Integrand size = 153, antiderivative size = 25 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=2 \left (-2+x-\frac {x}{e^{\frac {1}{x}}+x (3+x)-\log (x)}\right ) \end {dmath*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=2 \left (x+\frac {x}{-e^{\frac {1}{x}}-3 x-x^2+\log (x)}\right ) \end {dmath*}
Integrate[(-2*x + 2*E^(2/x)*x + 20*x^3 + 12*x^4 + 2*x^5 + E^x^(-1)*(-2 - 2 *x + 12*x^2 + 4*x^3) + (2*x - 4*E^x^(-1)*x - 12*x^2 - 4*x^3)*Log[x] + 2*x* Log[x]^2)/(E^(2/x)*x + 9*x^3 + 6*x^4 + x^5 + E^x^(-1)*(6*x^2 + 2*x^3) + (- 2*E^x^(-1)*x - 6*x^2 - 2*x^3)*Log[x] + x*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.
\(\displaystyle \int \frac {2 x^5+12 x^4+20 x^3+e^{\frac {1}{x}} \left (4 x^3+12 x^2-2 x-2\right )+\left (-4 x^3-12 x^2-4 e^{\frac {1}{x}} x+2 x\right ) \log (x)+2 e^{2/x} x-2 x+2 x \log ^2(x)}{x^5+6 x^4+9 x^3+e^{\frac {1}{x}} \left (2 x^3+6 x^2\right )+\left (-2 x^3-6 x^2-2 e^{\frac {1}{x}} x\right ) \log (x)+e^{2/x} x+x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^5+12 x^4+20 x^3+e^{\frac {1}{x}} \left (4 x^3+12 x^2-2 x-2\right )+\left (-4 x^3-12 x^2-4 e^{\frac {1}{x}} x+2 x\right ) \log (x)+2 e^{2/x} x-2 x+2 x \log ^2(x)}{x \left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 (x+1)}{x \left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )}+\frac {2 \left (2 x^3+4 x^2+2 x-\log (x)\right )}{x \left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{\left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}dx+8 \int \frac {x}{\left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}dx+4 \int \frac {x^2}{\left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}dx-2 \int \frac {1}{x^2+3 x+e^{\frac {1}{x}}-\log (x)}dx-2 \int \frac {1}{x \left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )}dx-2 \int \frac {\log (x)}{x \left (x^2+3 x+e^{\frac {1}{x}}-\log (x)\right )^2}dx+2 x\) |
Int[(-2*x + 2*E^(2/x)*x + 20*x^3 + 12*x^4 + 2*x^5 + E^x^(-1)*(-2 - 2*x + 1 2*x^2 + 4*x^3) + (2*x - 4*E^x^(-1)*x - 12*x^2 - 4*x^3)*Log[x] + 2*x*Log[x] ^2)/(E^(2/x)*x + 9*x^3 + 6*x^4 + x^5 + E^x^(-1)*(6*x^2 + 2*x^3) + (-2*E^x^ (-1)*x - 6*x^2 - 2*x^3)*Log[x] + x*Log[x]^2),x]
3.13.22.3.1 Defintions of rubi rules used
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(2 x -\frac {2 x}{x^{2}+3 x +{\mathrm e}^{\frac {1}{x}}-\ln \left (x \right )}\) | \(25\) |
parallelrisch | \(\frac {2 x^{3}+6 x^{2}-2 x \ln \left (x \right )+2 x \,{\mathrm e}^{\frac {1}{x}}-2 x}{x^{2}+3 x +{\mathrm e}^{\frac {1}{x}}-\ln \left (x \right )}\) | \(45\) |
