Optimal. Leaf size=25 \[ x+\frac {x}{2 (3+\log (-3+x)) \left (2+\frac {x}{\log (x)}\right )} \]
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Rubi [F]
time = 3.02, 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 {-9 x-51 x^2+18 x^3+\left (-3 x-35 x^2+12 x^3\right ) \log (-3+x)+\left (-6 x^2+2 x^3\right ) \log ^2(-3+x)+\left (-216 x+71 x^2+\left (-144 x+48 x^2\right ) \log (-3+x)+\left (-24 x+8 x^2\right ) \log ^2(-3+x)\right ) \log (x)+\left (-234+76 x+(-150+50 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)\right ) \log ^2(x)}{-54 x^2+18 x^3+\left (-36 x^2+12 x^3\right ) \log (-3+x)+\left (-6 x^2+2 x^3\right ) \log ^2(-3+x)+\left (-216 x+72 x^2+\left (-144 x+48 x^2\right ) \log (-3+x)+\left (-24 x+8 x^2\right ) \log ^2(-3+x)\right ) \log (x)+\left (-216+72 x+(-144+48 x) \log (-3+x)+(-24+8 x) \log ^2(-3+x)\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 {-3 x \left (-3-17 x+6 x^2\right )-x (-216+71 x) \log (x)-(-234+76 x) \log ^2(x)-2 (-3+x) \log ^2(-3+x) (x+2 \log (x))^2-(-3+x) \log (-3+x) \left (x+12 x^2+48 x \log (x)+50 \log ^2(x)\right )}{2 (3-x) (3+\log (-3+x))^2 (x+2 \log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {-3 x \left (-3-17 x+6 x^2\right )-x (-216+71 x) \log (x)-(-234+76 x) \log ^2(x)-2 (-3+x) \log ^2(-3+x) (x+2 \log (x))^2-(-3+x) \log (-3+x) \left (x+12 x^2+48 x \log (x)+50 \log ^2(x)\right )}{(3-x) (3+\log (-3+x))^2 (x+2 \log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-117+38 x-75 \log (-3+x)+25 x \log (-3+x)-12 \log ^2(-3+x)+4 x \log ^2(-3+x)}{2 (-3+x) (3+\log (-3+x))^2}+\frac {x (2+x)}{2 (3+\log (-3+x)) (x+2 \log (x))^2}-\frac {x (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{2 (-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-117+38 x-75 \log (-3+x)+25 x \log (-3+x)-12 \log ^2(-3+x)+4 x \log ^2(-3+x)}{(-3+x) (3+\log (-3+x))^2} \, dx+\frac {1}{4} \int \frac {x (2+x)}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \frac {x (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=\frac {1}{4} \int \left (4-\frac {x}{(-3+x) (3+\log (-3+x))^2}+\frac {1}{3+\log (-3+x)}\right ) \, dx+\frac {1}{4} \int \left (\frac {2 x}{(3+\log (-3+x)) (x+2 \log (x))^2}+\frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2}\right ) \, dx-\frac {1}{4} \int \left (\frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3 (-18+5 x-6 \log (-3+x)+2 x \log (-3+x))}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=x-\frac {1}{4} \int \frac {x}{(-3+x) (3+\log (-3+x))^2} \, dx+\frac {1}{4} \int \frac {1}{3+\log (-3+x)} \, dx+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {3}{4} \int \frac {-18+5 x-6 \log (-3+x)+2 x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \int \left (-\frac {18}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {5 x}{(3+\log (-3+x))^2 (x+2 \log (x))}-\frac {6 \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {2 x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx-\frac {1}{4} \text {Subst}\left (\int \frac {3+x}{x (3+\log (x))^2} \, dx,x,-3+x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{3+\log (x)} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {3}{4} \int \left (-\frac {18}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}+\frac {5 x}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}-\frac {6 \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}+\frac {2 x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx\\ &=x+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx+\frac {1}{4} \text {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (-3+x)\right )-\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{(3+\log (x))^2}+\frac {3}{x (3+\log (x))^2}\right ) \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {3}{2} \int \frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{2} \int \frac {x \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {x}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {\log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \text {Subst}\left (\int \frac {1}{(3+\log (x))^2} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {1}{x (3+\log (x))^2} \, dx,x,-3+x\right )-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {3}{2} \int \frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{2} \int \left (\frac {\log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3 \log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx-\frac {15}{4} \int \left (\frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))}+\frac {3}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))}\right ) \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {\log (-3+x)}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \text {Subst}\left (\int \frac {1}{3+\log (x)} \, dx,x,-3+x\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,3+\log (-3+x)\right )-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {\text {Ei}(3+\log (-3+x))}{4 e^3}+\frac {3}{4 (3+\log (-3+x))}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{4} \text {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (-3+x)\right )+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ &=x+\frac {3}{4 (3+\log (-3+x))}-\frac {3-x}{4 (3+\log (-3+x))}+\frac {1}{4} \int \frac {x^2}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx+\frac {1}{2} \int \frac {x}{(3+\log (-3+x)) (x+2 \log (x))^2} \, dx-\frac {1}{2} \int \frac {x \log (-3+x)}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {5}{4} \int \frac {x}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {15}{4} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {9}{2} \int \frac {1}{(3+\log (-3+x))^2 (x+2 \log (x))} \, dx-\frac {45}{4} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx+\frac {27}{2} \int \frac {1}{(-3+x) (3+\log (-3+x))^2 (x+2 \log (x))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 44, normalized size = 1.76 \begin {gather*} \frac {1}{2} \left (2 x+\frac {x}{2 (3+\log (-3+x))}-\frac {x^2}{2 (3+\log (-3+x)) (x+2 \log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.16, size = 24, normalized size = 0.96
method | result | size |
default | \(x +\frac {x \ln \left (x \right )}{2 \left (2 \ln \left (x \right )+x \right ) \left (\ln \left (x -3\right )+3\right )}\) | \(24\) |
risch | \(x +\frac {x \ln \left (x \right )}{2 \left (2 \ln \left (x \right )+x \right ) \left (\ln \left (x -3\right )+3\right )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (23) = 46\).
time = 0.32, size = 49, normalized size = 1.96 \begin {gather*} \frac {6 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x \log \left (x\right )\right )} \log \left (x - 3\right ) + 13 \, x \log \left (x\right )}{2 \, {\left ({\left (x + 2 \, \log \left (x\right )\right )} \log \left (x - 3\right ) + 3 \, x + 6 \, \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (23) = 46\).
time = 0.37, size = 53, normalized size = 2.12 \begin {gather*} \frac {2 \, x^{2} \log \left (x - 3\right ) + 6 \, x^{2} + {\left (4 \, x \log \left (x - 3\right ) + 13 \, x\right )} \log \left (x\right )}{2 \, {\left (x \log \left (x - 3\right ) + 2 \, {\left (\log \left (x - 3\right ) + 3\right )} \log \left (x\right ) + 3 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 27, normalized size = 1.08 \begin {gather*} x + \frac {x \log {\left (x \right )}}{6 x + \left (2 x + 4 \log {\left (x \right )}\right ) \log {\left (x - 3 \right )} + 12 \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 31, normalized size = 1.24 \begin {gather*} x + \frac {x \log \left (x\right )}{2 \, {\left (x \log \left (x - 3\right ) + 2 \, \log \left (x - 3\right ) \log \left (x\right ) + 3 \, x + 6 \, \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.38, size = 292, normalized size = 11.68 \begin {gather*} \frac {x}{2}-\ln \left (x\right )+\frac {\frac {x\,\left (3\,x^4+14\,x^3-2\,x^2+12\,x+24\right )}{8\,{\left (x+2\right )}^3}+\frac {2\,x\,{\ln \left (x\right )}^2\,\left (x^2+4\,x-3\right )}{{\left (x+2\right )}^3}-\frac {x\,\ln \left (x\right )\,\left (-3\,x^3-13\,x^2+12\,x+6\right )}{2\,{\left (x+2\right )}^3}}{x+2\,\ln \left (x\right )}+\frac {\frac {x^2\,\ln \left (x\right )-3\,x^2-4\,x\,{\ln \left (x\right )}^2+9\,x+18\,{\ln \left (x\right )}^2}{2\,{\left (x+2\,\ln \left (x\right )\right )}^2}-\frac {\ln \left (x-3\right )\,\left (2\,{\ln \left (x\right )}^2+x\right )\,\left (x-3\right )}{2\,{\left (x+2\,\ln \left (x\right )\right )}^2}}{\ln \left (x-3\right )+3}-\frac {\frac {x^2}{2}+\frac {27\,x}{2}+16}{x^3+6\,x^2+12\,x+8}+\frac {\frac {x\,\left (x^3+2\,x-24\right )}{8\,\left (x+2\right )}+\frac {x\,\ln \left (x\right )\,\left (3\,x^2-8\,x+18\right )}{4\,\left (x+2\right )}+\frac {x\,{\ln \left (x\right )}^2\,\left (2\,x-3\right )}{x+2}}{x^2+4\,x\,\ln \left (x\right )+4\,{\ln \left (x\right )}^2}+\frac {\ln \left (x\right )\,\left (2\,x^2+15\,x+8\right )}{x^3+6\,x^2+12\,x+8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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