Integrand size = 123, antiderivative size = 37 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=\log \left (-2+\frac {e^{-e^{x^2} (-4+x)} \log \left (\frac {e^4}{\frac {e^2}{3}-x}\right )}{x}\right ) \] Output:
ln(ln(exp(4)/(1/3*exp(2)-x))/x/exp((-4+x)*exp(x^2))-2)
\[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=\int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx \] Input:
Integrate[(-3*x + (E^2 - 3*x + E^x^2*(-3*x^2 + 24*x^3 - 6*x^4 + E^2*(x - 8 *x^2 + 2*x^3)))*Log[(3*E^4)/(E^2 - 3*x)])/(E^(E^x^2*(-4 + x))*(2*E^2*x^2 - 6*x^3) + (-(E^2*x) + 3*x^2)*Log[(3*E^4)/(E^2 - 3*x)]),x]
Output:
Integrate[(-3*x + (E^2 - 3*x + E^x^2*(-3*x^2 + 24*x^3 - 6*x^4 + E^2*(x - 8 *x^2 + 2*x^3)))*Log[(3*E^4)/(E^2 - 3*x)])/(E^(E^x^2*(-4 + x))*(2*E^2*x^2 - 6*x^3) + (-(E^2*x) + 3*x^2)*Log[(3*E^4)/(E^2 - 3*x)]), 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.
| \(\displaystyle \int \frac {\left (e^{x^2} \left (-6 x^4+24 x^3-3 x^2+e^2 \left (2 x^3-8 x^2+x\right )\right )-3 x+e^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )-3 x}{\left (3 x^2-e^2 x\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )+e^{e^{x^2} (x-4)} \left (2 e^2 x^2-6 x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 x-\left (e^{x^2} \left (-6 x^4+24 x^3-3 x^2+e^2 \left (2 x^3-8 x^2+x\right )\right )-3 x+e^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{\left (e^2-3 x\right ) x \left (-2 e^{e^{x^2} (x-4)} x+\log \left (\frac {3}{e^2-3 x}\right )+4\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{x^2} \left (2 x^2-8 x+1\right ) \left (-\log \left (\frac {1}{e^2-3 x}\right )-4 \left (1+\frac {\log (3)}{4}\right )\right )}{-2 e^{e^{x^2} (x-4)} x+\log \left (\frac {3}{e^2-3 x}\right )+4}+\frac {3 x \log \left (\frac {1}{e^2-3 x}\right )+15 x \left (1+\frac {\log (3)}{5}\right )-e^2 \log \left (\frac {1}{e^2-3 x}\right )-4 e^2 \left (1+\frac {\log (3)}{4}\right )}{\left (e^2-3 x\right ) x \left (-2 e^{e^{x^2} (x-4)} x+\log \left (\frac {3}{e^2-3 x}\right )+4\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {e^{x^2} \left (2 x^2-8 x+1\right ) \left (-\log \left (\frac {1}{e^2-3 x}\right )-4 \left (1+\frac {\log (3)}{4}\right )\right )}{-2 e^{e^{x^2} (x-4)} x+\log \left (\frac {3}{e^2-3 x}\right )+4}+\frac {3 x \log \left (\frac {1}{e^2-3 x}\right )+15 x \left (1+\frac {\log (3)}{5}\right )-e^2 \log \left (\frac {1}{e^2-3 x}\right )-4 e^2 \left (1+\frac {\log (3)}{4}\right )}{\left (e^2-3 x\right ) x \left (-2 e^{e^{x^2} (x-4)} x+\log \left (\frac {3}{e^2-3 x}\right )+4\right )}\right )dx\) |
