\(\int \frac {(-5 x-4 x^2) \log (x)+(-16-20 x-8 x^2) \log (x) \log (x^2)+(4+5 x+2 x^2) \log ^2(x^2)+(-4 x-5 x^2-2 x^3) \log ^4(x^2)+(4+5 x+2 x^2+(-8 x-10 x^2-4 x^3) \log ^2(x^2)) \log (4+5 x+2 x^2)+(-4 x-5 x^2-2 x^3) \log ^2(4+5 x+2 x^2)}{(4 x+5 x^2+2 x^3) \log ^4(x^2)+(8 x+10 x^2+4 x^3) \log ^2(x^2) \log (4+5 x+2 x^2)+(4 x+5 x^2+2 x^3) \log ^2(4+5 x+2 x^2)} \, dx\) [8020]

Optimal result

Integrand size = 224, antiderivative size = 27 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=-x+\frac {\log (x)}{\log ^2\left (x^2\right )+\log (4+x+x (4+2 x))} \]

[Out]

ln(x)/(ln(x^2)^2+ln(x+4+x*(4+2*x)))-x

Rubi [F]

\[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=\int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx \]

[In]

Int[((-5*x - 4*x^2)*Log[x] + (-16 - 20*x - 8*x^2)*Log[x]*Log[x^2] + (4 + 5*x + 2*x^2)*Log[x^2]^2 + (-4*x - 5*x
^2 - 2*x^3)*Log[x^2]^4 + (4 + 5*x + 2*x^2 + (-8*x - 10*x^2 - 4*x^3)*Log[x^2]^2)*Log[4 + 5*x + 2*x^2] + (-4*x -
5*x^2 - 2*x^3)*Log[4 + 5*x + 2*x^2]^2)/((4*x + 5*x^2 + 2*x^3)*Log[x^2]^4 + (8*x + 10*x^2 + 4*x^3)*Log[x^2]^2*
Log[4 + 5*x + 2*x^2] + (4*x + 5*x^2 + 2*x^3)*Log[4 + 5*x + 2*x^2]^2),x]

[Out]

