Integrand size = 161, antiderivative size = 33 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=\frac {\frac {1}{2}+x}{-3+\frac {e^{-\left (\left (x-x^2\right ) (x+\log (x))\right )} \log \left (\frac {21}{5}\right )}{x}} \] Output:
(1/2+x)/(ln(21/5)/x/exp((x+ln(x))*(-x^2+x))-3)
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=\frac {e^{x^2} x^{1+x} (1+2 x)}{2 \left (-3 e^{x^2} x^{1+x}+e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )} \] Input:
Integrate[(-6*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 + E^(x^2 - x^3 + (x - x^2)*Log[x])*((1 + 5*x + 3*x^2 - x^3 - 6*x^4)*Log[21/5] + (x - 4*x^3) *Log[21/5]*Log[x]))/(18*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 - 12*E^ (x^2 - x^3 + (x - x^2)*Log[x])*x*Log[21/5] + 2*Log[21/5]^2),x]
Output:
(E^x^2*x^(1 + x)*(1 + 2*x))/(2*(-3*E^x^2*x^(1 + x) + E^x^3*x^x^2*Log[21/5] ))
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.
| \(\displaystyle \int \frac {e^{-x^3+x^2+\left (x-x^2\right ) \log (x)} \left (\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)+\left (-6 x^4-x^3+3 x^2+5 x+1\right ) \log \left (\frac {21}{5}\right )\right )-6 x^2 e^{-2 x^3+2 x^2+2 \left (x-x^2\right ) \log (x)}}{18 x^2 e^{-2 x^3+2 x^2+2 \left (x-x^2\right ) \log (x)}-12 x \log \left (\frac {21}{5}\right ) e^{-x^3+x^2+\left (x-x^2\right ) \log (x)}+2 \log ^2\left (\frac {21}{5}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x^3} x^{2 x^2} \left (e^{-x^3+x^2+\left (x-x^2\right ) \log (x)} \left (\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)+\left (-6 x^4-x^3+3 x^2+5 x+1\right ) \log \left (\frac {21}{5}\right )\right )-6 x^2 e^{-2 x^3+2 x^2+2 \left (x-x^2\right ) \log (x)}\right )}{2 \left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {e^{2 x^3} x^{2 x^2} \left (6 e^{2 x^2-2 x^3} x^{2 \left (x-x^2\right )+2}-e^{x^2-x^3} x^{x-x^2} \left (\log \left (\frac {21}{5}\right ) \left (-6 x^4-x^3+3 x^2+5 x+1\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )\right )}{\left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{2 x^3} x^{2 x^2} \left (6 e^{2 x^2-2 x^3} x^{2 \left (x-x^2\right )+2}-e^{x^2-x^3} x^{x-x^2} \left (\log \left (\frac {21}{5}\right ) \left (-6 x^4-x^3+3 x^2+5 x+1\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )\right )}{\left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {1}{9} e^{x^3-x^2} \log \left (\frac {21}{5}\right ) \left (6 x^4+4 \log (x) x^3+x^3-3 x^2-\log (x) x-x-1\right ) x^{x^2-x-2}+\frac {e^{2 x^3} (2 x+1) \log ^2\left (\frac {21}{5}\right ) \left (3 x^3+2 \log (x) x^2-x^2-\log (x) x-x-1\right ) x^{2 x^2-1}}{3 \left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}+\frac {e^{2 x^3-x^2} \log ^2\left (\frac {21}{5}\right ) \left (6 x^4+4 \log (x) x^3+x^3-3 x^2-\log (x) x-x-1\right ) x^{2 x^2-x-2}}{9 \left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )}+\frac {2}{3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{x^2} x^x \left (e^{x^3} \left (6 x^4+x^3-3 x^2-5 x-1\right ) \log \left (\frac {21}{5}\right ) x^{x^2}+6 e^{x^2} x^{x+2}+e^{x^3} \left (4 x^2-1\right ) \log \left (\frac {21}{5}\right ) \log (x) x^{x^2+1}\right )}{\left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {3 e^{2 