3.36.0 ∫ −2+15x−8x2+(1−6x+3x2) log(x) ______________________________________________________________________ (−3x+9x2−3x3+(x−3x2+x3) log(x)) log(−54x+162x2−54x3+(18x−54x2+18x3) log(4)+(18x−54x2+18x3+(−6x+18x2−6x3) log(4)) log(x)) dx [3540]

Integrand size = 121, antiderivative size = 22 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log \left (\log \left (6 x \left (-(-1+x)^2+x\right ) (-3+\log (4)) (-3+\log (x))\right )\right ) \]

Rubi [F]

\[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx \]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2-15 x+8 x^2-\left (1-6 x+3 x^2\right ) \log (x)}{x \left (1-3 x+x^2\right ) (3-\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx \\ & = \int \left (\frac {-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {(-3+x) \left (-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)\right )}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx \\ & = \int \frac {-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-\int \frac {(-3+x) \left (-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)\right )}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx \\ & = \int \left (\frac {15}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {2}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {8 x}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {6 \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {\log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {3 x \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx-\int \left (\frac {6}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {47 x}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {39 x^2}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {8 x^3}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {19 x \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {15 x^2 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {3 x^3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx\right )+3 \int \frac {x \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+3 \int \frac {\log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-3 \int \frac {x^3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-6 \int \frac {1}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-6 \int \frac {\log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-8 \int \frac {x}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+8 \int \frac {x^3}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+15 \int \frac {1}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+15 \int \frac {x^2 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-19 \int \frac {x \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-39 \int \frac {x^2}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+47 \int \frac {x}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+\int \frac {\log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log \left (\log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )\right ) \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(24)=48\).

Time = 2.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.18


method	result	size

parallelrisch	\(\ln \left (\ln \left (\left (2 \left (-6 x^{3}+18 x^{2}-6 x \right ) \ln \left (2\right )+18 x^{3}-54 x^{2}+18 x \right ) \ln \left (x \right )+2 \left (18 x^{3}-54 x^{2}+18 x \right ) \ln \left (2\right )-54 x^{3}+162 x^{2}-54 x \right )\right )\)	\(70\)
default	\(\ln \left (\ln \left (6\right )+\ln \left (-2 \ln \left (2\right ) \ln \left (x \right ) x^{3}+6 x^{2} \ln \left (2\right ) \ln \left (x \right )+3 x^{3} \ln \left (x \right )+6 x^{3} \ln \left (2\right )-2 x \ln \left (2\right ) \ln \left (x \right )-9 x^{2} \ln \left (x \right )-18 x^{2} \ln \left (2\right )-9 x^{3}+3 x \ln \left (x \right )+6 x \ln \left (2\right )+27 x^{2}-9 x \right )\right )\)	\(83\)
risch	\(\ln \left (\ln \left (x^{2}-3 x +1\right )+\frac {i \left (-2 \pi {\operatorname {csgn}\left (i x \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )+\pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right )\right ) \operatorname {csgn}\left (i \left (x^{2}-3 x +1\right )\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )+\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right )\right ) {\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{2}-3 x +1\right )\right ) {\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{3}+\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right ) {\operatorname {csgn}\left (i x \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{2}+\pi {\operatorname {csgn}\left (i x \left (\ln \left (x \right )-3\right ) \left (x^{2}-3 x +1\right )\right )}^{3}-2 i \ln \left (2\right )-2 i \ln \left (\ln \left (x \right )-3\right )-2 i \ln \left (x \right )-2 i \ln \left (3\right )+2 \pi \right )}{2}\right )\)	\(309\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (24) = 48\).

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log \left (\log \left (-54 \, x^{3} + 162 \, x^{2} + 36 \, {\left (x^{3} - 3 \, x^{2} + x\right )} \log \left (2\right ) + 6 \, {\left (3 \, x^{3} - 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2} + x\right )} \log \left (2\right ) + 3 \, x\right )} \log \left (x\right ) - 54 \, x\right )\right ) \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).

Time = 0.57 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.00 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log {\left (\log {\left (- 54 x^{3} + 162 x^{2} - 54 x + \left (36 x^{3} - 108 x^{2} + 36 x\right ) \log {\left (2 \right )} + \left (18 x^{3} - 54 x^{2} + 18 x + \left (- 12 x^{3} + 36 x^{2} - 12 x\right ) \log {\left (2 \right )}\right ) \log {\left (x \right )} \right )} \right )} \]

Maxima [C] (veriﬁcation not implemented)

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log \left (i \, \pi + \log \left (3\right ) + \log \left (2\right ) + \log \left (x^{2} - 3 \, x + 1\right ) + \log \left (x\right ) + \log \left (2 \, \log \left (2\right ) - 3\right ) + \log \left (\log \left (x\right ) - 3\right )\right ) \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).

Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\log \left (\log \left (2\right ) + \log \left (-6 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 18 \, x^{2} \log \left (2\right ) + 9 \, x^{2} \log \left (x\right ) + 18 \, x \log \left (2\right ) \log \left (x\right ) - 27 \, x^{2} - 54 \, x \log \left (2\right ) - 27 \, x \log \left (x\right ) - 6 \, \log \left (2\right ) \log \left (x\right ) + 81 \, x + 18 \, \log \left (2\right ) + 9 \, \log \left (x\right ) - 27\right ) + \log \left (x\right )\right ) \]

Mupad [B] (veriﬁcation not implemented)

Time = 10.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {-2+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx=\ln \left (\ln \left (\ln \left (x\right )\,\left (18\,x-2\,\ln \left (2\right )\,\left (6\,x^3-18\,x^2+6\,x\right )-54\,x^2+18\,x^3\right )-54\,x+2\,\ln \left (2\right )\,\left (18\,x^3-54\,x^2+18\,x\right )+162\,x^2-54\,x^3\right )\right ) \]

\(\int \frac {-2+15 x-8 x^2+(1-6 x+3 x^2) \log (x)}{(-3 x+9 x^2-3 x^3+(x-3 x^2+x^3) \log (x)) \log (-54 x+162 x^2-54 x^3+(18 x-54 x^2+18 x^3) \log (4)+(18 x-54 x^2+18 x^3+(-6 x+18 x^2-6 x^3) \log (4)) \log (x))} \, dx\) [3540]

Optimal result