-x - ((20*I)*Defer[Int][Log[x]/((-5 + I*Sqrt[7] - 4*x)*(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])^2), x])/Sqrt[7] - (
4*(7 + (5*I)*Sqrt[7])*Defer[Int][Log[x]/((5 - I*Sqrt[7] + 4*x)*(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])^2), x])/7 -
((20*I)*Defer[Int][Log[x]/((5 + I*Sqrt[7] + 4*x)*(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])^2), x])/Sqrt[7] - (4*(7
- (5*I)*Sqrt[7])*Defer[Int][Log[x]/((5 + I*Sqrt[7] + 4*x)*(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])^2), x])/7 - 4*De
fer[Int][(Log[x]*Log[x^2])/(x*(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])^2), x] + Defer[Int][1/(x*(Log[x^2]^2 + Log[4
+ 5*x + 2*x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (x) \left (-\frac {x (5+4 x)}{4+5 x+2 x^2}-4 \log \left (x^2\right )\right )-\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right ) \left (-1+x \log ^2\left (x^2\right )+x \log \left (4+5 x+2 x^2\right )\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx \\ & = \int \left (-1-\frac {\log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{x \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )}\right ) \, dx \\ & = -x-\int \frac {\log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{x \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = -x+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx-\int \left (\frac {\log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{4 x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}-\frac {(5+2 x) \log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{4 \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx \\ & = -x-\frac {1}{4} \int \frac {\log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {1}{4} \int \frac {(5+2 x) \log (x) \left (5 x+4 x^2+16 \log \left (x^2\right )+20 x \log \left (x^2\right )+8 x^2 \log \left (x^2\right )\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = -x-\frac {1}{4} \int \left (\frac {5 \log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {4 x \log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {20 \log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {16 \log (x) \log \left (x^2\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {8 x \log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+\frac {1}{4} \int \left (\frac {25 x \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {30 x^2 \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {8 x^3 \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {80 \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {132 x \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {80 x^2 \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {16 x^3 \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = -x-\frac {5}{4} \int \frac {\log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+2 \int \frac {x^3 \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-2 \int \frac {x \log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-4 \int \frac {\log (x) \log \left (x^2\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+4 \int \frac {x^3 \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-5 \int \frac {\log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {25}{4} \int \frac {x \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {15}{2} \int \frac {x^2 \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+20 \int \frac {\log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+20 \int \frac {x^2 \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+33 \int \frac {x \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-\int \frac {x \log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = -x-\frac {5}{4} \int \frac {\log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-2 \int \frac {x \log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+2 \int \left (-\frac {5 \log (x)}{4 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {x \log (x)}{2 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {(20+17 x) \log (x)}{4 \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx-4 \int \frac {\log (x) \log \left (x^2\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+4 \int \left (-\frac {5 \log (x) \log \left (x^2\right )}{4 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {x \log (x) \log \left (x^2\right )}{2 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {(20+17 x) \log (x) \log \left (x^2\right )}{4 \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx-5 \int \frac {\log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {25}{4} \int \left (\frac {\left (1+\frac {5 i}{\sqrt {7}}\right ) \log (x)}{\left (5-i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {\left (1-\frac {5 i}{\sqrt {7}}\right ) \log (x)}{\left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+\frac {15}{2} \int \left (\frac {\log (x)}{2 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}-\frac {(4+5 x) \log (x)}{2 \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+20 \int \left (\frac {4 i \log (x) \log \left (x^2\right )}{\sqrt {7} \left (-5+i \sqrt {7}-4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {4 i \log (x) \log \left (x^2\right )}{\sqrt {7} \left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+20 \int \left (\frac {\log (x) \log \left (x^2\right )}{2 \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}-\frac {(4+5 x) \log (x) \log \left (x^2\right )}{2 \left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx+33 \int \left (\frac {\left (1+\frac {5 i}{\sqrt {7}}\right ) \log (x) \log \left (x^2\right )}{\left (5-i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}+\frac {\left (1-\frac {5 i}{\sqrt {7}}\right ) \log (x) \log \left (x^2\right )}{\left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2}\right ) \, dx-\int \frac {x \log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = -x+\frac {1}{2} \int \frac {(20+17 x) \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-\frac {5}{4} \int \frac {\log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-\frac {5}{2} \int \frac {\log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {15}{4} \int \frac {\log (x)}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-\frac {15}{4} \int \frac {(4+5 x) \log (x)}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-4 \int \frac {\log (x) \log \left (x^2\right )}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-2 \left (5 \int \frac {\log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx\right )+10 \int \frac {\log (x) \log \left (x^2\right )}{\left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx-10 \int \frac {(4+5 x) \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {(80 i) \int \frac {\log (x) \log \left (x^2\right )}{\left (-5+i \sqrt {7}-4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx}{\sqrt {7}}+\frac {(80 i) \int \frac {\log (x) \log \left (x^2\right )}{\left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx}{\sqrt {7}}+\frac {1}{28} \left (25 \left (7-5 i \sqrt {7}\right )\right ) \int \frac {\log (x)}{\left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {1}{7} \left (33 \left (7-5 i \sqrt {7}\right )\right ) \int \frac {\log (x) \log \left (x^2\right )}{\left (5+i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {1}{28} \left (25 \left (7+5 i \sqrt {7}\right )\right ) \int \frac {\log (x)}{\left (5-i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\frac {1}{7} \left (33 \left (7+5 i \sqrt {7}\right )\right ) \int \frac {\log (x) \log \left (x^2\right )}{\left (5-i \sqrt {7}+4 x\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {(20+17 x) \log (x) \log \left (x^2\right )}{\left (4+5 x+2 x^2\right ) \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )^2} \, dx+\int \frac {1}{x \left (\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=-x+\frac {\log (x)}{\log ^2\left (x^2\right )+\log \left (4+5 x+2 x^2\right )} \]

[In]

Integrate[((-5*x - 4*x^2)*Log[x] + (-16 - 20*x - 8*x^2)*Log[x]*Log[x^2] + (4 + 5*x + 2*x^2)*Log[x^2]^2 + (-4*x
- 5*x^2 - 2*x^3)*Log[x^2]^4 + (4 + 5*x + 2*x^2 + (-8*x - 10*x^2 - 4*x^3)*Log[x^2]^2)*Log[4 + 5*x + 2*x^2] + (
-4*x - 5*x^2 - 2*x^3)*Log[4 + 5*x + 2*x^2]^2)/((4*x + 5*x^2 + 2*x^3)*Log[x^2]^4 + (8*x + 10*x^2 + 4*x^3)*Log[x
^2]^2*Log[4 + 5*x + 2*x^2] + (4*x + 5*x^2 + 2*x^3)*Log[4 + 5*x + 2*x^2]^2),x]

[Out]

-x + Log[x]/(Log[x^2]^2 + Log[4 + 5*x + 2*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(27)=54\).

