Optimal. Leaf size=24 \[ \log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \]
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Rubi [F]
time = 8.72, 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 {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {\log (-2+x) \left (-4 (-2+x) x+\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log ^2\left (x^2\right )\right )}{x \log \left (x^2\right ) \left (2 x+\left (x^2-\log (3)\right ) \log \left (x^2\right )\right )}-\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{2-x} \, dx\\ &=\int \left (\frac {\log (-2+x) \left (8 x-4 x^2-2 x^2 \log ^2\left (x^2\right )+x^3 \log ^2\left (x^2\right )+x \log (3) \log ^2\left (x^2\right )-\log (9) \log ^2\left (x^2\right )\right )}{(-2+x) x \log \left (x^2\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}+\frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x}\right ) \, dx\\ &=\int \frac {\log (-2+x) \left (8 x-4 x^2-2 x^2 \log ^2\left (x^2\right )+x^3 \log ^2\left (x^2\right )+x \log (3) \log ^2\left (x^2\right )-\log (9) \log ^2\left (x^2\right )\right )}{(-2+x) x \log \left (x^2\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=\int \left (\frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) x \left (x^2-\log (3)\right )}-\frac {2 \log (-2+x)}{x \log \left (x^2\right )}+\frac {2 \left (x^2-\log (3)\right ) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}-\frac {2 \left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) \left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )+2 \int \frac {\left (x^2-\log (3)\right ) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-2 \int \frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) \left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) x \left (x^2-\log (3)\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )-2 \int \frac {\left (x^2+\log (3)\right ) \log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+2 \int \left (\frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )}-\frac {\log (3) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\left (x^2+\log (3)\right ) \log (-2+x)}{x \left (x^2-\log (3)\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-2 \int \left (\frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )}+\frac {\log (9) \log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \left (-\frac {\log (-2+x)}{x}+\frac {2 x \log (-2+x)}{x^2-\log (3)}\right ) \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=2 \int \frac {x \log (-2+x)}{x^2-\log (3)} \, dx-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-(2 \log (9)) \int \frac {\log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-\int \frac {\log (-2+x)}{x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\log (-2+x) \log \left (\frac {x}{2}\right )+2 \int \left (-\frac {\log (-2+x)}{2 \left (-x+\sqrt {\log (3)}\right )}+\frac {\log (-2+x)}{2 \left (x+\sqrt {\log (3)}\right )}\right ) \, dx-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-(2 \log (9)) \int \left (-\frac {\log (-2+x)}{2 \left (-x+\sqrt {\log (3)}\right ) \sqrt {\log (3)} \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}-\frac {\log (-2+x)}{2 \left (x+\sqrt {\log (3)}\right ) \sqrt {\log (3)} \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\log (-2+x) \log \left (\frac {x}{2}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}-\int \frac {\log (-2+x)}{-x+\sqrt {\log (3)}} \, dx+\int \frac {\log (-2+x)}{x+\sqrt {\log (3)}} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}-\int \frac {\log \left (\frac {-x+\sqrt {\log (3)}}{-2+\sqrt {\log (3)}}\right )}{-2+x} \, dx-\int \frac {\log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )}{-2+x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ &=-\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx-\text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-2+\sqrt {\log (3)}}\right )}{x} \, dx,x,-2+x\right )-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2+\sqrt {\log (3)}}\right )}{x} \, dx,x,-2+x\right )\\ &=-\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )+\text {Li}_2\left (\frac {-2+x}{-2+\sqrt {\log (3)}}\right )+\text {Li}_2\left (\frac {2-x}{2+\sqrt {\log (3)}}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 24, normalized size = 1.00 \begin {gather*} \log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.40, size = 2480, normalized size = 103.33
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2480\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 35, normalized size = 1.46 \begin {gather*} -{\left (\log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x - 2\right ) + \log \left (-{\left (x^{2} - \log \left (3\right )\right )} \log \left (x\right ) - x\right ) \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 1.42 \begin {gather*} \log \left (x - 2\right ) \log \left (-\frac {{\left (x^{2} - \log \left (3\right )\right )} \log \left (x^{2}\right ) + 2 \, x}{x \log \left (x^{2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.64, size = 27, normalized size = 1.12 \begin {gather*} \log {\left (\frac {- 2 x + \left (- x^{2} + \log {\left (3 \right )}\right ) \log {\left (x^{2} \right )}}{x \log {\left (x^{2} \right )}} \right )} \log {\left (x - 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 46, normalized size = 1.92 \begin {gather*} \log \left (-x^{2} \log \left (x^{2}\right ) + \log \left (3\right ) \log \left (x^{2}\right ) - 2 \, x\right ) \log \left (x - 2\right ) - \log \left (x - 2\right ) \log \left (x\right ) - \log \left (x - 2\right ) \log \left (\log \left (x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 35, normalized size = 1.46 \begin {gather*} \ln \left (-\frac {2\,x-\ln \left (x^2\right )\,\left (\ln \left (3\right )-x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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