Integrand size = 134, antiderivative size = 23 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=13-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]
Integrate[(-2*x - x^2 + (-2 - x - 2*x^2 - 2*x^3 - x^4)*Log[x] + (-2*x - 4* x^2 - 2*x^3)*Log[x]^2 + (-2*x - x^2)*Log[x]^3 + (2 + 3*x + x^2)*Log[x]*Log [(8 + 8*x + 2*x^2)*Log[x]])/((2*x^3 + x^4)*Log[x] + (4*x^2 + 2*x^3)*Log[x] ^2 + (2*x + x^2)*Log[x]^3),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-x^2+\left (-x^2-2 x\right ) \log ^3(x)+\left (x^2+3 x+2\right ) \log \left (\left (2 x^2+8 x+8\right ) \log (x)\right ) \log (x)+\left (-2 x^3-4 x^2-2 x\right ) \log ^2(x)+\left (-x^4-2 x^3-2 x^2-x-2\right ) \log (x)-2 x}{\left (x^2+2 x\right ) \log ^3(x)+\left (x^4+2 x^3\right ) \log (x)+\left (2 x^3+4 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x^2+\left (-x^2-2 x\right ) \log ^3(x)+\left (x^2+3 x+2\right ) \log \left (\left (2 x^2+8 x+8\right ) \log (x)\right ) \log (x)+\left (-2 x^3-4 x^2-2 x\right ) \log ^2(x)+\left (-x^4-2 x^3-2 x^2-x-2\right ) \log (x)-2 x}{x (x+2) \log (x) (x+\log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-x^4-2 x^3-2 x^2-x-2}{x (x+2) (x+\log (x))^2}-\frac {\log ^2(x)}{(x+\log (x))^2}-\frac {2 (x+1)^2 \log (x)}{(x+2) (x+\log (x))^2}+\frac {(x+1) \log \left (2 (x+2)^2 \log (x)\right )}{x (x+\log (x))^2}-\frac {x}{(x+2) \log (x) (x+\log (x))^2}-\frac {2}{(x+2) \log (x) (x+\log (x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{x^2 (x+2) \log (x)}dx+\int \frac {1}{x^2 (x+\log (x))}dx-\int \frac {1}{x (x+2) \log (x)}dx-2 \int \frac {1}{(x+2) (x+\log (x))}dx+\int \frac {\log \left (2 (x+2)^2 \log (x)\right )}{(x+\log (x))^2}dx+\int \frac {\log \left (2 (x+2)^2 \log (x)\right )}{x (x+\log (x))^2}dx-x\) |
Int[(-2*x - x^2 + (-2 - x - 2*x^2 - 2*x^3 - x^4)*Log[x] + (-2*x - 4*x^2 - 2*x^3)*Log[x]^2 + (-2*x - x^2)*Log[x]^3 + (2 + 3*x + x^2)*Log[x]*Log[(8 + 8*x + 2*x^2)*Log[x]])/((2*x^3 + x^4)*Log[x] + (4*x^2 + 2*x^3)*Log[x]^2 + ( 2*x + x^2)*Log[x]^3),x]
3.1.63.3.1 Defintions of rubi rules used
Time = 2.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57
method | result | size |
parallelrisch | \(\frac {-4 x^{2}-4 x \ln \left (x \right )-4 \ln \left (\left (2 x^{2}+8 x +8\right ) \ln \left (x \right )\right )}{4 x +4 \ln \left (x \right )}\) | \(36\) |
risch | \(-\frac {2 \ln \left (2+x \right )}{x +\ln \left (x \right )}-\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )+i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )-i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (2+x \right )^{2}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )-i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (2+x \right )^{2}\right )^{3}-i \pi \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{3}+2 x^{2}+2 x \ln \left (x \right )+2 \ln \left (2\right )+2 \ln \left (\ln \left (x \right )\right )}{2 \left (x +\ln \left (x \right )\right )}\) | \(197\) |
int(((x^2+3*x+2)*ln(x)*ln((2*x^2+8*x+8)*ln(x))+(-x^2-2*x)*ln(x)^3+(-2*x^3- 4*x^2-2*x)*ln(x)^2+(-x^4-2*x^3-2*x^2-x-2)*ln(x)-x^2-2*x)/((x^2+2*x)*ln(x)^ 3+(2*x^3+4*x^2)*ln(x)^2+(x^4+2*x^3)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-\frac {x^{2} + x \log \left (x\right ) + \log \left (2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )\right )}{x + \log \left (x\right )} \]
integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^ 3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2 +2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm=\
Exception generated. \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]
integrate(((x**2+3*x+2)*ln(x)*ln((2*x**2+8*x+8)*ln(x))+(-x**2-2*x)*ln(x)** 3+(-2*x**3-4*x**2-2*x)*ln(x)**2+(-x**4-2*x**3-2*x**2-x-2)*ln(x)-x**2-2*x)/ ((x**2+2*x)*ln(x)**3+(2*x**3+4*x**2)*ln(x)**2+(x**4+2*x**3)*ln(x)),x)
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-\frac {x^{2} + x \log \left (x\right ) + \log \left (2\right ) + 2 \, \log \left (x + 2\right ) + \log \left (\log \left (x\right )\right )}{x + \log \left (x\right )} \]
integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^ 3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2 +2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x - \frac {\log \left (2 \, x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 8 \, \log \left (x\right )\right )}{x + \log \left (x\right )} \]
integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^ 3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2 +2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm=\
Time = 9.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx=-x-\frac {\ln \left (\ln \left (x\right )\,\left (2\,x^2+8\,x+8\right )\right )}{x+\ln \left (x\right )} \]