Optimal. Leaf size=23 \[ 13-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]
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Rubi [F]
time = 3.36, 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 {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(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 {-x (2+x)-2 x (1+x)^2 \log ^2(x)-x (2+x) \log ^3(x)-\log (x) \left (2+x+2 x^2+2 x^3+x^4-\left (2+3 x+x^2\right ) \log \left (2 (2+x)^2 \log (x)\right )\right )}{x (2+x) \log (x) (x+\log (x))^2} \, dx\\ &=\int \left (-\frac {1}{(2+x) (x+\log (x))^2}-\frac {2}{x (2+x) (x+\log (x))^2}-\frac {2 x}{(2+x) (x+\log (x))^2}-\frac {2 x^2}{(2+x) (x+\log (x))^2}-\frac {x^3}{(2+x) (x+\log (x))^2}-\frac {1}{\log (x) (x+\log (x))^2}-\frac {2 (1+x)^2 \log (x)}{(2+x) (x+\log (x))^2}-\frac {\log ^2(x)}{(x+\log (x))^2}+\frac {(1+x) \log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x (2+x) (x+\log (x))^2} \, dx\right )-2 \int \frac {x}{(2+x) (x+\log (x))^2} \, dx-2 \int \frac {x^2}{(2+x) (x+\log (x))^2} \, dx-2 \int \frac {(1+x)^2 \log (x)}{(2+x) (x+\log (x))^2} \, dx-\int \frac {1}{(2+x) (x+\log (x))^2} \, dx-\int \frac {x^3}{(2+x) (x+\log (x))^2} \, dx-\int \frac {1}{\log (x) (x+\log (x))^2} \, dx-\int \frac {\log ^2(x)}{(x+\log (x))^2} \, dx+\int \frac {(1+x) \log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2} \, dx\\ &=-\left (2 \int \left (\frac {1}{(x+\log (x))^2}-\frac {2}{(2+x) (x+\log (x))^2}\right ) \, dx\right )-2 \int \left (\frac {1}{2 x (x+\log (x))^2}-\frac {1}{2 (2+x) (x+\log (x))^2}\right ) \, dx-2 \int \left (-\frac {2}{(x+\log (x))^2}+\frac {x}{(x+\log (x))^2}+\frac {4}{(2+x) (x+\log (x))^2}\right ) \, dx-2 \int \left (-\frac {x (1+x)^2}{(2+x) (x+\log (x))^2}+\frac {(1+x)^2}{(2+x) (x+\log (x))}\right ) \, dx-\int \frac {1}{(2+x) (x+\log (x))^2} \, dx-\int \left (\frac {4}{(x+\log (x))^2}-\frac {2 x}{(x+\log (x))^2}+\frac {x^2}{(x+\log (x))^2}-\frac {8}{(2+x) (x+\log (x))^2}\right ) \, dx-\int \left (\frac {1}{x^2 \log (x)}-\frac {1}{x (x+\log (x))^2}-\frac {1}{x^2 (x+\log (x))}\right ) \, dx-\int \left (1+\frac {x^2}{(x+\log (x))^2}-\frac {2 x}{x+\log (x)}\right ) \, dx+\int \left (\frac {\log \left (2 (2+x)^2 \log (x)\right )}{(x+\log (x))^2}+\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2}\right ) \, dx\\ &=-x-2 \int \frac {1}{(x+\log (x))^2} \, dx+2 \int \frac {x (1+x)^2}{(2+x) (x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \int \frac {(1+x)^2}{(2+x) (x+\log (x))} \, dx+4 \int \frac {1}{(2+x) (x+\log (x))^2} \, dx-\int \frac {1}{x^2 \log (x)} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {1}{x^2 (x+\log (x))} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2} \, dx\\ &=-x-2 \int \frac {1}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx+2 \int \left (\frac {1}{(x+\log (x))^2}+\frac {x^2}{(x+\log (x))^2}-\frac {2}{(2+x) (x+\log (x))^2}\right ) \, dx-2 \int \left (\frac {x}{x+\log (x)}+\frac {1}{(2+x) (x+\log (x))}\right ) \, dx+4 \int \frac {1}{(2+x) (x+\log (x))^2} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {1}{x^2 (x+\log (x))} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2} \, dx-\text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=-x-\text {Ei}(-\log (x))+2 \int \frac {x^2}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(2+x) (x+\log (x))} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {1}{x^2 (x+\log (x))} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 (2+x)^2 \log (x)\right )}{x (x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 22, normalized size = 0.96 \begin {gather*} -x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \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 = 2.36, size = 197, normalized size = 8.57
method | result | size |
risch | \(-\frac {2 \ln \left (2+x \right )}{x +\ln \left (x \right )}-\frac {i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )+i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )-i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{3}-i \pi \mathrm {csgn}\left (i \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (2+x \right )\right ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{3}+2 x^{2}+2 x \ln \left (x \right )+2 \ln \left (2\right )+2 \ln \left (\ln \left (x \right )\right )}{2 \left (x +\ln \left (x \right )\right )}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 27, normalized size = 1.17 \begin {gather*} -\frac {x^{2} + x \log \left (x\right ) + \log \left (2\right ) + 2 \, \log \left (x + 2\right ) + \log \left (\log \left (x\right )\right )}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 29, normalized size = 1.26 \begin {gather*} -\frac {x^{2} + x \log \left (x\right ) + \log \left (2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )\right )}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.10, size = 26, normalized size = 1.13 \begin {gather*} -x-\frac {\ln \left (\ln \left (x\right )\,\left (2\,x^2+8\,x+8\right )\right )}{x+\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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