Integrand size = 113, antiderivative size = 30 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=\frac {2 x \log (x)}{e^{e^{\log ^2(2)}-e^{x^2} (-3+x)}-x} \]
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=2 \left (-\log (x)-\frac {e^{e^{\log ^2(2)}} \log (x)}{-e^{e^{\log ^2(2)}}+e^{e^{x^2} (-3+x)} x}\right ) \]
Integrate[(-2*E^(2*E^x^2*(-3 + x))*x + E^(E^Log[2]^2 + E^x^2*(-3 + x))*(2 + (2 + E^x^2*(2*x - 12*x^2 + 4*x^3))*Log[x]))/(E^(2*E^Log[2]^2) - 2*E^(E^L og[2]^2 + E^x^2*(-3 + x))*x + E^(2*E^x^2*(-3 + x))*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.
\(\displaystyle \int \frac {e^{e^{x^2} (x-3)+e^{\log ^2(2)}} \left (\left (e^{x^2} \left (4 x^3-12 x^2+2 x\right )+2\right ) \log (x)+2\right )-2 e^{2 e^{x^2} (x-3)} x}{e^{2 e^{x^2} (x-3)} x^2-2 x e^{e^{x^2} (x-3)+e^{\log ^2(2)}}+e^{2 e^{\log ^2(2)}}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{e^{x^2} (x-3)+e^{\log ^2(2)}} \left (\left (e^{x^2} \left (4 x^3-12 x^2+2 x\right )+2\right ) \log (x)+2\right )-2 e^{2 e^{x^2} (x-3)} x}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x \left (2 x^2-6 x+1\right ) e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} \log (x)}{\left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}}\right )^2}-\frac {2 e^{e^{x^2} (x-3)} \left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}} \log (x)-e^{e^{\log ^2(2)}}\right )}{\left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \log (x) \int \frac {e^{e^{x^2} (x-3)+e^{\log ^2(2)}}}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx+2 \int \frac {e^{e^{x^2} (x-3)}}{e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x}dx+2 \log (x) \int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x}{\left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}}\right )^2}dx-12 \log (x) \int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x^2}{\left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}}\right )^2}dx-2 \int \frac {\int \frac {e^{e^{x^2} (x-3)+e^{\log ^2(2)}}}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx}{x}dx-2 \int \frac {\int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx}{x}dx+12 \int \frac {\int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x^2}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx}{x}dx+4 \log (x) \int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x^3}{\left (e^{e^{x^2} (x-3)} x-e^{e^{\log ^2(2)}}\right )^2}dx-4 \int \frac {\int \frac {e^{x^2+e^{x^2} (x-3)+e^{\log ^2(2)}} x^3}{\left (e^{e^{\log ^2(2)}}-e^{e^{x^2} (x-3)} x\right )^2}dx}{x}dx\) |
Int[(-2*E^(2*E^x^2*(-3 + x))*x + E^(E^Log[2]^2 + E^x^2*(-3 + x))*(2 + (2 + E^x^2*(2*x - 12*x^2 + 4*x^3))*Log[x]))/(E^(2*E^Log[2]^2) - 2*E^(E^Log[2]^ 2 + E^x^2*(-3 + x))*x + E^(2*E^x^2*(-3 + x))*x^2),x]
3.28.17.3.1 Defintions of rubi rules used
Time = 6.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
parallelrisch | \(\frac {2 x \ln \left (x \right ) {\mathrm e}^{\left (-3+x \right ) {\mathrm e}^{x^{2}}}}{-x \,{\mathrm e}^{\left (-3+x \right ) {\mathrm e}^{x^{2}}}+{\mathrm e}^{{\mathrm e}^{\ln \left (2\right )^{2}}}}\) | \(36\) |
risch | \(-2 \ln \left (x \right )+\frac {2 \,{\mathrm e}^{{\mathrm e}^{\ln \left (2\right )^{2}}} \ln \left (x \right )}{-x \,{\mathrm e}^{\left (-3+x \right ) {\mathrm e}^{x^{2}}}+{\mathrm e}^{{\mathrm e}^{\ln \left (2\right )^{2}}}}\) | \(37\) |
int(((((4*x^3-12*x^2+2*x)*exp(x^2)+2)*ln(x)+2)*exp((-3+x)*exp(x^2))*exp(ex p(ln(2)^2))-2*x*exp((-3+x)*exp(x^2))^2)/(exp(exp(ln(2)^2))^2-2*x*exp((-3+x )*exp(x^2))*exp(exp(ln(2)^2))+x^2*exp((-3+x)*exp(x^2))^2),x,method=_RETURN VERBOSE)
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=-\frac {2 \, x e^{\left ({\left (x - 3\right )} e^{\left (x^{2}\right )} + e^{\left (\log \left (2\right )^{2}\right )}\right )} \log \left (x\right )}{x e^{\left ({\left (x - 3\right )} e^{\left (x^{2}\right )} + e^{\left (\log \left (2\right )^{2}\right )}\right )} - e^{\left (2 \, e^{\left (\log \left (2\right )^{2}\right )}\right )}} \]
