Optimal. Leaf size=33 \[ \frac {1}{3} \log \left (-x+\frac {x \log \left (\frac {25}{\log ^2(2 x)}\right )}{(3+x) (-2+3 \log (2))}\right ) \]
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Rubi [F] time = 6.26, 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 {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\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 {2 (3+x)+\log (2 x) \left ((3+x)^2 (-2+\log (8))-3 \log \left (\frac {25}{\log ^2(2 x)}\right )\right )}{3 x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {2 (3+x)+\log (2 x) \left ((3+x)^2 (-2+\log (8))-3 \log \left (\frac {25}{\log ^2(2 x)}\right )\right )}{x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {1}{3} \int \left (\frac {3}{x (3+x)}+\frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\int \frac {1}{x (3+x)} \, dx\\ &=\frac {1}{3} \int \frac {1}{x} \, dx-\frac {1}{3} \int \frac {1}{3+x} \, dx+\frac {1}{3} \int \frac {2 (3+x)+x (-6+x (-2+\log (8))+\log (512)) \log (2 x)}{x (3+x) \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{3} \int \left (\frac {6+2 x-2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)-6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{3 (3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{3 x \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \frac {6+2 x-2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)-6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} \int \frac {-6-2 x+2 x^2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x)+6 x \left (1-\frac {\log (512)}{6}\right ) \log (2 x)}{x \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \frac {2 (3+x)+x (-6+x (-2+\log (8))+\log (512)) \log (2 x)}{x \log (2 x) \left (x (-2+\log (8))-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} \int \left (\frac {x^2 (-2+\log (8))}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {x (-6+\log (512))}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {6}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {2 x}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}\right ) \, dx\\ &=\frac {\log (x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{9} \int \left (\frac {\left (1-\frac {6}{\log (512)}\right ) \log (512)}{-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )}+\frac {2}{\log (2 x) \left (-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {6}{x \log (2 x) \left (-2 x \left (1-\frac {\log (8)}{2}\right )-6 \left (1-\frac {\log (512)}{6}\right )-\log \left (\frac {25}{\log ^2(2 x)}\right )\right )}+\frac {x (2-\log (8))}{2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )}\right ) \, dx+\frac {2}{9} \int \frac {x}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {2}{3} \int \frac {1}{(3+x) \log (2 x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} (-2+\log (8)) \int \frac {x^2}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx+\frac {1}{9} (-6+\log (512)) \int \frac {x}{(3+x) \left (2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )+\log \left (\frac {25}{\log ^2(2 x)}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] time = 0.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.01, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{3} \, \log \left (-3 \, {\left (x + 3\right )} \log \relax (2) + 2 \, x + \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + 6\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{2} - 3 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (2) + 12 \, x + 18\right )} \log \left (2 \, x\right ) + 3 \, \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) - 2 \, x - 6}{3 \, {\left ({\left (x^{2} + 3 \, x\right )} \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + {\left (2 \, x^{3} + 12 \, x^{2} - 3 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \relax (2) + 18 \, x\right )} \log \left (2 \, x\right )\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 108, normalized size = 3.27
method | result | size |
risch | \(\frac {\ln \relax (x )}{3}-\frac {\ln \left (3+x \right )}{3}+\frac {\ln \left (\ln \left (\ln \left (2 x \right )\right )+\frac {i \left (-\pi \mathrm {csgn}\left (i \ln \left (2 x \right )\right )^{2} \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \left (2 x \right )\right ) \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{3}-6 i x \ln \relax (2)-18 i \ln \relax (2)+4 i \ln \relax (5)+4 i x +12 i\right )}{4}\right )}{3}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 39, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \log \left (\frac {1}{2} \, x {\left (3 \, \log \relax (2) - 2\right )} - \log \relax (5) + \frac {9}{2} \, \log \relax (2) + \log \left (\log \relax (2) + \log \relax (x)\right ) - 3\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {2\,x-3\,\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )-\ln \left (2\,x\right )\,\left (12\,x-\ln \relax (2)\,\left (3\,x^2+18\,x+27\right )+2\,x^2+18\right )+6}{\ln \left (2\,x\right )\,\left (54\,x-\ln \relax (2)\,\left (9\,x^3+54\,x^2+81\,x\right )+36\,x^2+6\,x^3\right )+\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )\,\left (3\,x^2+9\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 41, normalized size = 1.24 \begin {gather*} \frac {\log {\relax (x )}}{3} - \frac {\log {\left (x + 3 \right )}}{3} + \frac {\log {\left (- 3 x \log {\relax (2 )} + 2 x + \log {\left (\frac {25}{\log {\left (2 x \right )}^{2}} \right )} - 9 \log {\relax (2 )} + 6 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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