Integrand size = 91, antiderivative size = 27 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=-x+\log \left (x \left (x-x \left (-1+625 x^4-\frac {4}{-4+\log (x)}\right )\right )\right ) \]
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=-x+2 \log (x)-\log (4-\log (x))+\log \left (4-2500 x^4-2 \log (x)+625 x^4 \log (x)\right ) \]
Integrate[(-28 + 16*x + 60000*x^4 - 10000*x^5 + (24 - 12*x - 30000*x^4 + 5 000*x^5)*Log[x] + (-4 + 2*x + 3750*x^4 - 625*x^5)*Log[x]^2)/(-16*x + 10000 *x^5 + (12*x - 5000*x^5)*Log[x] + (-2*x + 625*x^5)*Log[x]^2),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.
\(\displaystyle \int \frac {-10000 x^5+60000 x^4+\left (-625 x^5+3750 x^4+2 x-4\right ) \log ^2(x)+\left (5000 x^5-30000 x^4-12 x+24\right ) \log (x)+16 x-28}{10000 x^5+\left (625 x^5-2 x\right ) \log ^2(x)+\left (12 x-5000 x^5\right ) \log (x)-16 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {10000 x^5-60000 x^4-\left (-625 x^5+3750 x^4+2 x-4\right ) \log ^2(x)-\left (5000 x^5-30000 x^4-12 x+24\right ) \log (x)-16 x+28}{x (4-\log (x)) \left (-2500 x^4+625 x^4 \log (x)-2 \log (x)+4\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {390625 x^8+7500 x^4+4}{x \left (625 x^4-2\right ) \left (-2500 x^4+625 x^4 \log (x)-2 \log (x)+4\right )}+\frac {-625 x^5+3750 x^4+2 x-4}{x \left (625 x^4-2\right )}-\frac {1}{x (\log (x)-4)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -20 \int \frac {1}{\left (i \sqrt [4]{2}-5 x\right ) \left (625 \log (x) x^4-2500 x^4-2 \log (x)+4\right )}dx-20 \int \frac {1}{\left (\sqrt [4]{2}-5 x\right ) \left (625 \log (x) x^4-2500 x^4-2 \log (x)+4\right )}dx-2 \int \frac {1}{x \left (625 \log (x) x^4-2500 x^4-2 \log (x)+4\right )}dx+20 \int \frac {1}{\left (5 x+i \sqrt [4]{2}\right ) \left (625 \log (x) x^4-2500 x^4-2 \log (x)+4\right )}dx+20 \int \frac {1}{\left (5 x+\sqrt [4]{2}\right ) \left (625 \log (x) x^4-2500 x^4-2 \log (x)+4\right )}dx+625 \int \frac {x^3}{625 \log (x) x^4-2500 x^4-2 \log (x)+4}dx+\log \left (2-625 x^4\right )-x+2 \log (x)-\log (4-\log (x))\) |
Int[(-28 + 16*x + 60000*x^4 - 10000*x^5 + (24 - 12*x - 30000*x^4 + 5000*x^ 5)*Log[x] + (-4 + 2*x + 3750*x^4 - 625*x^5)*Log[x]^2)/(-16*x + 10000*x^5 + (12*x - 5000*x^5)*Log[x] + (-2*x + 625*x^5)*Log[x]^2),x]
3.13.95.3.1 Defintions of rubi rules used
Time = 2.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26
method | result | size |
parallelrisch | \(-\ln \left (\ln \left (x \right )-4\right )+\ln \left (x^{4} \ln \left (x \right )-4 x^{4}-\frac {2 \ln \left (x \right )}{625}+\frac {4}{625}\right )-x +2 \ln \left (x \right )\) | \(34\) |
default | \(2 \ln \left (x \right )-x +\ln \left (625 x^{4} \ln \left (x \right )-2500 x^{4}-2 \ln \left (x \right )+4\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(35\) |
norman | \(2 \ln \left (x \right )-x +\ln \left (625 x^{4} \ln \left (x \right )-2500 x^{4}-2 \ln \left (x \right )+4\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(35\) |
risch | \(-x +2 \ln \left (x \right )+\ln \left (625 x^{4}-2\right )+\ln \left (\ln \left (x \right )-\frac {4 \left (625 x^{4}-1\right )}{625 x^{4}-2}\right )-\ln \left (\ln \left (x \right )-4\right )\) | \(46\) |
