3.41.0 ∫ (−2x−x2+2x3+ex(−1−x+x2)) log(x)+(−exx−x2) log(x) log(exx+x2)+(−2exx−2x2+(2ex+2x) log(exx+x2)) log(log(x))+(ex+2x−x2) log(x) log2(log(x)) (2exx+2x2) log(x) dx [4076]

Integrand size = 130, antiderivative size = 26 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {1}{2} \left (-x+\log \left (x \left (e^x+x\right )\right )\right ) \left (-1-x+\log ^2(\log (x))\right ) \]

Rubi [F]

\[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx \]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{2 x \left (e^x+x\right ) \log (x)} \, dx \\ & = \frac {1}{2} \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{x \left (e^x+x\right ) \log (x)} \, dx \\ & = \frac {1}{2} \int \frac {-\frac {2 \left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{\log (x)}-\frac {e^x+2 x+e^x x+x^2-e^x x^2-2 x^3+x \left (e^x+x\right ) \log \left (x \left (e^x+x\right )\right )+\left (-e^x+(-2+x) x\right ) \log ^2(\log (x))}{e^x+x}}{x} \, dx \\ & = \frac {1}{2} \int \left (\frac {(-1+x) \left (1+x-\log ^2(\log (x))\right )}{e^x+x}+\frac {-\log (x)-x \log (x)+x^2 \log (x)-x \log (x) \log \left (x \left (e^x+x\right )\right )-2 x \log (\log (x))+2 \log \left (x \left (e^x+x\right )\right ) \log (\log (x))+\log (x) \log ^2(\log (x))}{x \log (x)}\right ) \, dx \\ & = \frac {1}{2} \int \frac {(-1+x) \left (1+x-\log ^2(\log (x))\right )}{e^x+x} \, dx+\frac {1}{2} \int \frac {-\log (x)-x \log (x)+x^2 \log (x)-x \log (x) \log \left (x \left (e^x+x\right )\right )-2 x \log (\log (x))+2 \log \left (x \left (e^x+x\right )\right ) \log (\log (x))+\log (x) \log ^2(\log (x))}{x \log (x)} \, dx \\ & = \frac {1}{2} \int \left (\frac {-1-x+x^2-x \log \left (x \left (e^x+x\right )\right )}{x}-\frac {2 \left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{x \log (x)}+\frac {\log ^2(\log (x))}{x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {1+x-\log ^2(\log (x))}{e^x+x}+\frac {x \left (1+x-\log ^2(\log (x))\right )}{e^x+x}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-1-x+x^2-x \log \left (x \left (e^x+x\right )\right )}{x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{x} \, dx-\frac {1}{2} \int \frac {1+x-\log ^2(\log (x))}{e^x+x} \, dx+\frac {1}{2} \int \frac {x \left (1+x-\log ^2(\log (x))\right )}{e^x+x} \, dx-\int \frac {\left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log (\log (x))}{x \log (x)} \, dx \\ & = \frac {1}{2} \int \left (\frac {-1-x+x^2}{x}-\log \left (e^x x+x^2\right )\right ) \, dx-\frac {1}{2} \int \left (\frac {1}{e^x+x}+\frac {x}{e^x+x}-\frac {\log ^2(\log (x))}{e^x+x}\right ) \, dx+\frac {1}{2} \int \left (\frac {x}{e^x+x}+\frac {x^2}{e^x+x}-\frac {x \log ^2(\log (x))}{e^x+x}\right ) \, dx+\frac {1}{2} \text {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )-\int \left (\frac {\log (\log (x))}{\log (x)}-\frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)}\right ) \, dx \\ & = \frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \frac {-1-x+x^2}{x} \, dx-\frac {1}{2} \int \log \left (e^x x+x^2\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx-\text {Subst}(\int \log (x) \, dx,x,\log (x)) \\ & = \log (x)-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \left (-1-\frac {1}{x}+x\right ) \, dx+\frac {1}{2} \int \frac {2 x+e^x (1+x)}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx \\ & = -\frac {x}{2}+\frac {x^2}{4}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \left (1+x-\frac {(-1+x) x}{e^x+x}\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx \\ & = \frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx-\frac {1}{2} \int \frac {(-1+x) x}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx \\ & = \frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x^2}{e^x+x} \, dx-\frac {1}{2} \int \left (-\frac {x}{e^x+x}+\frac {x^2}{e^x+x}\right ) \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx \\ & = \frac {x^2}{2}+\frac {\log (x)}{2}-\frac {1}{2} x \log \left (e^x x+x^2\right )-\log (x) \log (\log (x))+\frac {1}{2} \log (x) \log ^2(\log (x))-\frac {1}{2} \int \frac {1}{e^x+x} \, dx+\frac {1}{2} \int \frac {x}{e^x+x} \, dx+\frac {1}{2} \int \frac {\log ^2(\log (x))}{e^x+x} \, dx-\frac {1}{2} \int \frac {x \log ^2(\log (x))}{e^x+x} \, dx-\int \frac {\log (\log (x))}{\log (x)} \, dx+\int \frac {\log \left (e^x x+x^2\right ) \log (\log (x))}{x \log (x)} \, dx \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {1}{2} \left (x+x^2-\log (x)-\log \left (e^x+x\right )-x \log \left (x \left (e^x+x\right )\right )-\left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log ^2(\log (x))\right ) \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).

