Integrand size = 103, antiderivative size = 27 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=5-\frac {25+\log \left (\frac {\left (e^{x^2}+x-\log (x)\right )^2}{x^2}\right )}{x} \]
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=-\frac {25}{x}-\frac {\log \left (\frac {\left (e^{x^2}+x-\log (x)\right )^2}{x^2}\right )}{x} \]
Integrate[(-2 - 25*x + E^x^2*(-27 + 4*x^2) + 27*Log[x] + (-E^x^2 - x + Log [x])*Log[(E^(2*x^2) + 2*E^x^2*x + x^2 + (-2*E^x^2 - 2*x)*Log[x] + Log[x]^2 )/x^2])/(-(E^x^2*x^2) - x^3 + x^2*Log[x]),x]
Time = 1.59 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {e^{x^2} \left (4 x^2-27\right )+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {x^2+2 e^{x^2} x+e^{2 x^2}+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )-25 x+27 \log (x)-2}{-x^3-e^{x^2} x^2+x^2 \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-4 x^2+\log \left (\frac {\left (e^{x^2}+x-\log (x)\right )^2}{x^2}\right )+27}{x^2}+\frac {2 \left (2 x^3-2 x^2 \log (x)-x+1\right )}{x^2 \left (e^{x^2}+x-\log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log \left (\frac {\left (e^{x^2}+x-\log (x)\right )^2}{x^2}\right )}{x}-\frac {25}{x}\) |
Int[(-2 - 25*x + E^x^2*(-27 + 4*x^2) + 27*Log[x] + (-E^x^2 - x + Log[x])*L og[(E^(2*x^2) + 2*E^x^2*x + x^2 + (-2*E^x^2 - 2*x)*Log[x] + Log[x]^2)/x^2] )/(-(E^x^2*x^2) - x^3 + x^2*Log[x]),x]
3.13.56.3.1 Defintions of rubi rules used
Time = 0.48 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81
method | result | size |
parallelrisch | \(\frac {-50-2 \ln \left (\frac {\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{x^{2}}-2 x \right ) \ln \left (x \right )+{\mathrm e}^{2 x^{2}}+2 \,{\mathrm e}^{x^{2}} x +x^{2}}{x^{2}}\right )}{2 x}\) | \(49\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{x^{2}}-\ln \left (x \right )+x \right )}{x}+\frac {i \pi {\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )\right )}^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )\right ) {\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}\right )}^{2}+i \pi {\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}\right )}^{3}-i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}}{x^{2}}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}}{x^{2}}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-{\mathrm e}^{x^{2}}-x \right )^{2}}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 \ln \left (x \right )-50}{2 x}\) | \(338\) |
int(((ln(x)-exp(x^2)-x)*ln((ln(x)^2+(-2*exp(x^2)-2*x)*ln(x)+exp(x^2)^2+2*e xp(x^2)*x+x^2)/x^2)+27*ln(x)+(4*x^2-27)*exp(x^2)-25*x-2)/(x^2*ln(x)-x^2*ex p(x^2)-x^3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=-\frac {\log \left (\frac {x^{2} + 2 \, x e^{\left (x^{2}\right )} - 2 \, {\left (x + e^{\left (x^{2}\right )}\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{\left (2 \, x^{2}\right )}}{x^{2}}\right ) + 25}{x} \]
integrate(((log(x)-exp(x^2)-x)*log((log(x)^2+(-2*exp(x^2)-2*x)*log(x)+exp( x^2)^2+2*exp(x^2)*x+x^2)/x^2)+27*log(x)+(4*x^2-27)*exp(x^2)-25*x-2)/(x^2*l og(x)-x^2*exp(x^2)-x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=- \frac {\log {\left (\frac {x^{2} + 2 x e^{x^{2}} + \left (- 2 x - 2 e^{x^{2}}\right ) \log {\left (x \right )} + e^{2 x^{2}} + \log {\left (x \right )}^{2}}{x^{2}} \right )}}{x} - \frac {25}{x} \]
integrate(((ln(x)-exp(x**2)-x)*ln((ln(x)**2+(-2*exp(x**2)-2*x)*ln(x)+exp(x **2)**2+2*exp(x**2)*x+x**2)/x**2)+27*ln(x)+(4*x**2-27)*exp(x**2)-25*x-2)/( x**2*ln(x)-x**2*exp(x**2)-x**3),x)
-log((x**2 + 2*x*exp(x**2) + (-2*x - 2*exp(x**2))*log(x) + exp(2*x**2) + l og(x)**2)/x**2)/x - 25/x
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=\frac {2 \, \log \left (x\right ) - 2 \, \log \left (-x - e^{\left (x^{2}\right )} + \log \left (x\right )\right ) - 25}{x} \]
integrate(((log(x)-exp(x^2)-x)*log((log(x)^2+(-2*exp(x^2)-2*x)*log(x)+exp( x^2)^2+2*exp(x^2)*x+x^2)/x^2)+27*log(x)+(4*x^2-27)*exp(x^2)-25*x-2)/(x^2*l og(x)-x^2*exp(x^2)-x^3),x, algorithm=\
Time = 0.46 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=-\frac {\log \left (x^{2} + 2 \, x e^{\left (x^{2}\right )} - 2 \, x \log \left (x\right ) - 2 \, e^{\left (x^{2}\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{\left (2 \, x^{2}\right )}\right ) - 2 \, \log \left (x\right ) + 25}{x} \]
integrate(((log(x)-exp(x^2)-x)*log((log(x)^2+(-2*exp(x^2)-2*x)*log(x)+exp( x^2)^2+2*exp(x^2)*x+x^2)/x^2)+27*log(x)+(4*x^2-27)*exp(x^2)-25*x-2)/(x^2*l og(x)-x^2*exp(x^2)-x^3),x, algorithm=\
-(log(x^2 + 2*x*e^(x^2) - 2*x*log(x) - 2*e^(x^2)*log(x) + log(x)^2 + e^(2* x^2)) - 2*log(x) + 25)/x
Time = 12.97 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {-2-25 x+e^{x^2} \left (-27+4 x^2\right )+27 \log (x)+\left (-e^{x^2}-x+\log (x)\right ) \log \left (\frac {e^{2 x^2}+2 e^{x^2} x+x^2+\left (-2 e^{x^2}-2 x\right ) \log (x)+\log ^2(x)}{x^2}\right )}{-e^{x^2} x^2-x^3+x^2 \log (x)} \, dx=-\frac {\ln \left (\frac {{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-\ln \left (x\right )\,\left (2\,x+2\,{\mathrm {e}}^{x^2}\right )+{\ln \left (x\right )}^2+x^2}{x^2}\right )+25}{x} \]