Time = 22.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67

method	result	size
parallelrisch	\(\frac {10086 \ln \left (2 x^{2}+5 x +4\right )-4920 x \ln \left (x^{2}\right )^{2}+10086 \ln \left (x^{2}\right )^{2}+4920 \ln \left (x \right )-4920 x \ln \left (2 x^{2}+5 x +4\right )}{4920 \ln \left (x^{2}\right )^{2}+4920 \ln \left (2 x^{2}+5 x +4\right )}\)	\(72\)
risch	\(-x +\frac {4 \ln \left (x \right )}{-\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-6 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+4 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+16 \ln \left (x \right )^{2}+4 \ln \left (2 x^{2}+5 x +4\right )}\)	\(182\)

[In]

int(((-2*x^3-5*x^2-4*x)*ln(2*x^2+5*x+4)^2+((-4*x^3-10*x^2-8*x)*ln(x^2)^2+2*x^2+5*x+4)*ln(2*x^2+5*x+4)+(-2*x^3-
5*x^2-4*x)*ln(x^2)^4+(2*x^2+5*x+4)*ln(x^2)^2+(-8*x^2-20*x-16)*ln(x)*ln(x^2)+(-4*x^2-5*x)*ln(x))/((2*x^3+5*x^2+
4*x)*ln(2*x^2+5*x+4)^2+(4*x^3+10*x^2+8*x)*ln(x^2)^2*ln(2*x^2+5*x+4)+(2*x^3+5*x^2+4*x)*ln(x^2)^4),x,method=_RET
URNVERBOSE)

[Out]

1/4920*(10086*ln(2*x^2+5*x+4)-4920*x*ln(x^2)^2+10086*ln(x^2)^2+4920*ln(x)-4920*x*ln(2*x^2+5*x+4))/(ln(x^2)^2+l
n(2*x^2+5*x+4))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=-\frac {4 \, x \log \left (x\right )^{2} + x \log \left (2 \, x^{2} + 5 \, x + 4\right ) - \log \left (x\right )}{4 \, \log \left (x\right )^{2} + \log \left (2 \, x^{2} + 5 \, x + 4\right )} \]

[In]

integrate(((-2*x^3-5*x^2-4*x)*log(2*x^2+5*x+4)^2+((-4*x^3-10*x^2-8*x)*log(x^2)^2+2*x^2+5*x+4)*log(2*x^2+5*x+4)
+(-2*x^3-5*x^2-4*x)*log(x^2)^4+(2*x^2+5*x+4)*log(x^2)^2+(-8*x^2-20*x-16)*log(x)*log(x^2)+(-4*x^2-5*x)*log(x))/
((2*x^3+5*x^2+4*x)*log(2*x^2+5*x+4)^2+(4*x^3+10*x^2+8*x)*log(x^2)^2*log(2*x^2+5*x+4)+(2*x^3+5*x^2+4*x)*log(x^2
)^4),x, algorithm="fricas")

[Out]

-(4*x*log(x)^2 + x*log(2*x^2 + 5*x + 4) - log(x))/(4*log(x)^2 + log(2*x^2 + 5*x + 4))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=- x + \frac {\log {\left (x \right )}}{4 \log {\left (x \right )}^{2} + \log {\left (2 x^{2} + 5 x + 4 \right )}} \]

[In]

integrate(((-2*x**3-5*x**2-4*x)*ln(2*x**2+5*x+4)**2+((-4*x**3-10*x**2-8*x)*ln(x**2)**2+2*x**2+5*x+4)*ln(2*x**2
+5*x+4)+(-2*x**3-5*x**2-4*x)*ln(x**2)**4+(2*x**2+5*x+4)*ln(x**2)**2+(-8*x**2-20*x-16)*ln(x)*ln(x**2)+(-4*x**2-
5*x)*ln(x))/((2*x**3+5*x**2+4*x)*ln(2*x**2+5*x+4)**2+(4*x**3+10*x**2+8*x)*ln(x**2)**2*ln(2*x**2+5*x+4)+(2*x**3
+5*x**2+4*x)*ln(x**2)**4),x)

[Out]

-x + log(x)/(4*log(x)**2 + log(2*x**2 + 5*x + 4))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=-\frac {4 \, x \log \left (x\right )^{2} + x \log \left (2 \, x^{2} + 5 \, x + 4\right ) - \log \left (x\right )}{4 \, \log \left (x\right )^{2} + \log \left (2 \, x^{2} + 5 \, x + 4\right )} \]

[In]

integrate(((-2*x^3-5*x^2-4*x)*log(2*x^2+5*x+4)^2+((-4*x^3-10*x^2-8*x)*log(x^2)^2+2*x^2+5*x+4)*log(2*x^2+5*x+4)
+(-2*x^3-5*x^2-4*x)*log(x^2)^4+(2*x^2+5*x+4)*log(x^2)^2+(-8*x^2-20*x-16)*log(x)*log(x^2)+(-4*x^2-5*x)*log(x))/
((2*x^3+5*x^2+4*x)*log(2*x^2+5*x+4)^2+(4*x^3+10*x^2+8*x)*log(x^2)^2*log(2*x^2+5*x+4)+(2*x^3+5*x^2+4*x)*log(x^2
)^4),x, algorithm="maxima")

[Out]

-(4*x*log(x)^2 + x*log(2*x^2 + 5*x + 4) - log(x))/(4*log(x)^2 + log(2*x^2 + 5*x + 4))