Time = 14.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96


method	result	size

parallelrisch	\(-\frac {x \ln \left (\ln \left (x \right )\right )^{2}}{2}+\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right ) \ln \left (\ln \left (x \right )\right )^{2}}{2}+\frac {x^{2}}{2}-\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right ) x}{2}+\frac {x}{2}-\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right )}{2}\)	\(51\)
risch	\(\left (\frac {\ln \left (\ln \left (x \right )\right )^{2}}{2}-\frac {x}{2}\right ) \ln \left ({\mathrm e}^{x}+x \right )-\frac {x \ln \left (\ln \left (x \right )\right )^{2}}{2}+\frac {\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}}{2}+\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}-\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )}{4}+\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )}{4}-\frac {i x \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )}{4}-\frac {x \ln \left (x \right )}{2}-\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{3}}{4}-\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}+\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )}{4}+\frac {i x \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{3}}{4}+\frac {x^{2}}{2}-\frac {\ln \left (x \right )}{2}+\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+x \right )}{2}\)	\(252\)

Fricas [A] (veriﬁcation not implemented)

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, {\left (x - \log \left (x^{2} + x e^{x}\right )\right )} \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, {\left (x + 1\right )} \log \left (x^{2} + x e^{x}\right ) + \frac {1}{2} \, x \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.86 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {x^{2}}{2} - \frac {x \log {\left (\log {\left (x \right )} \right )}^{2}}{2} + \frac {x}{2} + \left (- \frac {x}{2} + \frac {\log {\left (\log {\left (x \right )} \right )}^{2}}{2}\right ) \log {\left (x^{2} + x e^{x} \right )} - \frac {\log {\left (x \right )}}{2} - \frac {\log {\left (x + e^{x} \right )}}{2} \]

Maxima [A] (veriﬁcation not implemented)

Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, {\left (x - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (\log \left (\log \left (x\right )\right )^{2} - x - 1\right )} \log \left (x + e^{x}\right ) - \frac {1}{2} \, {\left (x + 1\right )} \log \left (x\right ) + \frac {1}{2} \, x \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).

Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, x \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, \log \left (x + e^{x}\right ) \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x \log \left (x + e^{x}\right ) - \frac {1}{2} \, x \log \left (x\right ) + \frac {1}{2} \, x - \frac {1}{2} \, \log \left (x + e^{x}\right ) - \frac {1}{2} \, \log \left (x\right ) \]

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\int -\frac {-\ln \left (x\right )\,\left (2\,x+{\mathrm {e}}^x-x^2\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\left (2\,x\,{\mathrm {e}}^x-\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\left (2\,x+2\,{\mathrm {e}}^x\right )+2\,x^2\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (-x^2+x+1\right )+x^2-2\,x^3\right )+\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\ln \left (x\right )\,\left (x\,{\mathrm {e}}^x+x^2\right )}{\ln \left (x\right )\,\left (2\,x\,{\mathrm {e}}^x+2\,x^2\right )} \,d x \]

\(\int \frac {(-2 x-x^2+2 x^3+e^x (-1-x+x^2)) \log (x)+(-e^x x-x^2) \log (x) \log (e^x x+x^2)+(-2 e^x x-2 x^2+(2 e^x+2 x) \log (e^x x+x^2)) \log (\log (x))+(e^x+2 x-x^2) \log (x) \log ^2(\log (x))}{(2 e^x x+2 x^2) \log (x)} \, dx\) [4076]

Optimal result