\(\int \frac {20+24 x^2+16 x^3-4 x^4+(-2 x^2-8 x^3+2 x^4+(-10-40 x+10 x^2) \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})) \log (\frac {1}{5} (x^2+5 \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})))}{(-x^2-4 x^3+x^4+(-5-20 x+5 x^2) \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})) \log ^2(\frac {1}{5} (x^2+5 \log (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4})))} \, dx\) [3283]

Optimal result

Integrand size = 209, antiderivative size = 33 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\frac {2 x}{\log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{\left (2-2 \left (-4 x+x^2\right )\right )^2}\right )\right )} \]

[Out]

2*x/ln(ln(25*x^2/(-2*x^2+8*x+2)^2)+1/5*x^2)

Rubi [F]

\[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx \]

[In]

Int[(20 + 24*x^2 + 16*x^3 - 4*x^4 + (-2*x^2 - 8*x^3 + 2*x^4 + (-10 - 40*x + 10*x^2)*Log[(25*x^2)/(4 + 32*x + 5
6*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5])/((-x^2 - 4*x^3
+ x^4 + (-5 - 20*x + 5*x^2)*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 +
32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5]^2),x]

[Out]

68*Defer[Int][1/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)
^2)])/5]^2), x] + 16*Defer[Int][x/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])/5]^2), x] + 4*Defer[Int][x^2/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2
+ 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + 88*Defer[Int][1/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x
^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + (288*(5 - 2*Sqrt[5])*Defer[Int][1/((4
- 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2
)^2)])/5]^2), x])/5 - 8*Sqrt[5]*Defer[Int][1/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x] + (288*(5 + 2*Sqrt[5])*Defer[Int][1/((4 + 2*
Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)
])/5]^2), x])/5 + 16*Defer[Int][x/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4
*(-1 - 4*x + x^2)^2)])/5]^2), x] + (368*(5 + 2*Sqrt[5])*Defer[Int][1/((-4 - 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*
x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x])/5 - 8*Sqrt[5]*Defe
r[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(
-1 - 4*x + x^2)^2)])/5]^2), x] + (368*(5 - 2*Sqrt[5])*Defer[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^
2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]^2), x])/5 + 34*Defer[Int][1/
((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] +
8*Defer[Int][x/((-x^2 - 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])/5]), x] + 34*Defer[Int][1/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-
1 - 4*x + x^2)^2)])/5]), x] + (144*(5 - 2*Sqrt[5])*Defer[Int][1/((4 - 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x])/5 + (144*(5 + 2*Sqrt[5])*D
efer[Int][1/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4
*(-1 - 4*x + x^2)^2)])/5]), x])/5 + 8*Defer[Int][x/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 +
5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 2*Defer[Int][x^2/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^
2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + (144*(5 + 2*Sqrt[5])*Defer[Int][1/((-4 - 2*S
qrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]
)/5]), x])/5 + (144*(5 - 2*Sqrt[5])*Defer[Int][1/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x +
x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x])/5 + 10*Defer[Int][Log[(25*x^2)/(4*(-1 -
4*x + x^2)^2)]/((x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2
)])/5]), x] + 8*(5 - 2*Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((4 - 2*Sqrt[5] - 2*x)*(x^2 +
5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 + 2*
Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((4 + 2*Sqrt[5] - 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 -
4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 + 2*Sqrt[5])*Defer[Int][Log
[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]/((-4 - 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(
x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])/5]), x] + 8*(5 - 2*Sqrt[5])*Defer[Int][Log[(25*x^2)/(4*(-1 - 4*x
+ x^2)^2)]/((-4 + 2*Sqrt[5] + 2*x)*(x^2 + 5*Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)])*Log[(x^2 + 5*Log[(25*x^2)/(
4*(-1 - 4*x + x^2)^2)])/5]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-20-24 x^2-16 x^3+4 x^4-2 \left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx \\ & = \int \left (\frac {4 x^4}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {20}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {24 x^2}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {16 x^3}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {2 x^2}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {8 x^3}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {2 x^4}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {10 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {40 x \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}+\frac {10 x^2 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )}\right ) \, dx \\ & = 2 \int \frac {x^2}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+2 \int \frac {x^4}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+4 \int \frac {x^4}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+8 \int \frac {x^3}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+10 \int \frac {\log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+10 \int \frac {x^2 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+16 \int \frac {x^3}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+20 \int \frac {1}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+24 \int \frac {x^2}{\left (-1-4 x+x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx+40 \int \frac {x \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )}{\left (1+4 x-x^2\right ) \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\frac {2 x}{\log \left (\frac {x^2}{5}+\log \left (\frac {25 x^2}{4 \left (-1-4 x+x^2\right )^2}\right )\right )} \]

[In]

Integrate[(20 + 24*x^2 + 16*x^3 - 4*x^4 + (-2*x^2 - 8*x^3 + 2*x^4 + (-10 - 40*x + 10*x^2)*Log[(25*x^2)/(4 + 32
*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5])/((-x^2 -
4*x^3 + x^4 + (-5 - 20*x + 5*x^2)*Log[(25*x^2)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])*Log[(x^2 + 5*Log[(25*x^2
)/(4 + 32*x + 56*x^2 - 32*x^3 + 4*x^4)])/5]^2),x]

[Out]

(2*x)/Log[x^2/5 + Log[(25*x^2)/(4*(-1 - 4*x + x^2)^2)]]

Maple [A] (verified)

Time = 7.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

method	result	size
parallelrisch	\(\frac {2 x}{\ln \left (\ln \left (\frac {25 x^{2}}{4 \left (x^{4}-8 x^{3}+14 x^{2}+8 x +1\right )}\right )+\frac {x^{2}}{5}\right )}\)	\(39\)

