Integrand size = 148, antiderivative size = 32 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=25-\frac {x-\frac {e^x}{\left (2-e^{x^2}+x^2\right ) \log (x)}}{x^2} \]
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=-\frac {1}{x}+\frac {e^x}{x^2 \left (2-e^{x^2}+x^2\right ) \log (x)} \]
Integrate[(E^(x + x^2) + E^x*(-2 - x^2) + (E^(x + x^2)*(2 - x + 2*x^2) + E ^x*(-4 + 2*x - 4*x^2 + x^3))*Log[x] + (4*x + E^(2*x^2)*x + 4*x^3 + x^5 + E ^x^2*(-4*x - 2*x^3))*Log[x]^2)/((4*x^3 + E^(2*x^2)*x^3 + 4*x^5 + x^7 + E^x ^2*(-4*x^3 - 2*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.
\(\displaystyle \int \frac {e^{x^2+x}+e^x \left (-x^2-2\right )+\left (e^{x^2+x} \left (2 x^2-x+2\right )+e^x \left (x^3-4 x^2+2 x-4\right )\right ) \log (x)+\left (x^5+4 x^3+e^{2 x^2} x+e^{x^2} \left (-2 x^3-4 x\right )+4 x\right ) \log ^2(x)}{\left (x^7+4 x^5+4 x^3+e^{2 x^2} x^3+e^{x^2} \left (-2 x^5-4 x^3\right )\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{x^2+x}+e^x \left (-x^2-2\right )+\left (e^{x^2+x} \left (2 x^2-x+2\right )+e^x \left (x^3-4 x^2+2 x-4\right )\right ) \log (x)+\left (x^5+4 x^3+e^{2 x^2} x+e^{x^2} \left (-2 x^3-4 x\right )+4 x\right ) \log ^2(x)}{x^3 \left (x^2-e^{x^2}+2\right )^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{x^2}+\frac {2 e^x \left (x^2+1\right )}{x \left (x^2-e^{x^2}+2\right )^2 \log (x)}-\frac {e^x \left (2 x^2 \log (x)-x \log (x)+2 \log (x)+1\right )}{x^3 \left (x^2-e^{x^2}+2\right ) \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {e^x}{x \left (x^2-e^{x^2}+2\right )^2 \log (x)}dx+2 \int \frac {e^x x}{\left (x^2-e^{x^2}+2\right )^2 \log (x)}dx+\int \frac {e^x}{x^2 \left (x^2-e^{x^2}+2\right ) \log (x)}dx-2 \int \frac {e^x}{x \left (x^2-e^{x^2}+2\right ) \log (x)}dx-\int \frac {e^x}{x^3 \left (x^2-e^{x^2}+2\right ) \log ^2(x)}dx-2 \int \frac {e^x}{x^3 \left (x^2-e^{x^2}+2\right ) \log (x)}dx-\frac {1}{x}\) |
Int[(E^(x + x^2) + E^x*(-2 - x^2) + (E^(x + x^2)*(2 - x + 2*x^2) + E^x*(-4 + 2*x - 4*x^2 + x^3))*Log[x] + (4*x + E^(2*x^2)*x + 4*x^3 + x^5 + E^x^2*( -4*x - 2*x^3))*Log[x]^2)/((4*x^3 + E^(2*x^2)*x^3 + 4*x^5 + x^7 + E^x^2*(-4 *x^3 - 2*x^5))*Log[x]^2),x]
3.4.90.3.1 Defintions of rubi rules used
Time = 9.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {1}{x}+\frac {{\mathrm e}^{x}}{x^{2} \left (2+x^{2}-{\mathrm e}^{x^{2}}\right ) \ln \left (x \right )}\) | \(30\) |
parallelrisch | \(-\frac {x^{3} \ln \left (x \right )-x \,{\mathrm e}^{x^{2}} \ln \left (x \right )+2 x \ln \left (x \right )-{\mathrm e}^{x}}{x^{2} \ln \left (x \right ) \left (2+x^{2}-{\mathrm e}^{x^{2}}\right )}\) | \(48\) |