integrate(((((4*x^3-12*x^2+2*x)*exp(x^2)+2)*log(x)+2)*exp((-3+x)*exp(x^2)) *exp(exp(log(2)^2))-2*x*exp((-3+x)*exp(x^2))^2)/(exp(exp(log(2)^2))^2-2*x* exp((-3+x)*exp(x^2))*exp(exp(log(2)^2))+x^2*exp((-3+x)*exp(x^2))^2),x, alg orithm=\
-2*x*e^((x - 3)*e^(x^2) + e^(log(2)^2))*log(x)/(x*e^((x - 3)*e^(x^2) + e^( log(2)^2)) - e^(2*e^(log(2)^2)))
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=- 2 \log {\left (x \right )} - \frac {2 e^{e^{\log {\left (2 \right )}^{2}}} \log {\left (x \right )}}{x e^{\left (x - 3\right ) e^{x^{2}}} - e^{e^{\log {\left (2 \right )}^{2}}}} \]
integrate(((((4*x**3-12*x**2+2*x)*exp(x**2)+2)*ln(x)+2)*exp((-3+x)*exp(x** 2))*exp(exp(ln(2)**2))-2*x*exp((-3+x)*exp(x**2))**2)/(exp(exp(ln(2)**2))** 2-2*x*exp((-3+x)*exp(x**2))*exp(exp(ln(2)**2))+x**2*exp((-3+x)*exp(x**2))* *2),x)
Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=-\frac {2 \, e^{\left (3 \, e^{\left (x^{2}\right )} + e^{\left (\log \left (2\right )^{2}\right )}\right )} \log \left (x\right )}{x e^{\left (x e^{\left (x^{2}\right )}\right )} - e^{\left (3 \, e^{\left (x^{2}\right )} + e^{\left (\log \left (2\right )^{2}\right )}\right )}} - 2 \, \log \left (x\right ) \]
integrate(((((4*x^3-12*x^2+2*x)*exp(x^2)+2)*log(x)+2)*exp((-3+x)*exp(x^2)) *exp(exp(log(2)^2))-2*x*exp((-3+x)*exp(x^2))^2)/(exp(exp(log(2)^2))^2-2*x* exp((-3+x)*exp(x^2))*exp(exp(log(2)^2))+x^2*exp((-3+x)*exp(x^2))^2),x, alg orithm=\
-2*e^(3*e^(x^2) + e^(log(2)^2))*log(x)/(x*e^(x*e^(x^2)) - e^(3*e^(x^2) + e ^(log(2)^2))) - 2*log(x)
Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=-\frac {2 \, x e^{\left (x e^{\left (x^{2}\right )} - 3 \, e^{\left (x^{2}\right )}\right )} \log \left (x\right )}{x e^{\left (x e^{\left (x^{2}\right )} - 3 \, e^{\left (x^{2}\right )}\right )} - e^{\left (e^{\left (\log \left (2\right )^{2}\right )}\right )}} \]
integrate(((((4*x^3-12*x^2+2*x)*exp(x^2)+2)*log(x)+2)*exp((-3+x)*exp(x^2)) *exp(exp(log(2)^2))-2*x*exp((-3+x)*exp(x^2))^2)/(exp(exp(log(2)^2))^2-2*x* exp((-3+x)*exp(x^2))*exp(exp(log(2)^2))+x^2*exp((-3+x)*exp(x^2))^2),x, alg orithm=\
Time = 9.54 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.37 \[ \int \frac {-2 e^{2 e^{x^2} (-3+x)} x+e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} \left (2+\left (2+e^{x^2} \left (2 x-12 x^2+4 x^3\right )\right ) \log (x)\right )}{e^{2 e^{\log ^2(2)}}-2 e^{e^{\log ^2(2)}+e^{x^2} (-3+x)} x+e^{2 e^{x^2} (-3+x)} x^2} \, dx=-2\,\ln \left (x\right )-\frac {2\,\left (x^2\,{\mathrm {e}}^{x^2+2\,{\mathrm {e}}^{{\ln \left (2\right )}^2}}\,\ln \left (x\right )-6\,x^3\,{\mathrm {e}}^{x^2+2\,{\mathrm {e}}^{{\ln \left (2\right )}^2}}\,\ln \left (x\right )+2\,x^4\,{\mathrm {e}}^{x^2+2\,{\mathrm {e}}^{{\ln \left (2\right )}^2}}\,\ln \left (x\right )+x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\ln \left (2\right )}^2}}\,\ln \left (x\right )\right )}{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}-3\,{\mathrm {e}}^{x^2}}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{{\ln \left (2\right )}^2}}}{x}\right )\,\left (x^3\,{\mathrm {e}}^{x^2+{\mathrm {e}}^{{\ln \left (2\right )}^2}}-6\,x^4\,{\mathrm {e}}^{x^2+{\mathrm {e}}^{{\ln \left (2\right )}^2}}+2\,x^5\,{\mathrm {e}}^{x^2+{\mathrm {e}}^{{\ln \left (2\right )}^2}}+x^2\,{\mathrm {e}}^{{\mathrm {e}}^{{\ln \left (2\right )}^2}}\right )} \]
int(-(2*x*exp(2*exp(x^2)*(x - 3)) - exp(exp(log(2)^2))*exp(exp(x^2)*(x - 3 ))*(log(x)*(exp(x^2)*(2*x - 12*x^2 + 4*x^3) + 2) + 2))/(exp(2*exp(log(2)^2 )) + x^2*exp(2*exp(x^2)*(x - 3)) - 2*x*exp(exp(log(2)^2))*exp(exp(x^2)*(x - 3))),x)
- 2*log(x) - (2*(x^2*exp(2*exp(log(2)^2) + x^2)*log(x) - 6*x^3*exp(2*exp(l og(2)^2) + x^2)*log(x) + 2*x^4*exp(2*exp(log(2)^2) + x^2)*log(x) + x*exp(2 *exp(log(2)^2))*log(x)))/((exp(x*exp(x^2) - 3*exp(x^2)) - exp(exp(log(2)^2 ))/x)*(x^3*exp(exp(log(2)^2) + x^2) - 6*x^4*exp(exp(log(2)^2) + x^2) + 2*x ^5*exp(exp(log(2)^2) + x^2) + x^2*exp(exp(log(2)^2))))