Integrand size = 103, antiderivative size = 26 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=x-\log (5)-\frac {\log (5)}{x+\left (3 e^2-x\right ) \log (x)} \]
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=x-\frac {\log (5)}{x+3 e^2 \log (x)-x \log (x)} \]
Integrate[(x^3 + 3*E^2*Log[5] + (6*E^2*x^2 - 2*x^3 - x*Log[5])*Log[x] + (9 *E^4*x - 6*E^2*x^2 + x^3)*Log[x]^2)/(x^3 + (6*E^2*x^2 - 2*x^3)*Log[x] + (9 *E^4*x - 6*E^2*x^2 + x^3)*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.
\(\displaystyle \int \frac {x^3+\left (x^3-6 e^2 x^2+9 e^4 x\right ) \log ^2(x)+\left (-2 x^3+6 e^2 x^2-x \log (5)\right ) \log (x)+3 e^2 \log (5)}{x^3+\left (x^3-6 e^2 x^2+9 e^4 x\right ) \log ^2(x)+\left (6 e^2 x^2-2 x^3\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^3+\left (x^3-6 e^2 x^2+9 e^4 x\right ) \log ^2(x)+\left (-2 x^3+6 e^2 x^2-x \log (5)\right ) \log (x)+3 e^2 \log (5)}{x \left (x+x (-\log (x))+3 e^2 \log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\left (x^2-3 e^2 x+9 e^4\right ) \log (5)}{x \left (x-3 e^2\right ) \left (-x+x \log (x)-3 e^2 \log (x)\right )^2}-\frac {\log (5)}{\left (3 e^2-x\right ) \left (x+x (-\log (x))+3 e^2 \log (x)\right )}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 e^2 \log (5) \int \frac {1}{\left (3 e^2-x\right ) \left (-\log (x) x+x+3 e^2 \log (x)\right )^2}dx-\log (5) \int \frac {1}{\left (3 e^2-x\right ) \left (-\log (x) x+x+3 e^2 \log (x)\right )}dx-\log (5) \int \frac {1}{\left (\log (x) x-x-3 e^2 \log (x)\right )^2}dx+3 e^2 \log (5) \int \frac {1}{x \left (\log (x) x-x-3 e^2 \log (x)\right )^2}dx+x\) |
Int[(x^3 + 3*E^2*Log[5] + (6*E^2*x^2 - 2*x^3 - x*Log[5])*Log[x] + (9*E^4*x - 6*E^2*x^2 + x^3)*Log[x]^2)/(x^3 + (6*E^2*x^2 - 2*x^3)*Log[x] + (9*E^4*x - 6*E^2*x^2 + x^3)*Log[x]^2),x]
3.26.99.3.1 Defintions of rubi rules used
Time = 0.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
method | result | size |
default | \(x -\frac {\ln \left (5\right )}{3 \,{\mathrm e}^{2} \ln \left (x \right )-x \ln \left (x \right )+x}\) | \(22\) |
risch | \(x -\frac {\ln \left (5\right )}{3 \,{\mathrm e}^{2} \ln \left (x \right )-x \ln \left (x \right )+x}\) | \(22\) |
norman | \(\frac {x^{2}+3 \,{\mathrm e}^{2} x -x^{2} \ln \left (x \right )+9 \,{\mathrm e}^{4} \ln \left (x \right )-\ln \left (5\right )}{3 \,{\mathrm e}^{2} \ln \left (x \right )-x \ln \left (x \right )+x}\) | \(45\) |
parallelrisch | \(\frac {x^{2}+3 \,{\mathrm e}^{2} x -x^{2} \ln \left (x \right )+9 \,{\mathrm e}^{4} \ln \left (x \right )-\ln \left (5\right )}{3 \,{\mathrm e}^{2} \ln \left (x \right )-x \ln \left (x \right )+x}\) | \(45\) |
int(((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*ln(x)^2+(-x*ln(5)+6*x^2*exp(2)-2*x^3) *ln(x)+3*exp(2)*ln(5)+x^3)/((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*ln(x)^2+(6*x^2 *exp(2)-2*x^3)*ln(x)+x^3),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=-\frac {x^{2} - {\left (x^{2} - 3 \, x e^{2}\right )} \log \left (x\right ) - \log \left (5\right )}{{\left (x - 3 \, e^{2}\right )} \log \left (x\right ) - x} \]
integrate(((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log(x)^2+(-x*log(5)+6*x^2*exp(2 )-2*x^3)*log(x)+3*exp(2)*log(5)+x^3)/((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log( x)^2+(6*x^2*exp(2)-2*x^3)*log(x)+x^3),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.58 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=x + \frac {\log {\left (5 \right )}}{- x + \left (x - 3 e^{2}\right ) \log {\left (x \right )}} \]
integrate(((9*x*exp(2)**2-6*x**2*exp(2)+x**3)*ln(x)**2+(-x*ln(5)+6*x**2*ex p(2)-2*x**3)*ln(x)+3*exp(2)*ln(5)+x**3)/((9*x*exp(2)**2-6*x**2*exp(2)+x**3 )*ln(x)**2+(6*x**2*exp(2)-2*x**3)*ln(x)+x**3),x)
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=-\frac {x^{2} - {\left (x^{2} - 3 \, x e^{2}\right )} \log \left (x\right ) - \log \left (5\right )}{{\left (x - 3 \, e^{2}\right )} \log \left (x\right ) - x} \]
integrate(((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log(x)^2+(-x*log(5)+6*x^2*exp(2 )-2*x^3)*log(x)+3*exp(2)*log(5)+x^3)/((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log( x)^2+(6*x^2*exp(2)-2*x^3)*log(x)+x^3),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) - 3 \, x e^{2} \log \left (x\right ) - x^{2} + \log \left (5\right )}{x \log \left (x\right ) - 3 \, e^{2} \log \left (x\right ) - x} \]
integrate(((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log(x)^2+(-x*log(5)+6*x^2*exp(2 )-2*x^3)*log(x)+3*exp(2)*log(5)+x^3)/((9*x*exp(2)^2-6*x^2*exp(2)+x^3)*log( x)^2+(6*x^2*exp(2)-2*x^3)*log(x)+x^3),x, algorithm=\
Time = 11.58 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {x^3+3 e^2 \log (5)+\left (6 e^2 x^2-2 x^3-x \log (5)\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)}{x^3+\left (6 e^2 x^2-2 x^3\right ) \log (x)+\left (9 e^4 x-6 e^2 x^2+x^3\right ) \log ^2(x)} \, dx=-\frac {\ln \left (5\right )+x^2\,\ln \left (x\right )-x^2-3\,x\,{\mathrm {e}}^2\,\ln \left (x\right )}{x+3\,{\mathrm {e}}^2\,\ln \left (x\right )-x\,\ln \left (x\right )} \]