Input:
Int[(-3*x + (E^2 - 3*x + E^x^2*(-3*x^2 + 24*x^3 - 6*x^4 + E^2*(x - 8*x^2 + 2*x^3)))*Log[(3*E^4)/(E^2 - 3*x)])/(E^(E^x^2*(-4 + x))*(2*E^2*x^2 - 6*x^3 ) + (-(E^2*x) + 3*x^2)*Log[(3*E^4)/(E^2 - 3*x)]),x]
Output:
$Aborted
Time = 110.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\left (x -4\right ) {\mathrm e}^{x^{2}}+\ln \left ({\mathrm e}^{\left (x -4\right ) {\mathrm e}^{x^{2}}}-\frac {8+2 \ln \left (3\right )-2 \ln \left ({\mathrm e}^{2}-3 x \right )}{4 x}\right )\) | \(42\) |
| parallelrisch | \(-{\mathrm e}^{x^{2}} x +4 \,{\mathrm e}^{x^{2}}-\ln \left (x \right )+\ln \left (x \,{\mathrm e}^{\left (x -4\right ) {\mathrm e}^{x^{2}}}-\frac {\ln \left (\frac {3 \,{\mathrm e}^{4}}{{\mathrm e}^{2}-3 x}\right )}{2}\right )\) | \(47\) |
Input:
int(((((2*x^3-8*x^2+x)*exp(2)-6*x^4+24*x^3-3*x^2)*exp(x^2)+exp(2)-3*x)*ln( 3*exp(4)/(exp(2)-3*x))-3*x)/((2*x^2*exp(2)-6*x^3)*exp((x-4)*exp(x^2))+(-ex p(2)*x+3*x^2)*ln(3*exp(4)/(exp(2)-3*x))),x,method=_RETURNVERBOSE)
Output:
-(x-4)*exp(x^2)+ln(exp((x-4)*exp(x^2))-1/4*(8+2*ln(3)-2*ln(exp(2)-3*x))/x)
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=-{\left (x - 4\right )} e^{\left (x^{2}\right )} + \log \left (\frac {2 \, x e^{\left ({\left (x - 4\right )} e^{\left (x^{2}\right )}\right )} - \log \left (-\frac {3 \, e^{4}}{3 \, x - e^{2}}\right )}{x}\right ) \] Input:
integrate(((((2*x^3-8*x^2+x)*exp(2)-6*x^4+24*x^3-3*x^2)*exp(x^2)+exp(2)-3* x)*log(3*exp(4)/(exp(2)-3*x))-3*x)/((2*x^2*exp(2)-6*x^3)*exp((-4+x)*exp(x^ 2))+(-exp(2)*x+3*x^2)*log(3*exp(4)/(exp(2)-3*x))),x, algorithm="fricas")
Output:
-(x - 4)*e^(x^2) + log((2*x*e^((x - 4)*e^(x^2)) - log(-3*e^4/(3*x - e^2))) /x)
Time = 0.54 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=\left (4 - x\right ) e^{x^{2}} + \log {\left (e^{\left (x - 4\right ) e^{x^{2}}} - \frac {\log {\left (\frac {3 e^{4}}{- 3 x + e^{2}} \right )}}{2 x} \right )} \] Input:
integrate(((((2*x**3-8*x**2+x)*exp(2)-6*x**4+24*x**3-3*x**2)*exp(x**2)+exp (2)-3*x)*ln(3*exp(4)/(exp(2)-3*x))-3*x)/((2*x**2*exp(2)-6*x**3)*exp((-4+x) *exp(x**2))+(-exp(2)*x+3*x**2)*ln(3*exp(4)/(exp(2)-3*x))),x)
Output:
(4 - x)*exp(x**2) + log(exp((x - 4)*exp(x**2)) - log(3*exp(4)/(-3*x + exp( 2)))/(2*x))
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=-x e^{\left (x^{2}\right )} + \log \left (\frac {2 \, x e^{\left (x e^{\left (x^{2}\right )}\right )} - {\left (\log \left (3\right ) + 4\right )} e^{\left (4 \, e^{\left (x^{2}\right )}\right )} + e^{\left (4 \, e^{\left (x^{2}\right )}\right )} \log \left (-3 \, x + e^{2}\right )}{2 \, x}\right ) \] Input:
