Integrand size = 108, antiderivative size = 26 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=4-\frac {3+x^2}{2+e^{-\frac {-7+x}{x^2}} \log (x)} \]
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=-\frac {e^{\frac {1}{x}} \left (3+x^2\right )}{2 e^{\frac {1}{x}}+e^{\frac {7}{x^2}} \log (x)} \]
Integrate[(-4*E^((2*(-7 + x))/x^2)*x^4 + E^((-7 + x)/x^2)*(3*x^2 + x^4) + E^((-7 + x)/x^2)*(-42 + 3*x - 14*x^2 + x^3 - 2*x^4)*Log[x])/(4*E^((2*(-7 + x))/x^2)*x^3 + 4*E^((-7 + x)/x^2)*x^3*Log[x] + 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.
\(\displaystyle \int \frac {-4 e^{\frac {2 (x-7)}{x^2}} x^4+e^{\frac {x-7}{x^2}} \left (x^4+3 x^2\right )+e^{\frac {x-7}{x^2}} \left (-2 x^4+x^3-14 x^2+3 x-42\right ) \log (x)}{x^3 \log ^2(x)+4 e^{\frac {2 (x-7)}{x^2}} x^3+4 e^{\frac {x-7}{x^2}} x^3 \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {14}{x^2}} \left (-4 e^{\frac {2 (x-7)}{x^2}} x^4+e^{\frac {x-7}{x^2}} \left (x^4+3 x^2\right )+e^{\frac {x-7}{x^2}} \left (-2 x^4+x^3-14 x^2+3 x-42\right ) \log (x)\right )}{x^3 \left (e^{\frac {7}{x^2}} \log (x)+2 e^{\frac {1}{x}}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^{\frac {14}{x^2}} \left (x^2+3\right ) \log (x) \left (x^2+x \log (x)-14 \log (x)\right )}{2 x^3 \left (e^{\frac {7}{x^2}} \log (x)+2 e^{\frac {1}{x}}\right )^2}+\frac {e^{\frac {7}{x^2}} \left (x^4+2 x^4 \log (x)+x^3 \log (x)+3 x^2-14 x^2 \log (x)+3 x \log (x)-42 \log (x)\right )}{2 x^3 \left (e^{\frac {7}{x^2}} \log (x)+2 e^{\frac {1}{x}}\right )}-x\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (-\frac {e^{\frac {14}{x^2}} \left (x^2+3\right ) \log (x) \left (x^2+x \log (x)-14 \log (x)\right )}{2 x^3 \left (e^{\frac {7}{x^2}} \log (x)+2 e^{\frac {1}{x}}\right )^2}+\frac {e^{\frac {7}{x^2}} \left (x^4+2 x^4 \log (x)+x^3 \log (x)+3 x^2-14 x^2 \log (x)+3 x \log (x)-42 \log (x)\right )}{2 x^3 \left (e^{\frac {7}{x^2}} \log (x)+2 e^{\frac {1}{x}}\right )}-x\right )dx\) |
Int[(-4*E^((2*(-7 + x))/x^2)*x^4 + E^((-7 + x)/x^2)*(3*x^2 + x^4) + E^((-7 + x)/x^2)*(-42 + 3*x - 14*x^2 + x^3 - 2*x^4)*Log[x])/(4*E^((2*(-7 + x))/x ^2)*x^3 + 4*E^((-7 + x)/x^2)*x^3*Log[x] + x^3*Log[x]^2),x]
3.13.11.3.1 Defintions of rubi rules used
Time = 0.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-\frac {\left (x^{2}+3\right ) {\mathrm e}^{\frac {-7+x}{x^{2}}}}{\ln \left (x \right )+2 \,{\mathrm e}^{\frac {-7+x}{x^{2}}}}\) | \(31\) |
