Optimal. Leaf size=22 \[ \frac {1}{12} (-4-5 x)+\log ^{\frac {9}{2+x}}(2 x) \]
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Rubi [F] time = 1.94, 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 (-20 x-20 x^2-5 x^3\right ) \log (2 x)+\log ^{\frac {9}{2+x}}(2 x) (216+108 x-108 x \log (2 x) \log (\log (2 x)))}{\left (48 x+48 x^2+12 x^3\right ) \log (2 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 {\left (-20 x-20 x^2-5 x^3\right ) \log (2 x)+\log ^{\frac {9}{2+x}}(2 x) (216+108 x-108 x \log (2 x) \log (\log (2 x)))}{x \left (48+48 x+12 x^2\right ) \log (2 x)} \, dx\\ &=\int \frac {\left (-20 x-20 x^2-5 x^3\right ) \log (2 x)+\log ^{\frac {9}{2+x}}(2 x) (216+108 x-108 x \log (2 x) \log (\log (2 x)))}{12 x (2+x)^2 \log (2 x)} \, dx\\ &=\frac {1}{12} \int \frac {\left (-20 x-20 x^2-5 x^3\right ) \log (2 x)+\log ^{\frac {9}{2+x}}(2 x) (216+108 x-108 x \log (2 x) \log (\log (2 x)))}{x (2+x)^2 \log (2 x)} \, dx\\ &=\frac {1}{12} \int \left (-5+\frac {108 \log ^{\frac {7-x}{2+x}}(2 x) (2+x-x \log (2 x) \log (\log (2 x)))}{x (2+x)^2}\right ) \, dx\\ &=-\frac {5 x}{12}+9 \int \frac {\log ^{\frac {7-x}{2+x}}(2 x) (2+x-x \log (2 x) \log (\log (2 x)))}{x (2+x)^2} \, dx\\ &=-\frac {5 x}{12}+9 \int \left (\frac {\log ^{\frac {7-x}{2+x}}(2 x)}{x (2+x)}-\frac {\log ^{1+\frac {7-x}{2+x}}(2 x) \log (\log (2 x))}{(2+x)^2}\right ) \, dx\\ &=-\frac {5 x}{12}+9 \int \frac {\log ^{\frac {7-x}{2+x}}(2 x)}{x (2+x)} \, dx-9 \int \frac {\log ^{1+\frac {7-x}{2+x}}(2 x) \log (\log (2 x))}{(2+x)^2} \, dx\\ &=-\frac {5 x}{12}+9 \int \left (\frac {\log ^{\frac {7-x}{2+x}}(2 x)}{2 x}-\frac {\log ^{\frac {7-x}{2+x}}(2 x)}{2 (2+x)}\right ) \, dx-9 \int \frac {\log ^{\frac {9}{2+x}}(2 x) \log (\log (2 x))}{(2+x)^2} \, dx\\ &=-\frac {5 x}{12}+\frac {9}{2} \int \frac {\log ^{\frac {7-x}{2+x}}(2 x)}{x} \, dx-\frac {9}{2} \int \frac {\log ^{\frac {7-x}{2+x}}(2 x)}{2+x} \, dx-9 \int \frac {\log ^{\frac {9}{2+x}}(2 x) \log (\log (2 x))}{(2+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 18, normalized size = 0.82 \begin {gather*} -\frac {5 x}{12}+\log ^{\frac {9}{2+x}}(2 x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 16, normalized size = 0.73 \begin {gather*} -\frac {5}{12} \, x + \log \left (2 \, x\right )^{\frac {9}{x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 17, normalized size = 0.77
method | result | size |
risch | \(-\frac {5 x}{12}+\ln \left (2 x \right )^{\frac {9}{2+x}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 36, normalized size = 1.64 \begin {gather*} {\left (\log \relax (2) + \log \relax (x)\right )}^{\frac {9}{x + 2}} - \frac {5 \, {\left (x^{2} + 2 \, x - 4\right )}}{12 \, {\left (x + 2\right )}} - \frac {5}{3 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.78, size = 16, normalized size = 0.73 \begin {gather*} {\ln \left (2\,x\right )}^{\frac {9}{x+2}}-\frac {5\,x}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 17, normalized size = 0.77 \begin {gather*} - \frac {5 x}{12} + e^{\frac {9 \log {\left (\log {\left (2 x \right )} \right )}}{x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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