int(((-625*x^5+3750*x^4+2*x-4)*ln(x)^2+(5000*x^5-30000*x^4-12*x+24)*ln(x)- 10000*x^5+60000*x^4+16*x-28)/((625*x^5-2*x)*ln(x)^2+(-5000*x^5+12*x)*ln(x) +10000*x^5-16*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=-x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \left (x\right ) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \left (x\right ) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \left (x\right ) - 4\right ) \]
integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24) *log(x)-10000*x^5+60000*x^4+16*x-28)/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12 *x)*log(x)+10000*x^5-16*x),x, algorithm=\
-x + log(625*x^4 - 2) + 2*log(x) + log(-(2500*x^4 - (625*x^4 - 2)*log(x) - 4)/(625*x^4 - 2)) - log(log(x) - 4)
Exception generated. \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((-625*x**5+3750*x**4+2*x-4)*ln(x)**2+(5000*x**5-30000*x**4-12*x +24)*ln(x)-10000*x**5+60000*x**4+16*x-28)/((625*x**5-2*x)*ln(x)**2+(-5000* x**5+12*x)*ln(x)+10000*x**5-16*x),x)
Exception raised: PolynomialError >> 1/(390625*x**9 - 2500*x**5 + 4*x) con tains an element of the set of generators.
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=-x + \log \left (625 \, x^{4} - 2\right ) + 2 \, \log \left (x\right ) + \log \left (-\frac {2500 \, x^{4} - {\left (625 \, x^{4} - 2\right )} \log \left (x\right ) - 4}{625 \, x^{4} - 2}\right ) - \log \left (\log \left (x\right ) - 4\right ) \]
integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24) *log(x)-10000*x^5+60000*x^4+16*x-28)/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12 *x)*log(x)+10000*x^5-16*x),x, algorithm=\
-x + log(625*x^4 - 2) + 2*log(x) + log(-(2500*x^4 - (625*x^4 - 2)*log(x) - 4)/(625*x^4 - 2)) - log(log(x) - 4)
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=-x + \log \left (625 \, x^{4} \log \left (x\right ) - 2500 \, x^{4} - 2 \, \log \left (x\right ) + 4\right ) + 2 \, \log \left (x\right ) - \log \left (\log \left (x\right ) - 4\right ) \]
integrate(((-625*x^5+3750*x^4+2*x-4)*log(x)^2+(5000*x^5-30000*x^4-12*x+24) *log(x)-10000*x^5+60000*x^4+16*x-28)/((625*x^5-2*x)*log(x)^2+(-5000*x^5+12 *x)*log(x)+10000*x^5-16*x),x, algorithm=\
Time = 14.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {-28+16 x+60000 x^4-10000 x^5+\left (24-12 x-30000 x^4+5000 x^5\right ) \log (x)+\left (-4+2 x+3750 x^4-625 x^5\right ) \log ^2(x)}{-16 x+10000 x^5+\left (12 x-5000 x^5\right ) \log (x)+\left (-2 x+625 x^5\right ) \log ^2(x)} \, dx=\ln \left (390625\,x^8+7500\,x^4+4\right )-x+\ln \left (\frac {4\,\ln \left (x\right )-1250\,x^4\,\ln \left (x\right )+5000\,x^4-8}{2\,x-625\,x^5}\right )-\ln \left (x^4-\frac {2}{625}\right )-\ln \left (\frac {\left (\ln \left (x\right )-4\right )\,\left (390625\,x^8+7500\,x^4+4\right )}{x\,{\left (625\,x^4-2\right )}^2}\right )+2\,\ln \left (x\right ) \]
int(-(16*x + log(x)^2*(2*x + 3750*x^4 - 625*x^5 - 4) + 60000*x^4 - 10000*x ^5 - log(x)*(12*x + 30000*x^4 - 5000*x^5 - 24) - 28)/(16*x + log(x)^2*(2*x - 625*x^5) - log(x)*(12*x - 5000*x^5) - 10000*x^5),x)