int(((x*exp(x^2)^2+(-2*x^3-4*x)*exp(x^2)+x^5+4*x^3+4*x)*ln(x)^2+((2*x^2-x+ 2)*exp(x)*exp(x^2)+(x^3-4*x^2+2*x-4)*exp(x))*ln(x)+exp(x)*exp(x^2)+(-x^2-2 )*exp(x))/(x^3*exp(x^2)^2+(-2*x^5-4*x^3)*exp(x^2)+x^7+4*x^5+4*x^3)/ln(x)^2 ,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=-\frac {{\left (x e^{\left (2 \, x^{2}\right )} - {\left (x^{3} + 2 \, x\right )} e^{\left (x^{2}\right )}\right )} \log \left (x\right ) + e^{\left (x^{2} + x\right )}}{{\left (x^{2} e^{\left (2 \, x^{2}\right )} - {\left (x^{4} + 2 \, x^{2}\right )} e^{\left (x^{2}\right )}\right )} \log \left (x\right )} \]
integrate(((x*exp(x^2)^2+(-2*x^3-4*x)*exp(x^2)+x^5+4*x^3+4*x)*log(x)^2+((2 *x^2-x+2)*exp(x)*exp(x^2)+(x^3-4*x^2+2*x-4)*exp(x))*log(x)+exp(x)*exp(x^2) +(-x^2-2)*exp(x))/(x^3*exp(x^2)^2+(-2*x^5-4*x^3)*exp(x^2)+x^7+4*x^5+4*x^3) /log(x)^2,x, algorithm=\
-((x*e^(2*x^2) - (x^3 + 2*x)*e^(x^2))*log(x) + e^(x^2 + x))/((x^2*e^(2*x^2 ) - (x^4 + 2*x^2)*e^(x^2))*log(x))
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=- \frac {e^{x}}{- x^{4} \log {\left (x \right )} + x^{2} e^{x^{2}} \log {\left (x \right )} - 2 x^{2} \log {\left (x \right )}} - \frac {1}{x} \]
integrate(((x*exp(x**2)**2+(-2*x**3-4*x)*exp(x**2)+x**5+4*x**3+4*x)*ln(x)* *2+((2*x**2-x+2)*exp(x)*exp(x**2)+(x**3-4*x**2+2*x-4)*exp(x))*ln(x)+exp(x) *exp(x**2)+(-x**2-2)*exp(x))/(x**3*exp(x**2)**2+(-2*x**5-4*x**3)*exp(x**2) +x**7+4*x**5+4*x**3)/ln(x)**2,x)
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=-\frac {x e^{\left (x^{2}\right )} \log \left (x\right ) - {\left (x^{3} + 2 \, x\right )} \log \left (x\right ) + e^{x}}{x^{2} e^{\left (x^{2}\right )} \log \left (x\right ) - {\left (x^{4} + 2 \, x^{2}\right )} \log \left (x\right )} \]
integrate(((x*exp(x^2)^2+(-2*x^3-4*x)*exp(x^2)+x^5+4*x^3+4*x)*log(x)^2+((2 *x^2-x+2)*exp(x)*exp(x^2)+(x^3-4*x^2+2*x-4)*exp(x))*log(x)+exp(x)*exp(x^2) +(-x^2-2)*exp(x))/(x^3*exp(x^2)^2+(-2*x^5-4*x^3)*exp(x^2)+x^7+4*x^5+4*x^3) /log(x)^2,x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=-\frac {x^{3} \log \left (x\right ) - x e^{\left (x^{2}\right )} \log \left (x\right ) + 2 \, x \log \left (x\right ) - e^{x}}{x^{4} \log \left (x\right ) - x^{2} e^{\left (x^{2}\right )} \log \left (x\right ) + 2 \, x^{2} \log \left (x\right )} \]
integrate(((x*exp(x^2)^2+(-2*x^3-4*x)*exp(x^2)+x^5+4*x^3+4*x)*log(x)^2+((2 *x^2-x+2)*exp(x)*exp(x^2)+(x^3-4*x^2+2*x-4)*exp(x))*log(x)+exp(x)*exp(x^2) +(-x^2-2)*exp(x))/(x^3*exp(x^2)^2+(-2*x^5-4*x^3)*exp(x^2)+x^7+4*x^5+4*x^3) /log(x)^2,x, algorithm=\
-(x^3*log(x) - x*e^(x^2)*log(x) + 2*x*log(x) - e^x)/(x^4*log(x) - x^2*e^(x ^2)*log(x) + 2*x^2*log(x))
Time = 13.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{x+x^2}+e^x \left (-2-x^2\right )+\left (e^{x+x^2} \left (2-x+2 x^2\right )+e^x \left (-4+2 x-4 x^2+x^3\right )\right ) \log (x)+\left (4 x+e^{2 x^2} x+4 x^3+x^5+e^{x^2} \left (-4 x-2 x^3\right )\right ) \log ^2(x)}{\left (4 x^3+e^{2 x^2} x^3+4 x^5+x^7+e^{x^2} \left (-4 x^3-2 x^5\right )\right ) \log ^2(x)} \, dx=-\frac {x^3\,\ln \left (x\right )-{\mathrm {e}}^x+x\,\left (2\,\ln \left (x\right )-{\mathrm {e}}^{x^2}\,\ln \left (x\right )\right )}{x^2\,\ln \left (x\right )\,\left (x^2-{\mathrm {e}}^{x^2}+2\right )} \]