\(\int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+(-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6) \log (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4})+(2-6 x+10 x^2-10 x^3+5 x^4-x^5) \log ^2(\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4})}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx\) [6601]

Optimal result

Integrand size = 214, antiderivative size = 26 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=4-\left (3-\frac {1}{x}\right ) \left (-5+x+\log ^2\left (x+\frac {x}{(-1+x)^4}\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=\int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx \]

[In]

Int[(-10 + 30*x - 44*x^2 + 32*x^3 + 5*x^4 - 25*x^5 + 15*x^6 - 3*x^7 + (-4 + 16*x - 32*x^2 + 80*x^3 - 70*x^4 +
32*x^5 - 6*x^6)*Log[(2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)] + (2 - 6*x + 10*x^2 -
10*x^3 + 5*x^4 - x^5)*Log[(2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)]^2)/(-2*x^2 + 6
*x^3 - 10*x^4 + 10*x^5 - 5*x^6 + x^7),x]

[Out]

-5/x - 3*x - 11*Log[1 + (-1 + x)^4] + 48*Log[-1 + x]^2 - 24*Log[-1 + x]*Log[x] + 3*Log[x]^2 + 11*Log[2 - 4*x +
6*x^2 - 4*x^3 + x^4] + 24*Log[-1 + x]*Log[(x*(2 - 4*x + 6*x^2 - 4*x^3 + x^4))/(1 - x)^4] - 6*Log[x]*Log[(x*(2
- 4*x + 6*x^2 - 4*x^3 + x^4))/(1 - x)^4] + Log[(x*(2 - 4*x + 6*x^2 - 4*x^3 + x^4))/(1 - x)^4]^2/x - 96*Defer[
Int][Log[-1 + x]/(-2 + 4*x - 6*x^2 + 4*x^3 - x^4), x] - 288*Defer[Int][(x*Log[-1 + x])/(2 - 4*x + 6*x^2 - 4*x^
3 + x^4), x] + 288*Defer[Int][(x^2*Log[-1 + x])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] - 96*Defer[Int][(x^3*Log[-
1 + x])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] + 24*Defer[Int][Log[x]/(-2 + 4*x - 6*x^2 + 4*x^3 - x^4), x] + 72*D
efer[Int][(x*Log[x])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] - 72*Defer[Int][(x^2*Log[x])/(2 - 4*x + 6*x^2 - 4*x^3
+ x^4), x] + 24*Defer[Int][(x^3*Log[x])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] + 24*Defer[Int][Log[(x*(2 - 4*x +
6*x^2 - 4*x^3 + x^4))/(-1 + x)^4]/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] - 72*Defer[Int][(x*Log[(x*(2 - 4*x + 6*
x^2 - 4*x^3 + x^4))/(-1 + x)^4])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] + 72*Defer[Int][(x^2*Log[(x*(2 - 4*x + 6*
x^2 - 4*x^3 + x^4))/(-1 + x)^4])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x] - 24*Defer[Int][(x^3*Log[(x*(2 - 4*x + 6*
x^2 - 4*x^3 + x^4))/(-1 + x)^4])/(2 - 4*x + 6*x^2 - 4*x^3 + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {10-30 x+44 x^2-32 x^3-5 x^4+25 x^5-15 x^6+3 x^7-\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )-\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{x^2 \left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right )} \, dx \\ & = \int \left (-\frac {44}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {10}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {30}{(-1+x) x \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {32 x}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {5 x^2}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {25 x^3}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {15 x^4}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {3 x^5}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {2 (-1+3 x) \left (-2+2 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}\right ) \, dx \\ & = -\left (2 \int \frac {(-1+3 x) \left (-2+2 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx\right )-3 \int \frac {x^5}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+5 \int \frac {x^2}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-10 \int \frac {1}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+15 \int \frac {x^4}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-25 \int \frac {x^3}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+30 \int \frac {1}{(-1+x) x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+32 \int \frac {x}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-44 \int \frac {1}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-\int \frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2} \, dx \\ & = \frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}-2 \int \frac {\left (2-2 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(1-x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-2 \int \left (-\frac {8 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x}-\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}+\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x}+\frac {2 \left (-8+18 x-16 x^2+5 x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-3 \int \left (1+\frac {1}{-1+x}+\frac {2 x \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+5 \int \left (\frac {1}{-1+x}+\frac {2-2 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-10 \int \left (\frac {1}{-1+x}-\frac {1}{2 x^2}-\frac {3}{2 x}+\frac {(-2+x)^2 (-1+x)}{2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}\right ) \, dx+15 \int \left (\frac {1}{-1+x}+\frac {2 \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-25 \int \left (\frac {1}{-1+x}+\frac {2-2 x+4 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+30 \int \left (\frac {1}{-1+x}-\frac {1}{2 x}+\frac {-2+2 x^2-x^3}{2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}\right ) \, dx+32 \int \left (\frac {1}{-1+x}+\frac {2-3 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-44 \int \left (\frac {1}{-1+x}-\frac {(-1+x)^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx \\ & = -\frac {5}{x}-3 x+\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}+2 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2} \, dx-2 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x} \, dx-2 \int \left (-\frac {4 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x}+\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}+\frac {2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x}+\frac {2 \left (2-2 x^2+x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-4 \int \frac {\left (-8+18 x-16 x^2+5 x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-5 \int \frac {(-2+x)^2 (-1+x)}{2-4 x+6 x^2-4 x^3+x^4} \, dx+5 \int \frac {2-2 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx-6 \int \frac {x \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+15 \int \frac {-2+2 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+16 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x} \, dx-25 \int \frac {2-2 x+4 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+30 \int \frac {1-x+2 x^2}{2-4 x+6 x^2-4 x^3+x^4} \, dx+32 \int \frac {2-3 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+44 \int \frac {(-1+x)^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx \\ & = -\frac {5}{x}-3 x+11 \log \left (2-4 x+6 x^2-4 x^3+x^4\right )-\frac {2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}+16 \log (-1+x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )-2 \log (x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )+\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}+2 \int \frac {2-2 x+10 x^2-10 x^3+5 x^4-x^5}{(1-x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+2 \int \frac {(-1+x)^4 \left (\frac {x \left (-4+12 x-12 x^2+4 x^3\right )}{(-1+x)^4}+\frac {2-4 x+6 x^2-4 x^3+x^4}{(-1+x)^4}-\frac {4 x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^5}\right ) \log (x)}{x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-2 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2} \, dx-4 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x} \, dx-4 \int \frac {\left (2-2 x^2+x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-4 \int \left (-\frac {8 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}+\frac {18 x \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}-\frac {16 x^2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}+\frac {5 x^3 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-5 \text {Subst}\left (\int \frac {(-1+x)^2 x}{1+x^4} \, dx,x,-1+x\right )+5 \text {Subst}\left (\int \frac {2+x-x^3}{1+x^4} \, dx,x,-1+x\right )-6 \text {Subst}\left (\int \frac {(1+x) \left (2+3 x+2 x^2\right )}{1+x^4} \, dx,x,-1+x\right )+8 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x} \, dx+15 \text {Subst}\left (\int \frac {-1+x-x^2-x^3}{1+x^4} \, dx,x,-1+x\right )-16 \int \frac {(-1+x)^4 \left (\frac {x \left (-4+12 x-12 x^2+4 x^3\right )}{(-1+x)^4}+\frac {2-4 x+6 x^2-4 x^3+x^4}{(-1+x)^4}-\frac {4 x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^5}\right ) \log (-1+x)}{x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-25 \text {Subst}\left (\int \frac {3+3 x+x^2-x^3}{1+x^4} \, dx,x,-1+x\right )+30 \text {Subst}\left (\int \frac {2+3 x+2 x^2}{1+x^4} \, dx,x,-1+x\right )+32 \text {Subst}\left (\int \frac {1-x^3}{1+x^4} \, dx,x,-1+x\right ) \\ & = -\frac {5}{x}-3 x+11 \log \left (2-4 x+6 x^2-4 x^3+x^4\right )+24 \log (-1+x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )-6 \log (x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )+\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}-2 \int \frac {2-2 x+10 x^2-10 x^3+5 x^4-x^5}{(1-x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+2 \int \left (-\frac {4}{-1+x}+\frac {1}{x^2}+\frac {2}{x}+\frac {2 \left (2-2 x^2+x^3\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+2 \int \left (-\frac {4 \log (x)}{-1+x}+\frac {\log (x)}{x}+\frac {4 (-1+x)^3 \log (x)}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+4 \int \frac {(-1+x)^4 \left (\frac {x \left (-4+12 x-12 x^2+4 x^3\right )}{(-1+x)^4}+\frac {2-4 x+6 x^2-4 x^3+x^4}{(-1+x)^4}-\frac {4 x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^5}\right ) \log (x)}{x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-4 \int \left (\frac {2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}-\frac {2 x^2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}+\frac {x^3 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+5 \text {Subst}\left (\int \left (\frac {2}{1+x^4}+\frac {x \left (1-x^2\right )}{1+x^4}\right ) \, dx,x,-1+x\right )-5 \text {Subst}\left (\int \left (-\frac {2 x^2}{1+x^4}+\frac {x \left (1+x^2\right )}{1+x^4}\right ) \, dx,x,-1+x\right )-6 \text {Subst}\left (\int \left (\frac {x \left (5+2 x^2\right )}{1+x^4}+\frac {2+5 x^2}{1+x^4}\right ) \, dx,x,-1+x\right )-8 \int \frac {(-1+x)^4 \left (\frac {x \left (-4+12 x-12 x^2+4 x^3\right )}{(-1+x)^4}+\frac {2-4 x+6 x^2-4 x^3+x^4}{(-1+x)^4}-\frac {4 x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^5}\right ) \log (-1+x)}{x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+15 \text {Subst}\left (\int \left (\frac {-1-x^2}{1+x^4}+\frac {x \left (1-x^2\right )}{1+x^4}\right ) \, dx,x,-1+x\right )-16 \int \left (-\frac {4 \log (-1+x)}{-1+x}+\frac {\log (-1+x)}{x}+\frac {4 (-1+x)^3 \log (-1+x)}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-20 \int \frac {x^3 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-25 \text {Subst}\left (\int \left (\frac {x \left (3-x^2\right )}{1+x^4}+\frac {3+x^2}{1+x^4}\right ) \, dx,x,-1+x\right )+30 \text {Subst}\left (\int \left (\frac {3 x}{1+x^4}+\frac {2+2 x^2}{1+x^4}\right ) \, dx,x,-1+x\right )+32 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+32 \text {Subst}\left (\int \left (\frac {1}{1+x^4}-\frac {x^3}{1+x^4}\right ) \, dx,x,-1+x\right )+64 \int \frac {x^2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-72 \int \frac {x \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx \\ & = -\frac {7}{x}-3 x-8 \log (1-x)+4 \log (x)+11 \log \left (2-4 x+6 x^2-4 x^3+x^4\right )+24 \log (-1+x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )-6 \log (x) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )+\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}-2 \int \left (-\frac {4}{-1+x}+\frac {1}{x^2}+\frac {2}{x}+\frac {2 \left (2-2 x^2+x^3\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+2 \int \frac {\log (x)}{x} \, dx+4 \int \frac {2-2 x^2+x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+4 \int \left (-\frac {4 \log (x)}{-1+x}+\frac {\log (x)}{x}+\frac {4 (-1+x)^3 \log (x)}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-4 \int \frac {x^3 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+5 \text {Subst}\left (\int \frac {x \left (1-x^2\right )}{1+x^4} \, dx,x,-1+x\right )-5 \text {Subst}\left (\int \frac {x \left (1+x^2\right )}{1+x^4} \, dx,x,-1+x\right )-6 \text {Subst}\left (\int \frac {x \left (5+2 x^2\right )}{1+x^4} \, dx,x,-1+x\right )-6 \text {Subst}\left (\int \frac {2+5 x^2}{1+x^4} \, dx,x,-1+x\right )-8 \int \left (-\frac {4 \log (-1+x)}{-1+x}+\frac {\log (-1+x)}{x}+\frac {4 (-1+x)^3 \log (-1+x)}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-8 \int \frac {\log (x)}{-1+x} \, dx+8 \int \frac {(-1+x)^3 \log (x)}{2-4 x+6 x^2-4 x^3+x^4} \, dx-8 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+8 \int \frac {x^2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+10 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,-1+x\right )+10 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,-1+x\right )+15 \text {Subst}\left (\int \frac {-1-x^2}{1+x^4} \, dx,x,-1+x\right )+15 \text {Subst}\left (\int \frac {x \left (1-x^2\right )}{1+x^4} \, dx,x,-1+x\right )-16 \int \frac {\log (-1+x)}{x} \, dx-20 \int \frac {x^3 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-25 \text {Subst}\left (\int \frac {x \left (3-x^2\right )}{1+x^4} \, dx,x,-1+x\right )-25 \text {Subst}\left (\int \frac {3+x^2}{1+x^4} \, dx,x,-1+x\right )+30 \text {Subst}\left (\int \frac {2+2 x^2}{1+x^4} \, dx,x,-1+x\right )+32 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+32 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,-1+x\right )-32 \text {Subst}\left (\int \frac {x^3}{1+x^4} \, dx,x,-1+x\right )+64 \int \frac {\log (-1+x)}{-1+x} \, dx-64 \int \frac {(-1+x)^3 \log (-1+x)}{2-4 x+6 x^2-4 x^3+x^4} \, dx+64 \int \frac {x^2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-72 \int \frac {x \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+90 \text {Subst}\left (\int \frac {x}{1+x^4} \, dx,x,-1+x\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=-\frac {5}{x}-3 x-\frac {(-1+3 x) \log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x} \]