integrate(((((2*x^3-8*x^2+x)*exp(2)-6*x^4+24*x^3-3*x^2)*exp(x^2)+exp(2)-3* x)*log(3*exp(4)/(exp(2)-3*x))-3*x)/((2*x^2*exp(2)-6*x^3)*exp((-4+x)*exp(x^ 2))+(-exp(2)*x+3*x^2)*log(3*exp(4)/(exp(2)-3*x))),x, algorithm="maxima")
Output:
-x*e^(x^2) + log(1/2*(2*x*e^(x*e^(x^2)) - (log(3) + 4)*e^(4*e^(x^2)) + e^( 4*e^(x^2))*log(-3*x + e^2))/x)
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=-x e^{\left (x^{2}\right )} + 4 \, e^{\left (x^{2}\right )} + \log \left (-2 \, x e^{\left (x e^{\left (x^{2}\right )} - 4 \, e^{\left (x^{2}\right )}\right )} + \log \left (-\frac {3}{3 \, x - e^{2}}\right ) + 4\right ) - \log \left (x\right ) \] Input:
integrate(((((2*x^3-8*x^2+x)*exp(2)-6*x^4+24*x^3-3*x^2)*exp(x^2)+exp(2)-3* x)*log(3*exp(4)/(exp(2)-3*x))-3*x)/((2*x^2*exp(2)-6*x^3)*exp((-4+x)*exp(x^ 2))+(-exp(2)*x+3*x^2)*log(3*exp(4)/(exp(2)-3*x))),x, algorithm="giac")
Output:
-x*e^(x^2) + 4*e^(x^2) + log(-2*x*e^(x*e^(x^2) - 4*e^(x^2)) + log(-3/(3*x - e^2)) + 4) - log(x)
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=\ln \left (\ln \left (-\frac {3\,{\mathrm {e}}^4}{3\,x-{\mathrm {e}}^2}\right )-2\,x\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}}\right )-\ln \left (x\right )-{\mathrm {e}}^{x^2}\,\left (x-4\right ) \] Input:
int(-(3*x - log(-(3*exp(4))/(3*x - exp(2)))*(exp(2) - 3*x + exp(x^2)*(exp( 2)*(x - 8*x^2 + 2*x^3) - 3*x^2 + 24*x^3 - 6*x^4)))/(exp(exp(x^2)*(x - 4))* (2*x^2*exp(2) - 6*x^3) - log(-(3*exp(4))/(3*x - exp(2)))*(x*exp(2) - 3*x^2 )),x)
Output:
log(log(-(3*exp(4))/(3*x - exp(2))) - 2*x*exp(-4*exp(x^2))*exp(x*exp(x^2)) ) - log(x) - exp(x^2)*(x - 4)
Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \frac {-3 x+\left (e^2-3 x+e^{x^2} \left (-3 x^2+24 x^3-6 x^4+e^2 \left (x-8 x^2+2 x^3\right )\right )\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )}{e^{e^{x^2} (-4+x)} \left (2 e^2 x^2-6 x^3\right )+\left (-e^2 x+3 x^2\right ) \log \left (\frac {3 e^4}{e^2-3 x}\right )} \, dx=-e^{x^{2}} x +\mathrm {log}\left (e^{4 e^{x^{2}}} \mathrm {log}\left (\frac {3 e^{4}}{e^{2}-3 x}\right )-2 e^{e^{x^{2}} x} x \right )-\mathrm {log}\left (x \right ) \] Input:
int(((((2*x^3-8*x^2+x)*exp(2)-6*x^4+24*x^3-3*x^2)*exp(x^2)+exp(2)-3*x)*log (3*exp(4)/(exp(2)-3*x))-3*x)/((2*x^2*exp(2)-6*x^3)*exp((-4+x)*exp(x^2))+(- exp(2)*x+3*x^2)*log(3*exp(4)/(exp(2)-3*x))),x)
Output:
- e**(x**2)*x + log(e**(4*e**(x**2))*log((3*e**4)/(e**2 - 3*x)) - 2*e**(e **(x**2)*x)*x) - log(x)