parallelrisch | \(-\frac {x^{2} {\mathrm e}^{\frac {-7+x}{x^{2}}}+3 \,{\mathrm e}^{\frac {-7+x}{x^{2}}}}{\ln \left (x \right )+2 \,{\mathrm e}^{\frac {-7+x}{x^{2}}}}\) | \(41\) |
int(((-2*x^4+x^3-14*x^2+3*x-42)*exp((-7+x)/x^2)*ln(x)-4*x^4*exp((-7+x)/x^2 )^2+(x^4+3*x^2)*exp((-7+x)/x^2))/(x^3*ln(x)^2+4*x^3*exp((-7+x)/x^2)*ln(x)+ 4*x^3*exp((-7+x)/x^2)^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=-\frac {{\left (x^{2} + 3\right )} e^{\left (\frac {x - 7}{x^{2}}\right )}}{2 \, e^{\left (\frac {x - 7}{x^{2}}\right )} + \log \left (x\right )} \]
integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((-7+x)/x^2)*log(x)-4*x^4*exp((-7 +x)/x^2)^2+(x^4+3*x^2)*exp((-7+x)/x^2))/(x^3*log(x)^2+4*x^3*exp((-7+x)/x^2 )*log(x)+4*x^3*exp((-7+x)/x^2)^2),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=- \frac {x^{2}}{2} + \frac {x^{2} \log {\left (x \right )} + 3 \log {\left (x \right )}}{4 e^{\frac {x - 7}{x^{2}}} + 2 \log {\left (x \right )}} \]
integrate(((-2*x**4+x**3-14*x**2+3*x-42)*exp((-7+x)/x**2)*ln(x)-4*x**4*exp ((-7+x)/x**2)**2+(x**4+3*x**2)*exp((-7+x)/x**2))/(x**3*ln(x)**2+4*x**3*exp ((-7+x)/x**2)*ln(x)+4*x**3*exp((-7+x)/x**2)**2),x)
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=-\frac {{\left (x^{2} + 3\right )} e^{\frac {1}{x}}}{e^{\left (\frac {7}{x^{2}}\right )} \log \left (x\right ) + 2 \, e^{\frac {1}{x}}} \]
integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((-7+x)/x^2)*log(x)-4*x^4*exp((-7 +x)/x^2)^2+(x^4+3*x^2)*exp((-7+x)/x^2))/(x^3*log(x)^2+4*x^3*exp((-7+x)/x^2 )*log(x)+4*x^3*exp((-7+x)/x^2)^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=-\frac {2 \, x^{2} e^{\left (\frac {x - 7}{x^{2}}\right )} - 3 \, \log \left (x\right )}{2 \, {\left (2 \, e^{\left (\frac {x - 7}{x^{2}}\right )} + \log \left (x\right )\right )}} \]
integrate(((-2*x^4+x^3-14*x^2+3*x-42)*exp((-7+x)/x^2)*log(x)-4*x^4*exp((-7 +x)/x^2)^2+(x^4+3*x^2)*exp((-7+x)/x^2))/(x^3*log(x)^2+4*x^3*exp((-7+x)/x^2 )*log(x)+4*x^3*exp((-7+x)/x^2)^2),x, algorithm=\
Time = 10.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-4 e^{\frac {2 (-7+x)}{x^2}} x^4+e^{\frac {-7+x}{x^2}} \left (3 x^2+x^4\right )+e^{\frac {-7+x}{x^2}} \left (-42+3 x-14 x^2+x^3-2 x^4\right ) \log (x)}{4 e^{\frac {2 (-7+x)}{x^2}} x^3+4 e^{\frac {-7+x}{x^2}} x^3 \log (x)+x^3 \log ^2(x)} \, dx=\frac {\frac {3\,\ln \left (x\right )}{2}+\frac {x^2\,\ln \left (x\right )}{2}}{2\,{\mathrm {e}}^{\frac {1}{x}-\frac {7}{x^2}}+\ln \left (x\right )}-\frac {x^2}{2} \]