \, x^{6} \log \relax (x) + 75 \, x^{6} + 125000 \, x^{5}}{4 \, {\left (32 \, x^{5} \log \relax (2)^{5} - 32 \, x^{5} \log \relax (2)^{4} \log \relax (x) + 8 \, x^{5} \log \relax (2)^{3} \log \relax (x)^{2} + 2000 \, x^{4} \log \relax (2)^{4} - 1600 \, x^{4} \log \relax (2)^{3} \log \relax (x) + 300 \, x^{4} \log \relax (2)^{2} \log \relax (x)^{2} + 50000 \, x^{3} \log \relax (2)^{3} - 30000 \, x^{3} \log \relax (2)^{2} \log \relax (x) + 3750 \, x^{3} \log \relax (2) \log \relax (x)^{2} + 625000 \, x^{2} \log \relax (2)^{2} - 250000 \, x^{2} \log \relax (2) \log \relax (x) + 15625 \, x^{2} \log \relax (x)^{2} + 3906250 \, x \log \relax (2) - 781250 \, x \log \relax (x) + 9765625\right )}} + \frac {32 \, x^{6} \log \relax (2)^{2} \log \relax (x) + 800 \, x^{5} \log \relax (2) \log \relax (x) + 2 \, x^{5} \log \relax (2) + 2 \, x^{5} \log \relax (x) + 5000 \, x^{4} \log \relax (x) + 25 \, x^{4}}{4 \, {\left (8 \, x^{3} \log \relax (2)^{3} \log \relax (x)^{2} + 300 \, x^{2} \log \relax (2)^{2} \log \relax (x)^{2} + 3750 \, x \log \relax (2) \log \relax (x)^{2} + 15625 \, \log \relax (x)^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 119.47, size = 50, normalized size = 1.67
method | result | size |
risch | \(4 x^{4}-\frac {x^{4} \left (-1-16 x \ln \relax (2) \ln \relax (x )+8 x \ln \relax (x )^{2}-200 \ln \relax (x )\right )}{\ln \relax (x )^{2} \left (-50-4 x \ln \relax (2)+2 x \ln \relax (x )\right )^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 131, normalized size = 4.37 \begin {gather*} \frac {16 \, x^{6} \log \relax (x)^{4} + x^{4} - 32 \, {\left (2 \, x^{6} \log \relax (2) + 25 \, x^{5}\right )} \log \relax (x)^{3} + 8 \, {\left (8 \, x^{6} \log \relax (2)^{2} + x^{5} {\left (200 \, \log \relax (2) - 1\right )} + 1250 \, x^{4}\right )} \log \relax (x)^{2} + 8 \, {\left (2 \, x^{5} \log \relax (2) + 25 \, x^{4}\right )} \log \relax (x)}{4 \, {\left (x^{2} \log \relax (x)^{4} - 2 \, {\left (2 \, x^{2} \log \relax (2) + 25 \, x\right )} \log \relax (x)^{3} + {\left (4 \, x^{2} \log \relax (2)^{2} + 100 \, x \log \relax (2) + 625\right )} \log \relax (x)^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 40, normalized size = 1.33 \begin {gather*} \frac {x^4\,{\left (100\,\ln \relax (x)+4\,x\,\ln \left (\frac {4}{x}\right )\,\ln \relax (x)+1\right )}^2}{4\,{\ln \relax (x)}^2\,{\left (x\,\ln \left (\frac {4}{x}\right )+25\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.54, size = 87, normalized size = 2.90 \begin {gather*} 4 x^{4} + \frac {- 8 x^{5} \log {\relax (x )}^{2} + x^{4} + \left (16 x^{5} \log {\relax (2 )} + 200 x^{4}\right ) \log {\relax (x )}}{4 x^{2} \log {\relax (x )}^{4} + \left (- 16 x^{2} \log {\relax (2 )} - 200 x\right ) \log {\relax (x )}^{3} + \left (16 x^{2} \log {\relax (2 )}^{2} + 400 x \log {\relax (2 )} + 2500\right ) \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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