\(\int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+(-9+45 x^2+18 x^3+x^4) \log (x)+(-x+9 x^3+6 x^4+x^5+(-1+9 x^2+6 x^3+x^4) \log (x)) \log (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8})}{-x+9 x^3+6 x^4+x^5+(-1+9 x^2+6 x^3+x^4) \log (x)} \, dx\) [7911]

Optimal result

Integrand size = 175, antiderivative size = 25 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x \left (5+\log \left (\frac {(x+\log (x))^2}{\left (-\frac {1}{x^2}+(3+x)^2\right )^2}\right )\right ) \]

[Out]

(ln((x+ln(x))^2/(-1/x^2+(3+x)^2)^2)+5)*x

Rubi [F]

\[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=\int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx \]

[In]

Int[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 + 6*x^4
+ x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3 + 79*x
^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x]

[Out]

5*x + x*Log[(x^4*(x + Log[x])^2)/(1 - 9*x^2 - 6*x^3 - x^4)^2] + 2*Defer[Int][(-x - Log[x])^(-1), x] + 2*Defer[
Int][(x + Log[x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2+11 x-18 x^2-75 x^3-32 x^4-3 x^5-\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)-\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x)) \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right )}{\left (1-9 x^2-6 x^3-x^4\right ) (x+\log (x))} \, dx \\ & = \int \left (-\frac {2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}-\frac {11 x}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {18 x^2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {75 x^3}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {32 x^4}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {3 x^5}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right )\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx\right )+3 \int \frac {x^5}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx-11 \int \frac {x}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+18 \int \frac {x^2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+32 \int \frac {x^4}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+75 \int \frac {x^3}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right ) \, dx \\ & = x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )-2 \int \left (\frac {1}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {1}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+3 \int \left (-\frac {6}{x+\log (x)}+\frac {x}{x+\log (x)}+\frac {-33+109 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {-21-55 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx-11 \int \left (\frac {x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+18 \int \left (\frac {1-3 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {1+3 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+32 \int \left (\frac {1}{x+\log (x)}+\frac {10-33 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {8+21 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+75 \int \left (\frac {-3+10 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {-3-8 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx-\int -\frac {2 \left (1+3 x-9 x^2-15 x^3-x^4+x^5+2 \left (1+3 x^3+x^4\right ) \log (x)\right )}{\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x))} \, dx+\int \left (\frac {-9+45 x^2+18 x^3+x^4}{-1+9 x^2+6 x^3+x^4}-\frac {x \left (-9+45 x^2+18 x^3+x^4\right )}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx \\ & = x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )+\frac {3}{2} \int \frac {-33+109 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {3}{2} \int \frac {-21-55 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+2 \int \frac {1+3 x-9 x^2-15 x^3-x^4+x^5+2 \left (1+3 x^3+x^4\right ) \log (x)}{\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x))} \, dx+3 \int \frac {x}{x+\log (x)} \, dx-\frac {11}{2} \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {11}{2} \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+9 \int \frac {1-3 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+9 \int \frac {1+3 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+16 \int \frac {10-33 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+16 \int \frac {8+21 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-18 \int \frac {1}{x+\log (x)} \, dx+32 \int \frac {1}{x+\log (x)} \, dx+\frac {75}{2} \int \frac {-3+10 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {75}{2} \int \frac {-3-8 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {-9+45 x^2+18 x^3+x^4}{-1+9 x^2+6 x^3+x^4} \, dx-\int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-\int \frac {x \left (-9+45 x^2+18 x^3+x^4\right )}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx \\ & = x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )+\frac {3}{2} \int \left (-\frac {33}{\left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {109 x}{\left (-1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+\frac {3}{2} \int \left (-\frac {21}{\left (1+3 x+x^2\right ) (x+\log (x))}-\frac {55 x}{\left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+2 \int \left (\frac {2 \left (1+3 x^3+x^4\right )}{-1+9 x^2+6 x^3+x^4}+\frac {-1-x}{x+\log (x)}\right ) \, dx+3 \int \frac {x}{x+\log (x)} \, dx+\frac {11}{2} \int \left (\frac {1-\frac {3}{\sqrt {5}}}{\left (3-\sqrt {5}+2 x\right ) (x+\log (x))}+\frac {1+\frac {3}{\sqrt {5}}}{\left (3+\sqrt {5}+2 x\right ) (x+\log (x))}\right ) \, dx-\frac {11}{2} \int \left (\frac {1-\frac {3}{\sqrt {13}}}{\left (3-\sqrt {13}+2 x\right ) (x+\log (x))}+\frac {1+\frac {3}{\sqrt {13}}}{\left (3+\sqrt {13}+2 x\right ) (x+\log (x))}\right ) \, dx+9 \int \left (\frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {3 x}{\left (-1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+9 \int \left (\frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))}+\frac {3 x}{\left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+16 \int \left (\frac {10}{\left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {33 x}{\left (-1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+16 \int \left (\frac {8}{\left (1+3 x+x^2\right ) (x+\log (x))}+\frac {21 x}{\left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx-18 \int \frac {1}{x+\log (x)} \, dx+32 \int \frac {1}{x+\log (x)} \, dx+\frac {75}{2} \int \left (-\frac {3}{\left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {10 x}{\left (-1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+\frac {75}{2} \int \left (-\frac {3}{\left (1+3 x+x^2\right ) (x+\log (x))}-\frac {8 x}{\left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+\int \left (-\frac {2}{\sqrt {5} \left (-3+\sqrt {5}-2 x\right ) (x+\log (x))}-\frac {2}{\sqrt {5} \left (3+\sqrt {5}+2 x\right ) (x+\log (x))}\right ) \, dx-\int \left (-\frac {2}{\sqrt {13} \left (-3+\sqrt {13}-2 x\right ) (x+\log (x))}-\frac {2}{\sqrt {13} \left (3+\sqrt {13}+2 x\right ) (x+\log (x))}\right ) \, dx-\int \left (\frac {12}{x+\log (x)}+\frac {x}{x+\log (x)}-\frac {2 (-3+11 x)}{\left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {2 (3+7 x)}{\left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+\text {Subst}\left (\int \frac {585-432 x-360 x^2+192 x^3+16 x^4}{65-72 x^2+16 x^4} \, dx,x,\frac {3}{2}+x\right ) \\ & = x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )+2 \int \frac {-1-x}{x+\log (x)} \, dx+2 \int \frac {-3+11 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+2 \int \frac {3+7 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+3 \int \frac {x}{x+\log (x)} \, dx+4 \int \frac {1+3 x^3+x^4}{-1+9 x^2+6 x^3+x^4} \, dx+9 \int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+9 \int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-12 \int \frac {1}{x+\log (x)} \, dx-18 \int \frac {1}{x+\log (x)} \, dx-27 \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+27 \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-\frac {63}{2} \int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+32 \int \frac {1}{x+\log (x)} \, dx-\frac {99}{2} \int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx-\frac {165}{2} \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-\frac {225}{2} \int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx-\frac {225}{2} \int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+128 \int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+160 \int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {327}{2} \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx-300 \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+336 \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+375 \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx-528 \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx-\frac {2 \int \frac {1}{\left (-3+\sqrt {5}-2 x\right ) (x+\log (x))} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\left (3+\sqrt {5}+2 x\right ) (x+\log (x))} \, dx}{\sqrt {5}}+\frac {2 \int \frac {1}{\left (-3+\sqrt {13}-2 x\right ) (x+\log (x))} \, dx}{\sqrt {13}}+\frac {2 \int \frac {1}{\left (3+\sqrt {13}+2 x\right ) (x+\log (x))} \, dx}{\sqrt {13}}+\frac {1}{10} \left (11 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3-\sqrt {5}+2 x\right ) (x+\log (x))} \, dx+\frac {1}{10} \left (11 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x\right ) (x+\log (x))} \, dx-\frac {1}{26} \left (11 \left (13-3 \sqrt {13}\right )\right ) \int \frac {1}{\left (3-\sqrt {13}+2 x\right ) (x+\log (x))} \, dx-\frac {1}{26} \left (11 \left (13+3 \sqrt {13}\right )\right ) \int \frac {1}{\left (3+\sqrt {13}+2 x\right ) (x+\log (x))} \, dx-\int \frac {x}{x+\log (x)} \, dx+\text {Subst}\left (\int \frac {x \left (-432+192 x^2\right )}{65-72 x^2+16 x^4} \, dx,x,\frac {3}{2}+x\right )+\text {Subst}\left (\int \frac {585-360 x^2+16 x^4}{65-72 x^2+16 x^4} \, dx,x,\frac {3}{2}+x\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=5 x+x \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right ) \]

[In]

Integrate[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 +
6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3
+ 79*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x
]

[Out]

5*x + x*Log[(x^4*(x + Log[x])^2)/(-1 + 9*x^2 + 6*x^3 + x^4)^2]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).

