Optimal. Leaf size=25 \[ \frac {\left (1+\frac {1}{20} x^2 \left (3+\log ^2\left (\frac {x}{4}\right )\right )\right )^2}{x^2} \]
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Rubi [B] time = 0.14, antiderivative size = 53, normalized size of antiderivative = 2.12, number of steps used = 18, number of rules used = 6, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2351, 2301, 2304, 2305} \begin {gather*} \frac {9 x^2}{400}+\frac {1}{x^2}+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{10} \log ^2\left (\frac {x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2301
Rule 2304
Rule 2305
Rule 2351
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{200} \int \frac {-400+9 x^4+\left (40 x^2+6 x^4\right ) \log \left (\frac {x}{4}\right )+6 x^4 \log ^2\left (\frac {x}{4}\right )+2 x^4 \log ^3\left (\frac {x}{4}\right )+x^4 \log ^4\left (\frac {x}{4}\right )}{x^3} \, dx\\ &=\frac {1}{200} \int \left (\frac {-400+9 x^4}{x^3}+\frac {2 \left (20+3 x^2\right ) \log \left (\frac {x}{4}\right )}{x}+6 x \log ^2\left (\frac {x}{4}\right )+2 x \log ^3\left (\frac {x}{4}\right )+x \log ^4\left (\frac {x}{4}\right )\right ) \, dx\\ &=\frac {1}{200} \int \frac {-400+9 x^4}{x^3} \, dx+\frac {1}{200} \int x \log ^4\left (\frac {x}{4}\right ) \, dx+\frac {1}{100} \int \frac {\left (20+3 x^2\right ) \log \left (\frac {x}{4}\right )}{x} \, dx+\frac {1}{100} \int x \log ^3\left (\frac {x}{4}\right ) \, dx+\frac {3}{100} \int x \log ^2\left (\frac {x}{4}\right ) \, dx\\ &=\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{200} x^2 \log ^3\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {1}{200} \int \left (-\frac {400}{x^3}+9 x\right ) \, dx-\frac {1}{100} \int x \log ^3\left (\frac {x}{4}\right ) \, dx+\frac {1}{100} \int \left (\frac {20 \log \left (\frac {x}{4}\right )}{x}+3 x \log \left (\frac {x}{4}\right )\right ) \, dx-\frac {3}{200} \int x \log ^2\left (\frac {x}{4}\right ) \, dx-\frac {3}{100} \int x \log \left (\frac {x}{4}\right ) \, dx\\ &=\frac {1}{x^2}+\frac {3 x^2}{100}-\frac {3}{200} x^2 \log \left (\frac {x}{4}\right )+\frac {3}{400} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {3}{200} \int x \log \left (\frac {x}{4}\right ) \, dx+\frac {3}{200} \int x \log ^2\left (\frac {x}{4}\right ) \, dx+\frac {3}{100} \int x \log \left (\frac {x}{4}\right ) \, dx+\frac {1}{5} \int \frac {\log \left (\frac {x}{4}\right )}{x} \, dx\\ &=\frac {1}{x^2}+\frac {3 x^2}{160}+\frac {3}{400} x^2 \log \left (\frac {x}{4}\right )+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )-\frac {3}{200} \int x \log \left (\frac {x}{4}\right ) \, dx\\ &=\frac {1}{x^2}+\frac {9 x^2}{400}+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.02, size = 53, normalized size = 2.12 \begin {gather*} \frac {1}{x^2}+\frac {9 x^2}{400}+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 41, normalized size = 1.64 \begin {gather*} \frac {x^{4} \log \left (\frac {1}{4} \, x\right )^{4} + 9 \, x^{4} + 2 \, {\left (3 \, x^{4} + 20 \, x^{2}\right )} \log \left (\frac {1}{4} \, x\right )^{2} + 400}{400 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 35, normalized size = 1.40 \begin {gather*} \frac {1}{400} \, x^{2} \log \left (\frac {1}{4} \, x\right )^{4} + \frac {1}{200} \, {\left (3 \, x^{2} + 20\right )} \log \left (\frac {1}{4} \, x\right )^{2} + \frac {9}{400} \, x^{2} + \frac {1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 40, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {3 x^{2} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{2}}{400}+\frac {\ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {1}{x^{2}}\) | \(40\) |
default | \(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {3 x^{2} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{2}}{400}+\frac {\ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {1}{x^{2}}\) | \(40\) |
risch | \(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {\left (3 x^{2}+20\right ) \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{4}+400}{400 x^{2}}\) | \(40\) |
norman | \(\frac {1+\frac {9 x^{4}}{400}+\frac {x^{2} \ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {3 x^{4} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {x^{4} \ln \left (\frac {x}{4}\right )^{4}}{400}}{x^{2}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 113, normalized size = 4.52 \begin {gather*} \frac {1}{800} \, {\left (2 \, \log \left (\frac {1}{4} \, x\right )^{4} - 4 \, \log \left (\frac {1}{4} \, x\right )^{3} + 6 \, \log \left (\frac {1}{4} \, x\right )^{2} - 6 \, \log \left (\frac {1}{4} \, x\right ) + 3\right )} x^{2} + \frac {1}{800} \, {\left (4 \, \log \left (\frac {1}{4} \, x\right )^{3} - 6 \, \log \left (\frac {1}{4} \, x\right )^{2} + 6 \, \log \left (\frac {1}{4} \, x\right ) - 3\right )} x^{2} + \frac {3}{400} \, {\left (2 \, \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, \log \left (\frac {1}{4} \, x\right ) + 1\right )} x^{2} + \frac {3}{200} \, x^{2} \log \left (\frac {1}{4} \, x\right ) + \frac {3}{200} \, x^{2} + \frac {1}{10} \, \log \left (\frac {1}{4} \, x\right )^{2} + \frac {1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.04, size = 34, normalized size = 1.36 \begin {gather*} x^2\,\left (\frac {{\ln \left (\frac {x}{4}\right )}^4}{400}+\frac {3\,{\ln \left (\frac {x}{4}\right )}^2}{200}+\frac {9}{400}\right )+\frac {{\ln \left (\frac {x}{4}\right )}^2}{10}+\frac {1}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 39, normalized size = 1.56 \begin {gather*} \frac {x^{2} \log {\left (\frac {x}{4} \right )}^{4}}{400} + \frac {9 x^{2}}{400} + \left (\frac {3 x^{2}}{200} + \frac {1}{10}\right ) \log {\left (\frac {x}{4} \right )}^{2} + \frac {1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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