Optimal. Leaf size=26 \[ \left (1+e^6\right )^2 \left (1-\frac {-x^2+\log (x)}{x+\log (3)}\right ) \]
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Rubi [B] time = 0.48, antiderivative size = 83, normalized size of antiderivative = 3.19, number of steps used = 14, number of rules used = 9, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1594, 27, 6688, 12, 6742, 43, 44, 2314, 31} \begin {gather*} \left (1+e^6\right )^2 x-\frac {\left (1+e^6\right )^2 \log ^2(3)}{x+\log (3)}+\frac {\left (1+e^6\right )^2 x \log (x)}{\log (3) (x+\log (3))}-\frac {\left (1+e^6\right )^2 \log (x)}{\log (3)}+\frac {\left (1+e^6\right )^2 \log (3) \log (9)}{x+\log (3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 43
Rule 44
Rule 1594
Rule 2314
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx\\ &=\int \frac {-x+x^3+e^{12} \left (-x+x^3\right )+e^6 \left (-2 x+2 x^3\right )+\left (-1+2 x^2+e^{12} \left (-1+2 x^2\right )+e^6 \left (-2+4 x^2\right )\right ) \log (3)+\left (x+2 e^6 x+e^{12} x\right ) \log (x)}{x (x+\log (3))^2} \, dx\\ &=\int \frac {\left (1+e^6\right )^2 \left (-x+x^3-\log (3)+x^2 \log (9)+x \log (x)\right )}{x (x+\log (3))^2} \, dx\\ &=\left (1+e^6\right )^2 \int \frac {-x+x^3-\log (3)+x^2 \log (9)+x \log (x)}{x (x+\log (3))^2} \, dx\\ &=\left (1+e^6\right )^2 \int \left (-\frac {1}{(x+\log (3))^2}+\frac {x^2}{(x+\log (3))^2}-\frac {\log (3)}{x (x+\log (3))^2}+\frac {x \log (9)}{(x+\log (3))^2}+\frac {\log (x)}{(x+\log (3))^2}\right ) \, dx\\ &=\frac {\left (1+e^6\right )^2}{x+\log (3)}+\left (1+e^6\right )^2 \int \frac {x^2}{(x+\log (3))^2} \, dx+\left (1+e^6\right )^2 \int \frac {\log (x)}{(x+\log (3))^2} \, dx-\left (\left (1+e^6\right )^2 \log (3)\right ) \int \frac {1}{x (x+\log (3))^2} \, dx+\left (\left (1+e^6\right )^2 \log (9)\right ) \int \frac {x}{(x+\log (3))^2} \, dx\\ &=\frac {\left (1+e^6\right )^2}{x+\log (3)}+\frac {\left (1+e^6\right )^2 x \log (x)}{\log (3) (x+\log (3))}+\left (1+e^6\right )^2 \int \left (1+\frac {\log ^2(3)}{(x+\log (3))^2}-\frac {\log (9)}{x+\log (3)}\right ) \, dx-\frac {\left (1+e^6\right )^2 \int \frac {1}{x+\log (3)} \, dx}{\log (3)}-\left (\left (1+e^6\right )^2 \log (3)\right ) \int \left (\frac {1}{x \log ^2(3)}-\frac {1}{\log (3) (x+\log (3))^2}-\frac {1}{\log ^2(3) (x+\log (3))}\right ) \, dx+\left (\left (1+e^6\right )^2 \log (9)\right ) \int \left (-\frac {\log (3)}{(x+\log (3))^2}+\frac {1}{x+\log (3)}\right ) \, dx\\ &=\left (1+e^6\right )^2 x-\frac {\left (1+e^6\right )^2 \log ^2(3)}{x+\log (3)}+\frac {\left (1+e^6\right )^2 \log (3) \log (9)}{x+\log (3)}-\frac {\left (1+e^6\right )^2 \log (x)}{\log (3)}+\frac {\left (1+e^6\right )^2 x \log (x)}{\log (3) (x+\log (3))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 30, normalized size = 1.15 \begin {gather*} \frac {\left (1+e^6\right )^2 \left (x^2+x \log (3)+\log ^2(3)-\log (x)\right )}{x+\log (3)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 63, normalized size = 2.42 \begin {gather*} \frac {x^{2} e^{12} + 2 \, x^{2} e^{6} + {\left (e^{12} + 2 \, e^{6} + 1\right )} \log \relax (3)^{2} + x^{2} + {\left (x e^{12} + 2 \, x e^{6} + x\right )} \log \relax (3) - {\left (e^{12} + 2 \, e^{6} + 1\right )} \log \relax (x)}{x + \log \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 76, normalized size = 2.92 \begin {gather*} \frac {x^{2} e^{12} + 2 \, x^{2} e^{6} + x e^{12} \log \relax (3) + 2 \, x e^{6} \log \relax (3) + e^{12} \log \relax (3)^{2} + 2 \, e^{6} \log \relax (3)^{2} + x^{2} + x \log \relax (3) + \log \relax (3)^{2} - e^{12} \log \relax (x) - 2 \, e^{6} \log \relax (x) - \log \relax (x)}{x + \log \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 42, normalized size = 1.62
method | result | size |
