\(\int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+(-60 x^3-5 x^4-10 x^7) \log ^2(x)+(-20 x+20 x^3 \log ^2(x)) \log (1-x^2 \log ^2(x))}{-16-8 x-x^2+8 x^4+2 x^5-x^8+(16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}) \log ^2(x)+(16+4 x-4 x^4+(-16 x^2-4 x^3+4 x^6) \log ^2(x)) \log (1-x^2 \log ^2(x))+(-4+4 x^2 \log ^2(x)) \log ^2(1-x^2 \log ^2(x))} \, dx\) [6494]

Optimal result

Integrand size = 198, antiderivative size = 29 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^2}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx \]

[In]

Int[(40*x + 5*x^2 + 10*x^5 - 20*x^3*Log[x] + (-60*x^3 - 5*x^4 - 10*x^7)*Log[x]^2 + (-20*x + 20*x^3*Log[x]^2)*L
og[1 - x^2*Log[x]^2])/(-16 - 8*x - x^2 + 8*x^4 + 2*x^5 - x^8 + (16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10)*L
og[x]^2 + (16 + 4*x - 4*x^4 + (-16*x^2 - 4*x^3 + 4*x^6)*Log[x]^2)*Log[1 - x^2*Log[x]^2] + (-4 + 4*x^2*Log[x]^2
)*Log[1 - x^2*Log[x]^2]^2),x]

[Out]

-5*Defer[Int][x^2/((-1 + x^2*Log[x]^2)*(-4 - x + x^4 + 2*Log[1 - x^2*Log[x]^2])^2), x] + 20*Defer[Int][x^5/((-
1 + x^2*Log[x]^2)*(-4 - x + x^4 + 2*Log[1 - x^2*Log[x]^2])^2), x] - 20*Defer[Int][(x^3*Log[x])/((-1 + x^2*Log[
x]^2)*(-4 - x + x^4 + 2*Log[1 - x^2*Log[x]^2])^2), x] - 20*Defer[Int][(x^3*Log[x]^2)/((-1 + x^2*Log[x]^2)*(-4
- x + x^4 + 2*Log[1 - x^2*Log[x]^2])^2), x] + 5*Defer[Int][(x^4*Log[x]^2)/((-1 + x^2*Log[x]^2)*(-4 - x + x^4 +
2*Log[1 - x^2*Log[x]^2])^2), x] - 20*Defer[Int][(x^7*Log[x]^2)/((-1 + x^2*Log[x]^2)*(-4 - x + x^4 + 2*Log[1 -
x^2*Log[x]^2])^2), x] + 10*Defer[Int][x/(-4 - x + x^4 + 2*Log[1 - x^2*Log[x]^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 x \left (-8-x-2 x^4+4 x^2 \log (x)+x^2 \log ^2(x) \left (12+x+2 x^4-4 \log \left (1-x^2 \log ^2(x)\right )\right )+4 \log \left (1-x^2 \log ^2(x)\right )\right )}{\left (1-x^2 \log ^2(x)\right ) \left (4+x-x^4-2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx \\ & = 5 \int \frac {x \left (-8-x-2 x^4+4 x^2 \log (x)+x^2 \log ^2(x) \left (12+x+2 x^4-4 \log \left (1-x^2 \log ^2(x)\right )\right )+4 \log \left (1-x^2 \log ^2(x)\right )\right )}{\left (1-x^2 \log ^2(x)\right ) \left (4+x-x^4-2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx \\ & = 5 \int \left (-\frac {x^2 \left (1-4 x^3+4 x \log (x)+4 x \log ^2(x)-x^2 \log ^2(x)+4 x^5 \log ^2(x)\right )}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}+\frac {2 x}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )}\right ) \, dx \\ & = -\left (5 \int \frac {x^2 \left (1-4 x^3+4 x \log (x)+4 x \log ^2(x)-x^2 \log ^2(x)+4 x^5 \log ^2(x)\right )}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx\right )+10 \int \frac {x}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \, dx \\ & = -\left (5 \int \left (\frac {x^2}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}-\frac {4 x^5}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}+\frac {4 x^3 \log (x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}+\frac {4 x^3 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}-\frac {x^4 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}+\frac {4 x^7 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2}\right ) \, dx\right )+10 \int \frac {x}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \, dx \\ & = -\left (5 \int \frac {x^2}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx\right )+5 \int \frac {x^4 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx+10 \int \frac {x}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \, dx+20 \int \frac {x^5}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx-20 \int \frac {x^3 \log (x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx-20 \int \frac {x^3 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx-20 \int \frac {x^7 \log ^2(x)}{\left (-1+x^2 \log ^2(x)\right ) \left (-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^2}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \]

[In]

