3.63.0 ∫ −2x−x3+(2x+3x2) log(x)−3x log2(x)+log3(x) _______________________________________________ (−x3−x4+(x2+3x3) log(x)−3x2 log2(x)+x log3(x)) log(x2+x3−2x2 log(x)+x log2(x) x2−2x log(x)+log2(x) ) log(log(x2+x3−2x2 log(x)+x log2(x) x2−2x log(x)+log2(x) )) dx [6234]

Integrand size = 152, antiderivative size = 17 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (x+\frac {x^2}{(-x+\log (x))^2}\right )\right )\right ) \]

Rubi [F]

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

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

Mathematica [A] (veriﬁed)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (\frac {x \left (x+x^2-2 x \log (x)+\log ^2(x)\right )}{(x-\log (x))^2}\right )\right )\right ) \]

Maple [A] (veriﬁed)

Time = 75.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06


method	result	size

parallelrisch	\(\ln \left (\ln \left (\ln \left (\frac {x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}\right )\right )\right )\)	\(35\)
default	\(\ln \left (\ln \left (\ln \left (x \right )-2 \ln \left (\ln \left (x \right )-x \right )+\ln \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right )+\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+x \right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right )}{2}\right )\right )\)	\(313\)
risch	\(\ln \left (\ln \left (\ln \left (x \right )-2 \ln \left (x -\ln \left (x \right )\right )+\ln \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )^{2}\right )-\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (-2 \ln \left (x \right )+1\right ) x +\ln \left (x \right )^{2}\right )}{\left (\ln \left (x \right )-x \right )^{2}}\right )\right )}{2}\right )\right )\)	\(333\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (\frac {x^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )\right )\right ) \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).

Time = 2.73 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log {\left (\log {\left (\log {\left (\frac {x^{3} - 2 x^{2} \log {\left (x \right )} + x^{2} + x \log {\left (x \right )}^{2}}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \right )} \right )} \right )} \]

Maxima [A] (veriﬁcation not implemented)

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + x\right ) + \log \left (x\right ) - 2 \, \log \left (-x + \log \left (x\right )\right )\right )\right ) \]

Giac [F]

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

Mupad [B] (veriﬁcation not implemented)

Time = 13.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {-2 x-x^3+\left (2 x+3 x^2\right ) \log (x)-3 x \log ^2(x)+\log ^3(x)}{\left (-x^3-x^4+\left (x^2+3 x^3\right ) \log (x)-3 x^2 \log ^2(x)+x \log ^3(x)\right ) \log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \log \left (\log \left (\frac {x^2+x^3-2 x^2 \log (x)+x \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (x+\frac {x^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}\right )\right )\right ) \]

Optimal result