[In]

Integrate[(-10 + 30*x - 44*x^2 + 32*x^3 + 5*x^4 - 25*x^5 + 15*x^6 - 3*x^7 + (-4 + 16*x - 32*x^2 + 80*x^3 - 70*
x^4 + 32*x^5 - 6*x^6)*Log[(2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)] + (2 - 6*x + 10
*x^2 - 10*x^3 + 5*x^4 - x^5)*Log[(2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)]^2)/(-2*x
^2 + 6*x^3 - 10*x^4 + 10*x^5 - 5*x^6 + x^7),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(26)=52\).

Time = 0.89 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.12

method	result	size
norman	\(\frac {-5+\ln \left (\frac {x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{2}-3 x^{2}-3 x \ln \left (\frac {x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{2}}{x}\)	\(107\)

[In]

int(((-x^5+5*x^4-10*x^3+10*x^2-6*x+2)*ln((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))^2+(-6*x^6+32*x^5
-70*x^4+80*x^3-32*x^2+16*x-4)*ln((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))-3*x^7+15*x^6-25*x^5+5*x^
4+32*x^3-44*x^2+30*x-10)/(x^7-5*x^6+10*x^5-10*x^4+6*x^3-2*x^2),x,method=_RETURNVERBOSE)

[Out]

(-5+ln((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))^2-3*x^2-3*x*ln((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*
x^3+6*x^2-4*x+1))^2)/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).

Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (\frac {x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{2} + 3 \, x^{2} + 5}{x} \]

[In]

integrate(((-x^5+5*x^4-10*x^3+10*x^2-6*x+2)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))^2+(-6*x^6
+32*x^5-70*x^4+80*x^3-32*x^2+16*x-4)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))-3*x^7+15*x^6-25*
x^5+5*x^4+32*x^3-44*x^2+30*x-10)/(x^7-5*x^6+10*x^5-10*x^4+6*x^3-2*x^2),x, algorithm="fricas")

[Out]

-((3*x - 1)*log((x^5 - 4*x^4 + 6*x^3 - 4*x^2 + 2*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))^2 + 3*x^2 + 5)/x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=- 3 x + \frac {\left (1 - 3 x\right ) \log {\left (\frac {x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + 2 x}{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1} \right )}^{2}}{x} - \frac {5}{x} \]

[In]

integrate(((-x**5+5*x**4-10*x**3+10*x**2-6*x+2)*ln((x**5-4*x**4+6*x**3-4*x**2+2*x)/(x**4-4*x**3+6*x**2-4*x+1))
**2+(-6*x**6+32*x**5-70*x**4+80*x**3-32*x**2+16*x-4)*ln((x**5-4*x**4+6*x**3-4*x**2+2*x)/(x**4-4*x**3+6*x**2-4*
x+1))-3*x**7+15*x**6-25*x**5+5*x**4+32*x**3-44*x**2+30*x-10)/(x**7-5*x**6+10*x**5-10*x**4+6*x**3-2*x**2),x)

[Out]