Time = 9.67 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64

method	result	size
parallelrisch	\(-60+\ln \left (\frac {x^{4} \ln \left (x \right )^{2}+2 x^{5} \ln \left (x \right )+x^{6}}{x^{8}+12 x^{7}+54 x^{6}+108 x^{5}+79 x^{4}-12 x^{3}-18 x^{2}+1}\right ) x +5 x\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(844\)

[In]

int((((x^4+6*x^3+9*x^2-1)*ln(x)+x^5+6*x^4+9*x^3-x)*ln((x^4*ln(x)^2+2*x^5*ln(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5
+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*ln(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+9*x^2-1)*
ln(x)+x^5+6*x^4+9*x^3-x),x,method=_RETURNVERBOSE)

[Out]

-60+ln((x^4*ln(x)^2+2*x^5*ln(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12*x^3-18*x^2+1))*x+5*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x \log \left (\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2}}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1}\right ) + 5 \, x \]

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="fricas")

[Out]

x*log((x^6 + 2*x^5*log(x) + x^4*log(x)^2)/(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1)) +
5*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (22) = 44\).

Time = 0.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.08 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=5 x + \left (x + \frac {1}{5}\right ) \log {\left (\frac {x^{6} + 2 x^{5} \log {\left (x \right )} + x^{4} \log {\left (x \right )}^{2}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 79 x^{4} - 12 x^{3} - 18 x^{2} + 1} \right )} - \frac {4 \log {\left (x \right )}}{5} - \frac {2 \log {\left (x + \log {\left (x \right )} \right )}}{5} + \frac {2 \log {\left (x^{4} + 6 x^{3} + 9 x^{2} - 1 \right )}}{5} \]

[In]

integrate((((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x)*ln((x**4*ln(x)**2+2*x**5*ln(x)+x**6)/(x**8+12*x
**7+54*x**6+108*x**5+79*x**4-12*x**3-18*x**2+1))+(x**4+18*x**3+45*x**2-9)*ln(x)+3*x**5+32*x**4+75*x**3+18*x**2
-11*x-2)/((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x),x)

[Out]

5*x + (x + 1/5)*log((x**6 + 2*x**5*log(x) + x**4*log(x)**2)/(x**8 + 12*x**7 + 54*x**6 + 108*x**5 + 79*x**4 - 1
2*x**3 - 18*x**2 + 1)) - 4*log(x)/5 - 2*log(x + log(x))/5 + 2*log(x**4 + 6*x**3 + 9*x**2 - 1)/5

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=-2 \, x \log \left (x^{2} + 3 \, x + 1\right ) - 2 \, x \log \left (x^{2} + 3 \, x - 1\right ) + 2 \, x \log \left (x + \log \left (x\right )\right ) + 4 \, x \log \left (x\right ) + 5 \, x \]

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="maxima")

[Out]

-2*x*log(x^2 + 3*x + 1) - 2*x*log(x^2 + 3*x - 1) + 2*x*log(x + log(x)) + 4*x*log(x) + 5*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.93 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=-x \log \left (x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1\right ) + x \log \left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 4 \, x \log \left (x\right ) + 5 \, x \]

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="giac")

[Out]

-x*log(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1) + x*log(x^2 + 2*x*log(x) + log(x)^2) +
4*x*log(x) + 5*x

Mupad [B] (verification not implemented)

Time = 12.95 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x\,\left (\ln \left (\frac {x^6+2\,x^5\,\ln \left (x\right )+x^4\,{\ln \left (x\right )}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+79\,x^4-12\,x^3-18\,x^2+1}\right )+5\right ) \]

[In]

int((log((2*x^5*log(x) + x^4*log(x)^2 + x^6)/(79*x^4 - 12*x^3 - 18*x^2 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + 1))
*(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9*x^2 + 6*x^3 + x^4 - 1)) - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + log(
x)*(45*x^2 + 18*x^3 + x^4 - 9) - 2)/(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9*x^2 + 6*x^3 + x^4 - 1)),x)

[Out]

x*(log((2*x^5*log(x) + x^4*log(x)^2 + x^6)/(79*x^4 - 12*x^3 - 18*x^2 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + 1)) +
5)