3.31.0 ∫ −2x−x2+(−2−x−2x2−2x3−x4) log(x)+(−2x−4x2−2x3) log2(x)+(−2x−x2) log3(x)+(2+3x+x2) log(x) log((8+8x+2x2) log(x)) (2x3+x4) log(x)+(4x2+2x3) log2(x)+(2x+x2) log3(x) dx [3063]

Integrand size = 134, antiderivative size = 23 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=13-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]

Rubi [F]

\[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=\int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx \]

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (veriﬁed)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]

Maple [A] (veriﬁed)

Time = 2.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57


method	result	size

parallelrisch	\(\frac {-4 x^{2}-4 x \ln \left (x \right )-4 \ln \left (\left (2 x^{2}+8 x +8\right ) \ln \left (x \right )\right )}{4 x +4 \ln \left (x \right )}\)	\(36\)
risch	\(-\frac {2 \ln \left (2+x \right )}{x +\ln \left (x \right )}-\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )+i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )-i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (2+x \right )^{2}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )-i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{3}-i \pi \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{3}+2 x^{2}+2 x \ln \left (x \right )+2 \ln \left (2\right )+2 \ln \left (\ln \left (x \right )\right )}{2 \left (x +\ln \left (x \right )\right )}\)	\(197\)

Fricas [A] (veriﬁcation not implemented)

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-\frac {x^{2} + x \log \left (x\right ) + \log \left (2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )\right )}{x + \log \left (x\right )} \]

Sympy [F(-2)]

Exception generated. \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]

Maxima [A] (veriﬁcation not implemented)

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-\frac {x^{2} + x \log \left (x\right ) + \log \left (2\right ) + 2 \, \log \left (x + 2\right ) + \log \left (\log \left (x\right )\right )}{x + \log \left (x\right )} \]

Giac [A] (veriﬁcation not implemented)

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x - \frac {\log \left (2 \, x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 8 \, \log \left (x\right )\right )}{x + \log \left (x\right )} \]

Mupad [B] (veriﬁcation not implemented)

Time = 9.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x-\frac {\ln \left (\ln \left (x\right )\,\left (2\,x^2+8\,x+8\right )\right )}{x+\ln \left (x\right )} \]

\(\int \frac {-2 x-x^2+(-2-x-2 x^2-2 x^3-x^4) \log (x)+(-2 x-4 x^2-2 x^3) \log ^2(x)+(-2 x-x^2) \log ^3(x)+(2+3 x+x^2) \log (x) \log ((8+8 x+2 x^2) \log (x))}{(2 x^3+x^4) \log (x)+(4 x^2+2 x^3) \log ^2(x)+(2 x+x^2) \log ^3(x)} \, dx\) [3063]

Optimal result