norman | \(\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) x^{2}+\left (-{\mathrm e}^{12}-2 \,{\mathrm e}^{6}-1\right ) \ln \relax (x )}{\ln \relax (3)+x}\) | \(42\) |
risch | \(-\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \ln \relax (x )}{\ln \relax (3)+x}+\frac {\left ({\mathrm e}^{12}+2 \,{\mathrm e}^{6}+1\right ) \left (\ln \relax (3)^{2}+x \ln \relax (3)+x^{2}\right )}{\ln \relax (3)+x}\) | \(47\) |
default | \(\frac {\ln \relax (x ) x \,{\mathrm e}^{12}}{\ln \relax (3) \left (\ln \relax (3)+x \right )}+\frac {2 \ln \relax (x ) x \,{\mathrm e}^{6}}{\ln \relax (3) \left (\ln \relax (3)+x \right )}+\frac {\ln \relax (x ) x}{\ln \relax (3) \left (\ln \relax (3)+x \right )}+x \,{\mathrm e}^{12}+2 x \,{\mathrm e}^{6}+x +\frac {\ln \relax (3)^{2} {\mathrm e}^{12}}{\ln \relax (3)+x}+\frac {2 \ln \relax (3)^{2} {\mathrm e}^{6}}{\ln \relax (3)+x}+\frac {\ln \relax (3)^{2}}{\ln \relax (3)+x}-\frac {\ln \relax (x ) {\mathrm e}^{12}}{\ln \relax (3)}-\frac {2 \ln \relax (x ) {\mathrm e}^{6}}{\ln \relax (3)}-\frac {\ln \relax (x )}{\ln \relax (3)}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 369, normalized size = 14.19 \begin {gather*} 2 \, {\left (\frac {\log \relax (3)}{x + \log \relax (3)} + \log \left (x + \log \relax (3)\right )\right )} e^{12} \log \relax (3) - {\left (\frac {1}{x \log \relax (3) + \log \relax (3)^{2}} - \frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)^{2}} + \frac {\log \relax (x)}{\log \relax (3)^{2}}\right )} e^{12} \log \relax (3) + 4 \, {\left (\frac {\log \relax (3)}{x + \log \relax (3)} + \log \left (x + \log \relax (3)\right )\right )} e^{6} \log \relax (3) - 2 \, {\left (\frac {1}{x \log \relax (3) + \log \relax (3)^{2}} - \frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)^{2}} + \frac {\log \relax (x)}{\log \relax (3)^{2}}\right )} e^{6} \log \relax (3) - {\left (2 \, \log \relax (3) \log \left (x + \log \relax (3)\right ) - x + \frac {\log \relax (3)^{2}}{x + \log \relax (3)}\right )} e^{12} - {\left (\frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)} - \frac {\log \relax (x)}{\log \relax (3)}\right )} e^{12} - 2 \, {\left (2 \, \log \relax (3) \log \left (x + \log \relax (3)\right ) - x + \frac {\log \relax (3)^{2}}{x + \log \relax (3)}\right )} e^{6} - 2 \, {\left (\frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)} - \frac {\log \relax (x)}{\log \relax (3)}\right )} e^{6} + 2 \, {\left (\frac {\log \relax (3)}{x + \log \relax (3)} + \log \left (x + \log \relax (3)\right )\right )} \log \relax (3) - {\left (\frac {1}{x \log \relax (3) + \log \relax (3)^{2}} - \frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)^{2}} + \frac {\log \relax (x)}{\log \relax (3)^{2}}\right )} \log \relax (3) - 2 \, \log \relax (3) \log \left (x + \log \relax (3)\right ) + x - \frac {\log \relax (3)^{2}}{x + \log \relax (3)} - \frac {e^{12} \log \relax (x)}{x + \log \relax (3)} - \frac {2 \, e^{6} \log \relax (x)}{x + \log \relax (3)} + \frac {e^{12}}{x + \log \relax (3)} + \frac {2 \, e^{6}}{x + \log \relax (3)} - \frac {\log \left (x + \log \relax (3)\right )}{\log \relax (3)} - \frac {\log \relax (x)}{x + \log \relax (3)} + \frac {\log \relax (x)}{\log \relax (3)} + \frac {1}{x + \log \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\left (\ln \relax (x)-x^2\right )\,{\left ({\mathrm {e}}^6+1\right )}^2}{x+\ln \relax (3)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 60, normalized size = 2.31 \begin {gather*} x \left (1 + 2 e^{6} + e^{12}\right ) + \frac {\left (- e^{12} - 2 e^{6} - 1\right ) \log {\relax (x )}}{x + \log {\relax (3 )}} + \frac {\log {\relax (3 )}^{2} + 2 e^{6} \log {\relax (3 )}^{2} + e^{12} \log {\relax (3 )}^{2}}{x + \log {\relax (3 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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