Optimal. Leaf size=24 \[ \left (2-\frac {1}{2} (630-x) x\right ) \log \left (\frac {x}{-1+\log (4 x)}\right ) \]
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Rubi [F] time = 0.84, 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 {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 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 {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (-2+2 \log (4 x))} \, dx\\ &=\int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{2 x (1-\log (4 x))} \, dx\\ &=\frac {1}{2} \int \frac {8-1260 x+2 x^2-\left (4-630 x+x^2\right ) \log (4 x)-\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{x (1-\log (4 x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))}+2 (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (4-630 x+x^2\right ) (-2+\log (4 x))}{x (-1+\log (4 x))} \, dx+\int (-315+x) \log \left (\frac {x}{-1+\log (4 x)}\right ) \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (\frac {4-630 x+x^2}{x}+\frac {-4+630 x-x^2}{x (-1+\log (4 x))}\right ) \, dx-\int \frac {(630-x) (-2+\log (4 x))}{2 (1-\log (4 x))} \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {4-630 x+x^2}{x} \, dx-\frac {1}{2} \int \frac {(630-x) (-2+\log (4 x))}{1-\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \left (-630+\frac {4}{x}+x\right ) \, dx-\frac {1}{2} \int \left (-630+x+\frac {630-x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \frac {630-x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-\frac {1}{2} \int \left (\frac {630}{-1+\log (4 x)}-\frac {x}{-1+\log (4 x)}\right ) \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {x}{-1+\log (4 x)} \, dx+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-315 \int \frac {1}{-1+\log (4 x)} \, dx\\ &=2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{32} \operatorname {Subst}\left (\int \frac {e^{2 x}}{-1+x} \, dx,x,\log (4 x)\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx-\frac {315}{4} \operatorname {Subst}\left (\int \frac {e^x}{-1+x} \, dx,x,\log (4 x)\right )\\ &=\frac {1}{32} e^2 \text {Ei}(-2 (1-\log (4 x)))-\frac {315}{4} e \text {Ei}(-1+\log (4 x))+2 \log (x)-315 x \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} x^2 \log \left (-\frac {x}{1-\log (4 x)}\right )+\frac {1}{2} \int \frac {-4+630 x-x^2}{x (-1+\log (4 x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.31, size = 36, normalized size = 1.50 \begin {gather*} \frac {1}{2} \left (4 \log (x)-4 \log (1-\log (4 x))+(-630+x) x \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 21, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (\frac {x}{\log \left (4 \, x\right ) - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \relax (x) - \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (\log \left (4 \, x\right ) - 1\right ) + 2 \, \log \relax (x) - 2 \, \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 45, normalized size = 1.88
method | result | size |
norman | \(-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )+\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}-2 \ln \left (\ln \left (4 x \right )-1\right )+2 \ln \relax (x )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \relax (x) - \frac {1}{2} \, {\left (x^{2} - 630 \, x - 4\right )} \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) - 4 \, \log \left (2 \, \log \relax (2) + \log \relax (x) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.40, size = 36, normalized size = 1.50 \begin {gather*} 2\,\ln \relax (x)-2\,\ln \left (\ln \left (4\,x\right )-1\right )-\ln \left (\frac {x}{\ln \left (4\,x\right )-1}\right )\,\left (315\,x-\frac {x^2}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 32, normalized size = 1.33 \begin {gather*} \left (\frac {x^{2}}{2} - 315 x\right ) \log {\left (\frac {x}{\log {\left (4 x \right )} - 1} \right )} + 2 \log {\relax (x )} - 2 \log {\left (\log {\left (4 x \right )} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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