-3*x + (1 - 3*x)*log((x**5 - 4*x**4 + 6*x**3 - 4*x**2 + 2*x)/(x**4 - 4*x**3 + 6*x**2 - 4*x + 1))**2/x - 5/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.50 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=-\frac {{\left (3 \, x - 1\right )} \log \left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 2\right )^{2} + 16 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right )^{2} - 8 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right ) \log \left (x\right ) + {\left (3 \, x - 1\right )} \log \left (x\right )^{2} + 3 \, x^{2} - 2 \, {\left (4 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right ) - {\left (3 \, x - 1\right )} \log \left (x\right )\right )} \log \left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 2\right ) + 5}{x} \]

[In]

integrate(((-x^5+5*x^4-10*x^3+10*x^2-6*x+2)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))^2+(-6*x^6
+32*x^5-70*x^4+80*x^3-32*x^2+16*x-4)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))-3*x^7+15*x^6-25*
x^5+5*x^4+32*x^3-44*x^2+30*x-10)/(x^7-5*x^6+10*x^5-10*x^4+6*x^3-2*x^2),x, algorithm="maxima")

[Out]

-((3*x - 1)*log(x^4 - 4*x^3 + 6*x^2 - 4*x + 2)^2 + 16*(3*x - 1)*log(x - 1)^2 - 8*(3*x - 1)*log(x - 1)*log(x) +
(3*x - 1)*log(x)^2 + 3*x^2 - 2*(4*(3*x - 1)*log(x - 1) - (3*x - 1)*log(x))*log(x^4 - 4*x^3 + 6*x^2 - 4*x + 2)
+ 5)/x

Giac [F]

\[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx=\int { -\frac {3 \, x^{7} - 15 \, x^{6} + 25 \, x^{5} - 5 \, x^{4} - 32 \, x^{3} + {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 6 \, x - 2\right )} \log \left (\frac {x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{2} + 44 \, x^{2} + 2 \, {\left (3 \, x^{6} - 16 \, x^{5} + 35 \, x^{4} - 40 \, x^{3} + 16 \, x^{2} - 8 \, x + 2\right )} \log \left (\frac {x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 30 \, x + 10}{x^{7} - 5 \, x^{6} + 10 \, x^{5} - 10 \, x^{4} + 6 \, x^{3} - 2 \, x^{2}} \,d x } \]

[In]

integrate(((-x^5+5*x^4-10*x^3+10*x^2-6*x+2)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))^2+(-6*x^6
+32*x^5-70*x^4+80*x^3-32*x^2+16*x-4)*log((x^5-4*x^4+6*x^3-4*x^2+2*x)/(x^4-4*x^3+6*x^2-4*x+1))-3*x^7+15*x^6-25*
x^5+5*x^4+32*x^3-44*x^2+30*x-10)/(x^7-5*x^6+10*x^5-10*x^4+6*x^3-2*x^2),x, algorithm="giac")

[Out]

integrate(-(3*x^7 - 15*x^6 + 25*x^5 - 5*x^4 - 32*x^3 + (x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 6*x - 2)*log((x^5 - 4*
x^4 + 6*x^3 - 4*x^2 + 2*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))^2 + 44*x^2 + 2*(3*x^6 - 16*x^5 + 35*x^4 - 40*x^3 +
16*x^2 - 8*x + 2)*log((x^5 - 4*x^4 + 6*x^3 - 4*x^2 + 2*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 30*x + 10)/(x^7
- 5*x^6 + 10*x^5 - 10*x^4 + 6*x^3 - 2*x^2), x)

Mupad [B] (verification not implemented)

Time = 12.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx={\ln \left (\frac {x^5-4\,x^4+6\,x^3-4\,x^2+2\,x}{x^4-4\,x^3+6\,x^2-4\,x+1}\right )}^2\,\left (\frac {1}{x}-3\right )-3\,x-\frac {5}{x} \]

[In]

int((log((2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))*(32*x^2 - 16*x - 80*x^3 + 70*x^4
- 32*x^5 + 6*x^6 + 4) - 30*x + log((2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))^2*(6*
x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 2) + 44*x^2 - 32*x^3 - 5*x^4 + 25*x^5 - 15*x^6 + 3*x^7 + 10)/(2*x^2 - 6*x^
3 + 10*x^4 - 10*x^5 + 5*x^6 - x^7),x)

[Out]

log((2*x - 4*x^2 + 6*x^3 - 4*x^4 + x^5)/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))^2*(1/x - 3) - 3*x - 5/x