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=-x + \frac {\log \left (x\right )}{4 \, \log \left (x\right )^{2} + \log \left (2 \, x^{2} + 5 \, x + 4\right )} \]

[In]

integrate(((-2*x^3-5*x^2-4*x)*log(2*x^2+5*x+4)^2+((-4*x^3-10*x^2-8*x)*log(x^2)^2+2*x^2+5*x+4)*log(2*x^2+5*x+4)
+(-2*x^3-5*x^2-4*x)*log(x^2)^4+(2*x^2+5*x+4)*log(x^2)^2+(-8*x^2-20*x-16)*log(x)*log(x^2)+(-4*x^2-5*x)*log(x))/
((2*x^3+5*x^2+4*x)*log(2*x^2+5*x+4)^2+(4*x^3+10*x^2+8*x)*log(x^2)^2*log(2*x^2+5*x+4)+(2*x^3+5*x^2+4*x)*log(x^2
)^4),x, algorithm="giac")

[Out]

-x + log(x)/(4*log(x)^2 + log(2*x^2 + 5*x + 4))

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-5 x-4 x^2\right ) \log (x)+\left (-16-20 x-8 x^2\right ) \log (x) \log \left (x^2\right )+\left (4+5 x+2 x^2\right ) \log ^2\left (x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^4\left (x^2\right )+\left (4+5 x+2 x^2+\left (-8 x-10 x^2-4 x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (4+5 x+2 x^2\right )+\left (-4 x-5 x^2-2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )}{\left (4 x+5 x^2+2 x^3\right ) \log ^4\left (x^2\right )+\left (8 x+10 x^2+4 x^3\right ) \log ^2\left (x^2\right ) \log \left (4+5 x+2 x^2\right )+\left (4 x+5 x^2+2 x^3\right ) \log ^2\left (4+5 x+2 x^2\right )} \, dx=\int -\frac {{\ln \left (2\,x^2+5\,x+4\right )}^2\,\left (2\,x^3+5\,x^2+4\,x\right )-\ln \left (2\,x^2+5\,x+4\right )\,\left (5\,x-{\ln \left (x^2\right )}^2\,\left (4\,x^3+10\,x^2+8\,x\right )+2\,x^2+4\right )-{\ln \left (x^2\right )}^2\,\left (2\,x^2+5\,x+4\right )+{\ln \left (x^2\right )}^4\,\left (2\,x^3+5\,x^2+4\,x\right )+\ln \left (x\right )\,\left (4\,x^2+5\,x\right )+\ln \left (x^2\right )\,\ln \left (x\right )\,\left (8\,x^2+20\,x+16\right )}{\left (2\,x^3+5\,x^2+4\,x\right )\,{\ln \left (x^2\right )}^4+\left (4\,x^3+10\,x^2+8\,x\right )\,{\ln \left (x^2\right )}^2\,\ln \left (2\,x^2+5\,x+4\right )+\left (2\,x^3+5\,x^2+4\,x\right )\,{\ln \left (2\,x^2+5\,x+4\right )}^2} \,d x \]

[In]

int(-(log(5*x + 2*x^2 + 4)^2*(4*x + 5*x^2 + 2*x^3) - log(5*x + 2*x^2 + 4)*(5*x - log(x^2)^2*(8*x + 10*x^2 + 4*
x^3) + 2*x^2 + 4) - log(x^2)^2*(5*x + 2*x^2 + 4) + log(x^2)^4*(4*x + 5*x^2 + 2*x^3) + log(x)*(5*x + 4*x^2) + l
og(x^2)*log(x)*(20*x + 8*x^2 + 16))/(log(5*x + 2*x^2 + 4)^2*(4*x + 5*x^2 + 2*x^3) + log(x^2)^4*(4*x + 5*x^2 +
2*x^3) + log(x^2)^2*log(5*x + 2*x^2 + 4)*(8*x + 10*x^2 + 4*x^3)),x)

[Out]

int(-(log(5*x + 2*x^2 + 4)^2*(4*x + 5*x^2 + 2*x^3) - log(5*x + 2*x^2 + 4)*(5*x - log(x^2)^2*(8*x + 10*x^2 + 4*
x^3) + 2*x^2 + 4) - log(x^2)^2*(5*x + 2*x^2 + 4) + log(x^2)^4*(4*x + 5*x^2 + 2*x^3) + log(x)*(5*x + 4*x^2) + l
og(x^2)*log(x)*(20*x + 8*x^2 + 16))/(log(5*x + 2*x^2 + 4)^2*(4*x + 5*x^2 + 2*x^3) + log(x^2)^4*(4*x + 5*x^2 +
2*x^3) + log(x^2)^2*log(5*x + 2*x^2 + 4)*(8*x + 10*x^2 + 4*x^3)), x)