\(\int \frac {4+8 x+(8 x+4 x^2) \log (x)+(-2+(-2-2 x) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})}{5 x^4+(5 x^4+5 x^5) \log (x)+(-10 x^3+(-10 x^3-10 x^4) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})+(5 x^2+(5 x^2+5 x^3) \log (x)) \log ^2(\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})} \, dx\) [2100]

Optimal result

Integrand size = 173, antiderivative size = 31 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=5-\frac {2}{5 x \left (x-\log \left (\frac {x^2}{(x+x (1+x) \log (x))^2}\right )\right )} \]

[Out]

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

Rubi [F]

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

[In]

Int[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log
[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 5*x^5)*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)*Log[x
] + (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x +
x^2)*Log[x]^2)^(-1)]^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 6.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=\frac {2}{5 x \left (-x+\log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )} \]

[In]

Integrate[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^
2)*Log[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 5*x^5)*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)
*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 +
2*x + x^2)*Log[x]^2)^(-1)]^2),x]

[Out]

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

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45

method	result	size
parallelrisch	\(-\frac {2}{5 x \left (x -\ln \left (\frac {1}{x^{2} \ln \left (x \right )^{2}+2 x \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+2 x \ln \left (x \right )+2 \ln \left (x \right )+1}\right )\right )}\)	\(45\)
default	\(-\frac {4 i}{5 \left (\pi \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right )^{2} \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right ) \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )^{3}+2 i x +4 i \ln \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right ) x}\)	\(105\)
risch	\(-\frac {4 i}{5 x \left (\pi \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )^{2} \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{3}+2 i x +4 i \ln \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )}\)	\(105\)

[In]

int((((-2-2*x)*ln(x)-2)*ln(1/((x^2+2*x+1)*ln(x)^2+(2+2*x)*ln(x)+1))+(4*x^2+8*x)*ln(x)+8*x+4)/(((5*x^3+5*x^2)*l
n(x)+5*x^2)*ln(1/((x^2+2*x+1)*ln(x)^2+(2+2*x)*ln(x)+1))^2+((-10*x^4-10*x^3)*ln(x)-10*x^3)*ln(1/((x^2+2*x+1)*ln
(x)^2+(2+2*x)*ln(x)+1))+(5*x^5+5*x^4)*ln(x)+5*x^4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} - x \log \left (\frac {1}{{\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (x + 1\right )} \log \left (x\right ) + 1}\right )\right )}} \]

[In]

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=\frac {2}{- 5 x^{2} + 5 x \log {\left (\frac {1}{\left (2 x + 2\right ) \log {\left (x \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}^{2} + 1} \right )}} \]

[In]

integrate((((-2-2*x)*ln(x)-2)*ln(1/((x**2+2*x+1)*ln(x)**2+(2+2*x)*ln(x)+1))+(4*x**2+8*x)*ln(x)+8*x+4)/(((5*x**
3+5*x**2)*ln(x)+5*x**2)*ln(1/((x**2+2*x+1)*ln(x)**2+(2+2*x)*ln(x)+1))**2+((-10*x**4-10*x**3)*ln(x)-10*x**3)*ln
(1/((x**2+2*x+1)*ln(x)**2+(2+2*x)*ln(x)+1))+(5*x**5+5*x**4)*ln(x)+5*x**4),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} + 2 \, x \log \left ({\left (x + 1\right )} \log \left (x\right ) + 1\right )\right )}} \]

[In]

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.89 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} + x \log \left (x^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )\right )}} \]

[In]

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(2*x + 2) + 2) + log(x)*(8*x + 4*x
^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 +
10*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2
) + 5*x^4),x)

[Out]

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(2*x + 2) + 2) + log(x)*(8*x + 4*x
^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 +
10*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2
) + 5*x^4), x)