Integrand size = 89, antiderivative size = 24 \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\frac {x-x^2+\log (x)-\log (\log (x))}{8+2 x+\log (x)} \]
Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {8+x+x^2+\log (\log (x))}{8+2 x+\log (x)} \]
Integrate[(-8 - 2*x + (7 + 9*x - 15*x^2 - 2*x^3)*Log[x] + (-x - 2*x^2)*Log [x]^2 + (1 + 2*x)*Log[x]*Log[Log[x]])/((64*x + 32*x^2 + 4*x^3)*Log[x] + (1 6*x + 4*x^2)*Log[x]^2 + x*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.
\(\displaystyle \int \frac {\left (-2 x^2-x\right ) \log ^2(x)+\left (-2 x^3-15 x^2+9 x+7\right ) \log (x)-2 x+(2 x+1) \log (\log (x)) \log (x)-8}{\left (4 x^2+16 x\right ) \log ^2(x)+\left (4 x^3+32 x^2+64 x\right ) \log (x)+x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-2 x^2-x\right ) \log ^2(x)+\left (-2 x^3-15 x^2+9 x+7\right ) \log (x)-2 x+(2 x+1) \log (\log (x)) \log (x)-8}{x \log (x) (2 x+\log (x)+8)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-2 x^3-15 x^2+9 x+7}{x (2 x+\log (x)+8)^2}+\frac {(2 x+1) \log (\log (x))}{x (2 x+\log (x)+8)^2}-\frac {(2 x+1) \log (x)}{(2 x+\log (x)+8)^2}-\frac {8}{x \log (x) (2 x+\log (x)+8)^2}-\frac {2}{\log (x) (2 x+\log (x)+8)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {x^2}{(2 x+\log (x)+8)^2}dx-\frac {1}{2} \int \frac {1}{(x+4)^2 \log (x)}dx-2 \int \frac {1}{x (x+4)^2 \log (x)}dx+17 \int \frac {1}{(2 x+\log (x)+8)^2}dx+8 \int \frac {1}{x (2 x+\log (x)+8)^2}dx+3 \int \frac {x}{(2 x+\log (x)+8)^2}dx-\int \frac {1}{2 x+\log (x)+8}dx+\frac {1}{8} \int \frac {1}{x (2 x+\log (x)+8)}dx-2 \int \frac {x}{2 x+\log (x)+8}dx-\frac {1}{8} \int \frac {1}{(x+4) (2 x+\log (x)+8)}dx+2 \int \frac {\log (\log (x))}{(2 x+\log (x)+8)^2}dx+\int \frac {\log (\log (x))}{x (2 x+\log (x)+8)^2}dx\) |
Int[(-8 - 2*x + (7 + 9*x - 15*x^2 - 2*x^3)*Log[x] + (-x - 2*x^2)*Log[x]^2 + (1 + 2*x)*Log[x]*Log[Log[x]])/((64*x + 32*x^2 + 4*x^3)*Log[x] + (16*x + 4*x^2)*Log[x]^2 + x*Log[x]^3),x]
3.10.17.3.1 Defintions of rubi rules used
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
parallelrisch | \(\frac {-8-x^{2}-x -\ln \left (\ln \left (x \right )\right )}{8+2 x +\ln \left (x \right )}\) | \(26\) |
risch | \(-\frac {\ln \left (\ln \left (x \right )\right )}{8+2 x +\ln \left (x \right )}-\frac {x^{2}+x +8}{8+2 x +\ln \left (x \right )}\) | \(33\) |
int(((1+2*x)*ln(x)*ln(ln(x))+(-2*x^2-x)*ln(x)^2+(-2*x^3-15*x^2+9*x+7)*ln(x )-2*x-8)/(x*ln(x)^3+(4*x^2+16*x)*ln(x)^2+(4*x^3+32*x^2+64*x)*ln(x)),x,meth od=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x^{2} + x + \log \left (\log \left (x\right )\right ) + 8}{2 \, x + \log \left (x\right ) + 8} \]
integrate(((1+2*x)*log(x)*log(log(x))+(-2*x^2-x)*log(x)^2+(-2*x^3-15*x^2+9 *x+7)*log(x)-2*x-8)/(x*log(x)^3+(4*x^2+16*x)*log(x)^2+(4*x^3+32*x^2+64*x)* log(x)),x, algorithm=\
Exception generated. \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]
integrate(((1+2*x)*ln(x)*ln(ln(x))+(-2*x**2-x)*ln(x)**2+(-2*x**3-15*x**2+9 *x+7)*ln(x)-2*x-8)/(x*ln(x)**3+(4*x**2+16*x)*ln(x)**2+(4*x**3+32*x**2+64*x )*ln(x)),x)
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x^{2} + x + \log \left (\log \left (x\right )\right ) + 8}{2 \, x + \log \left (x\right ) + 8} \]
integrate(((1+2*x)*log(x)*log(log(x))+(-2*x^2-x)*log(x)^2+(-2*x^3-15*x^2+9 *x+7)*log(x)-2*x-8)/(x*log(x)^3+(4*x^2+16*x)*log(x)^2+(4*x^3+32*x^2+64*x)* log(x)),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x^{2} + x + 8}{2 \, x + \log \left (x\right ) + 8} - \frac {\log \left (\log \left (x\right )\right )}{2 \, x + \log \left (x\right ) + 8} \]
integrate(((1+2*x)*log(x)*log(log(x))+(-2*x^2-x)*log(x)^2+(-2*x^3-15*x^2+9 *x+7)*log(x)-2*x-8)/(x*log(x)^3+(4*x^2+16*x)*log(x)^2+(4*x^3+32*x^2+64*x)* log(x)),x, algorithm=\
Timed out. \[ \int \frac {-8-2 x+\left (7+9 x-15 x^2-2 x^3\right ) \log (x)+\left (-x-2 x^2\right ) \log ^2(x)+(1+2 x) \log (x) \log (\log (x))}{\left (64 x+32 x^2+4 x^3\right ) \log (x)+\left (16 x+4 x^2\right ) \log ^2(x)+x \log ^3(x)} \, dx=\int -\frac {2\,x+{\ln \left (x\right )}^2\,\left (2\,x^2+x\right )-\ln \left (x\right )\,\left (-2\,x^3-15\,x^2+9\,x+7\right )-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x+1\right )+8}{x\,{\ln \left (x\right )}^3+\left (4\,x^2+16\,x\right )\,{\ln \left (x\right )}^2+\left (4\,x^3+32\,x^2+64\,x\right )\,\ln \left (x\right )} \,d x \]
int(-(2*x + log(x)^2*(x + 2*x^2) - log(x)*(9*x - 15*x^2 - 2*x^3 + 7) - log (log(x))*log(x)*(2*x + 1) + 8)/(log(x)^2*(16*x + 4*x^2) + x*log(x)^3 + log (x)*(64*x + 32*x^2 + 4*x^3)),x)