int((2*x*ln(x)^2+(-4*x*exp(1/x)-4*x^3-12*x^2+2*x)*ln(x)+2*x*exp(1/x)^2+(4* x^3+12*x^2-2*x-2)*exp(1/x)+2*x^5+12*x^4+20*x^3-2*x)/(x*ln(x)^2+(-2*x*exp(1 /x)-2*x^3-6*x^2)*ln(x)+x*exp(1/x)^2+(2*x^3+6*x^2)*exp(1/x)+x^5+6*x^4+9*x^3 ),x,method=_RETURNVERBOSE)
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {2 \, {\left (x^{3} + 3 \, x^{2} + x e^{\frac {1}{x}} - x \log \left (x\right ) - x\right )}}{x^{2} + 3 \, x + e^{\frac {1}{x}} - \log \left (x\right )} \end {dmath*}
integrate((2*x*log(x)^2+(-4*x*exp(1/x)-4*x^3-12*x^2+2*x)*log(x)+2*x*exp(1/ x)^2+(4*x^3+12*x^2-2*x-2)*exp(1/x)+2*x^5+12*x^4+20*x^3-2*x)/(x*log(x)^2+(- 2*x*exp(1/x)-2*x^3-6*x^2)*log(x)+x*exp(1/x)^2+(2*x^3+6*x^2)*exp(1/x)+x^5+6 *x^4+9*x^3),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=2 x - \frac {2 x}{x^{2} + 3 x + e^{\frac {1}{x}} - \log {\left (x \right )}} \end {dmath*}
integrate((2*x*ln(x)**2+(-4*x*exp(1/x)-4*x**3-12*x**2+2*x)*ln(x)+2*x*exp(1 /x)**2+(4*x**3+12*x**2-2*x-2)*exp(1/x)+2*x**5+12*x**4+20*x**3-2*x)/(x*ln(x )**2+(-2*x*exp(1/x)-2*x**3-6*x**2)*ln(x)+x*exp(1/x)**2+(2*x**3+6*x**2)*exp (1/x)+x**5+6*x**4+9*x**3),x)
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {2 \, {\left (x^{3} + 3 \, x^{2} + x e^{\frac {1}{x}} - x \log \left (x\right ) - x\right )}}{x^{2} + 3 \, x + e^{\frac {1}{x}} - \log \left (x\right )} \end {dmath*}
integrate((2*x*log(x)^2+(-4*x*exp(1/x)-4*x^3-12*x^2+2*x)*log(x)+2*x*exp(1/ x)^2+(4*x^3+12*x^2-2*x-2)*exp(1/x)+2*x^5+12*x^4+20*x^3-2*x)/(x*log(x)^2+(- 2*x*exp(1/x)-2*x^3-6*x^2)*log(x)+x*exp(1/x)^2+(2*x^3+6*x^2)*exp(1/x)+x^5+6 *x^4+9*x^3),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {2 \, {\left (x^{3} + 3 \, x^{2} + x e^{\frac {1}{x}} - x \log \left (x\right ) - x\right )}}{x^{2} + 3 \, x + e^{\frac {1}{x}} - \log \left (x\right )} \end {dmath*}
integrate((2*x*log(x)^2+(-4*x*exp(1/x)-4*x^3-12*x^2+2*x)*log(x)+2*x*exp(1/ x)^2+(4*x^3+12*x^2-2*x-2)*exp(1/x)+2*x^5+12*x^4+20*x^3-2*x)/(x*log(x)^2+(- 2*x*exp(1/x)-2*x^3-6*x^2)*log(x)+x*exp(1/x)^2+(2*x^3+6*x^2)*exp(1/x)+x^5+6 *x^4+9*x^3),x, algorithm=\
Time = 15.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {-2 x+2 e^{2/x} x+20 x^3+12 x^4+2 x^5+e^{\frac {1}{x}} \left (-2-2 x+12 x^2+4 x^3\right )+\left (2 x-4 e^{\frac {1}{x}} x-12 x^2-4 x^3\right ) \log (x)+2 x \log ^2(x)}{e^{2/x} x+9 x^3+6 x^4+x^5+e^{\frac {1}{x}} \left (6 x^2+2 x^3\right )+\left (-2 e^{\frac {1}{x}} x-6 x^2-2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=2\,x-\frac {2\,x}{3\,x+{\mathrm {e}}^{1/x}-\ln \left (x\right )+x^2} \end {dmath*}