[In]

int((((10*x^2-40*x-10)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*ln(ln(25*x^2/(4*x^4-32*x^3+5
6*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+x^4-4*
x^3-x^2)/ln(ln(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*x/ln(ln(25/4*x^2/(x^4-8*x^3+14*x^2+8*x+1))+1/5*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\frac {2 \, x}{\log \left (\frac {1}{5} \, x^{2} + \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right )} \]

[In]

integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="fricas")

[Out]

2*x/log(1/5*x^2 + log(25/4*x^2/(x^4 - 8*x^3 + 14*x^2 + 8*x + 1)))

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\frac {2 x}{\log {\left (\frac {x^{2}}{5} + \log {\left (\frac {25 x^{2}}{4 x^{4} - 32 x^{3} + 56 x^{2} + 32 x + 4} \right )} \right )}} \]

[In]

integrate((((10*x**2-40*x-10)*ln(25*x**2/(4*x**4-32*x**3+56*x**2+32*x+4))+2*x**4-8*x**3-2*x**2)*ln(ln(25*x**2/
(4*x**4-32*x**3+56*x**2+32*x+4))+1/5*x**2)-4*x**4+16*x**3+24*x**2+20)/((5*x**2-20*x-5)*ln(25*x**2/(4*x**4-32*x
**3+56*x**2+32*x+4))+x**4-4*x**3-x**2)/ln(ln(25*x**2/(4*x**4-32*x**3+56*x**2+32*x+4))+1/5*x**2)**2,x)

[Out]

2*x/log(x**2/5 + log(25*x**2/(4*x**4 - 32*x**3 + 56*x**2 + 32*x + 4)))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=-\frac {2 \, x}{\log \left (5\right ) - \log \left (x^{2} + 10 \, \log \left (5\right ) - 10 \, \log \left (2\right ) - 10 \, \log \left (x^{2} - 4 \, x - 1\right ) + 10 \, \log \left (x\right )\right )} \]

[In]

integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="maxima")

[Out]

-2*x/(log(5) - log(x^2 + 10*log(5) - 10*log(2) - 10*log(x^2 - 4*x - 1) + 10*log(x)))

Giac [F]

\[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=\int { -\frac {2 \, {\left (2 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} - {\left (x^{4} - 4 \, x^{3} - x^{2} + 5 \, {\left (x^{2} - 4 \, x - 1\right )} \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right )} \log \left (\frac {1}{5} \, x^{2} + \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right ) - 10\right )}}{{\left (x^{4} - 4 \, x^{3} - x^{2} + 5 \, {\left (x^{2} - 4 \, x - 1\right )} \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right )} \log \left (\frac {1}{5} \, x^{2} + \log \left (\frac {25 \, x^{2}}{4 \, {\left (x^{4} - 8 \, x^{3} + 14 \, x^{2} + 8 \, x + 1\right )}}\right )\right )^{2}} \,d x } \]

[In]

integrate((((10*x^2-40*x-10)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+2*x^4-8*x^3-2*x^2)*log(log(25*x^2/(4*x^4
-32*x^3+56*x^2+32*x+4))+1/5*x^2)-4*x^4+16*x^3+24*x^2+20)/((5*x^2-20*x-5)*log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+
4))+x^4-4*x^3-x^2)/log(log(25*x^2/(4*x^4-32*x^3+56*x^2+32*x+4))+1/5*x^2)^2,x, algorithm="giac")

[Out]

sage2

Mupad [B] (verification not implemented)

Time = 8.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.18 \[ \int \frac {20+24 x^2+16 x^3-4 x^4+\left (-2 x^2-8 x^3+2 x^4+\left (-10-40 x+10 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log \left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )}{\left (-x^2-4 x^3+x^4+\left (-5-20 x+5 x^2\right ) \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right ) \log ^2\left (\frac {1}{5} \left (x^2+5 \log \left (\frac {25 x^2}{4+32 x+56 x^2-32 x^3+4 x^4}\right )\right )\right )} \, dx=x-\frac {5\,x}{-x^4+4\,x^3+6\,x^2+5}-\frac {6\,x^3}{-x^4+4\,x^3+6\,x^2+5}-\frac {4\,x^4}{-x^4+4\,x^3+6\,x^2+5}+\frac {x^5}{-x^4+4\,x^3+6\,x^2+5}+\frac {2\,x}{\ln \left (\ln \left (x^2\right )+2\,\ln \left (5\right )+\ln \left (\frac {1}{4\,x^4-32\,x^3+56\,x^2+32\,x+4}\right )+\frac {x^2}{5}\right )} \]

[In]

int(-(24*x^2 + 16*x^3 - 4*x^4 - log(log((25*x^2)/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) + x^2/5)*(log((25*x^2)/
(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4))*(40*x - 10*x^2 + 10) + 2*x^2 + 8*x^3 - 2*x^4) + 20)/(log(log((25*x^2)/(3
2*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) + x^2/5)^2*(log((25*x^2)/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4))*(20*x - 5*x
^2 + 5) + x^2 + 4*x^3 - x^4)),x)

[Out]

x - (5*x)/(6*x^2 + 4*x^3 - x^4 + 5) - (6*x^3)/(6*x^2 + 4*x^3 - x^4 + 5) - (4*x^4)/(6*x^2 + 4*x^3 - x^4 + 5) +
x^5/(6*x^2 + 4*x^3 - x^4 + 5) + (2*x)/log(log(x^2) + 2*log(5) + log(1/(32*x + 56*x^2 - 32*x^3 + 4*x^4 + 4)) +
x^2/5)