Integrate[(40*x + 5*x^2 + 10*x^5 - 20*x^3*Log[x] + (-60*x^3 - 5*x^4 - 10*x^7)*Log[x]^2 + (-20*x + 20*x^3*Log[x
]^2)*Log[1 - x^2*Log[x]^2])/(-16 - 8*x - x^2 + 8*x^4 + 2*x^5 - x^8 + (16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x
^10)*Log[x]^2 + (16 + 4*x - 4*x^4 + (-16*x^2 - 4*x^3 + 4*x^6)*Log[x]^2)*Log[1 - x^2*Log[x]^2] + (-4 + 4*x^2*Lo
g[x]^2)*Log[1 - x^2*Log[x]^2]^2),x]

[Out]

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

Maple [A] (verified)

Time = 71.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)
risch	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)
parallelrisch	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)

[In]

int(((20*x^3*ln(x)^2-20*x)*ln(-x^2*ln(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*ln(x)^2-20*x^3*ln(x)+10*x^5+5*x^2+40*x)/(
(4*x^2*ln(x)^2-4)*ln(-x^2*ln(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*ln(x)^2-4*x^4+4*x+16)*ln(-x^2*ln(x)^2+1)+(x^10-2*
x^7-8*x^6+x^4+8*x^3+16*x^2)*ln(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (-x^{2} \log \left (x\right )^{2} + 1\right ) - 4} \]

[In]

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*
x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log
(x)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^{2}}{x^{4} - x + 2 \log {\left (- x^{2} \log {\left (x \right )}^{2} + 1 \right )} - 4} \]

[In]

integrate(((20*x**3*ln(x)**2-20*x)*ln(-x**2*ln(x)**2+1)+(-10*x**7-5*x**4-60*x**3)*ln(x)**2-20*x**3*ln(x)+10*x*
*5+5*x**2+40*x)/((4*x**2*ln(x)**2-4)*ln(-x**2*ln(x)**2+1)**2+((4*x**6-4*x**3-16*x**2)*ln(x)**2-4*x**4+4*x+16)*
ln(-x**2*ln(x)**2+1)+(x**10-2*x**7-8*x**6+x**4+8*x**3+16*x**2)*ln(x)**2-x**8+2*x**5+8*x**4-x**2-8*x-16),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (x \log \left (x\right ) + 1\right ) + 2 \, \log \left (-x \log \left (x\right ) + 1\right ) - 4} \]

[In]

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*
x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log
(x)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (-x^{2} \log \left (x\right )^{2} + 1\right ) - 4} \]

[In]

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*
x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log
(x)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\int \frac {40\,x-20\,x^3\,\ln \left (x\right )-\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )\,\left (20\,x-20\,x^3\,{\ln \left (x\right )}^2\right )-{\ln \left (x\right )}^2\,\left (10\,x^7+5\,x^4+60\,x^3\right )+5\,x^2+10\,x^5}{{\ln \left (x\right )}^2\,\left (x^{10}-2\,x^7-8\,x^6+x^4+8\,x^3+16\,x^2\right )-8\,x+\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )\,\left (4\,x-{\ln \left (x\right )}^2\,\left (-4\,x^6+4\,x^3+16\,x^2\right )-4\,x^4+16\right )-x^2+8\,x^4+2\,x^5-x^8+{\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )}^2\,\left (4\,x^2\,{\ln \left (x\right )}^2-4\right )-16} \,d x \]

[In]

int((40*x - 20*x^3*log(x) - log(1 - x^2*log(x)^2)*(20*x - 20*x^3*log(x)^2) - log(x)^2*(60*x^3 + 5*x^4 + 10*x^7
) + 5*x^2 + 10*x^5)/(log(x)^2*(16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10) - 8*x + log(1 - x^2*log(x)^2)*(4*x
- log(x)^2*(16*x^2 + 4*x^3 - 4*x^6) - 4*x^4 + 16) - x^2 + 8*x^4 + 2*x^5 - x^8 + log(1 - x^2*log(x)^2)^2*(4*x^
2*log(x)^2 - 4) - 16),x)

[Out]

int((40*x - 20*x^3*log(x) - log(1 - x^2*log(x)^2)*(20*x - 20*x^3*log(x)^2) - log(x)^2*(60*x^3 + 5*x^4 + 10*x^7
) + 5*x^2 + 10*x^5)/(log(x)^2*(16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10) - 8*x + log(1 - x^2*log(x)^2)*(4*x
- log(x)^2*(16*x^2 + 4*x^3 - 4*x^6) - 4*x^4 + 16) - x^2 + 8*x^4 + 2*x^5 - x^8 + log(1 - x^2*log(x)^2)^2*(4*x^
2*log(x)^2 - 4) - 16), x)