x^2} x^{2 x+1} (2 x+1) \left (3 x^3+2 \log (x) x^2-x^2-\log (x) x-x-1\right )}{\left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}-\frac {e^{x^2} x^x \left (6 x^4+4 \log (x) x^3+x^3-3 x^2-\log (x) x-5 x-1\right )}{3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {3 e^{2 x^2} x^{2 x+1} (2 x+1) \left (3 x^3+2 \log (x) x^2-x^2-\log (x) x-x-1\right )}{\left (3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )\right )^2}-\frac {e^{x^2} x^x \left (6 x^4+4 \log (x) x^3+x^3-3 x^2-\log (x) x-5 x-1\right )}{3 e^{x^2} x^{x+1}-e^{x^3} x^{x^2} \log \left (\frac {21}{5}\right )}\right )dx\) |
Input:
Int[(-6*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 + E^(x^2 - x^3 + (x - x ^2)*Log[x])*((1 + 5*x + 3*x^2 - x^3 - 6*x^4)*Log[21/5] + (x - 4*x^3)*Log[2 1/5]*Log[x]))/(18*E^(2*x^2 - 2*x^3 + 2*(x - x^2)*Log[x])*x^2 - 12*E^(x^2 - x^3 + (x - x^2)*Log[x])*x*Log[21/5] + 2*Log[21/5]^2),x]
Output:
$Aborted
Time = 1.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79
| method | result | size |
| parallelrisch | \(\frac {6 x^{2} {\mathrm e}^{\left (-x^{2}+x \right ) \ln \left (x \right )-x^{3}+x^{2}}+\ln \left (\frac {21}{5}\right )}{-18 x \,{\mathrm e}^{\left (-x^{2}+x \right ) \ln \left (x \right )-x^{3}+x^{2}}+6 \ln \left (\frac {21}{5}\right )}\) | \(59\) |
| risch | \(-\frac {x}{3}+\frac {2 x \ln \left (5\right )-2 \ln \left (7\right ) x -2 x \ln \left (3\right )+\ln \left (5\right )-\ln \left (7\right )-\ln \left (3\right )}{18 x \,x^{-x \left (-1+x \right )} {\mathrm e}^{-x^{2} \left (-1+x \right )}+6 \ln \left (5\right )-6 \ln \left (7\right )-6 \ln \left (3\right )}\) | \(66\) |
Input:
int((-6*x^2*exp((-x^2+x)*ln(x)-x^3+x^2)^2+((-4*x^3+x)*ln(21/5)*ln(x)+(-6*x ^4-x^3+3*x^2+5*x+1)*ln(21/5))*exp((-x^2+x)*ln(x)-x^3+x^2))/(18*x^2*exp((-x ^2+x)*ln(x)-x^3+x^2)^2-12*x*ln(21/5)*exp((-x^2+x)*ln(x)-x^3+x^2)+2*ln(21/5 )^2),x,method=_RETURNVERBOSE)
Output:
1/6*(6*x^2*exp((-x^2+x)*ln(x)-x^3+x^2)+ln(21/5))/(-3*x*exp((-x^2+x)*ln(x)- x^3+x^2)+ln(21/5))
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=-\frac {6 \, x^{2} e^{\left (-x^{3} + x^{2} - {\left (x^{2} - x\right )} \log \left (x\right )\right )} + \log \left (\frac {21}{5}\right )}{6 \, {\left (3 \, x e^{\left (-x^{3} + x^{2} - {\left (x^{2} - x\right )} \log \left (x\right )\right )} - \log \left (\frac {21}{5}\right )\right )}} \] Input:
integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log (x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18* x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^3+ x^2)+2*log(21/5)^2),x, algorithm="fricas")
Output:
-1/6*(6*x^2*e^(-x^3 + x^2 - (x^2 - x)*log(x)) + log(21/5))/(3*x*e^(-x^3 + x^2 - (x^2 - x)*log(x)) - log(21/5))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=- \frac {x}{3} + \frac {- 2 x \log {\left (21 \right )} + 2 x \log {\left (5 \right )} - \log {\left (21 \right )} + \log {\left (5 \right )}}{18 x e^{- x^{3} + x^{2} + \left (- x^{2} + x\right ) \log {\left (x \right )}} - 6 \log {\left (21 \right )} + 6 \log {\left (5 \right )}} \] Input:
integrate((-6*x**2*exp((-x**2+x)*ln(x)-x**3+x**2)**2+((-4*x**3+x)*ln(21/5) *ln(x)+(-6*x**4-x**3+3*x**2+5*x+1)*ln(21/5))*exp((-x**2+x)*ln(x)-x**3+x**2 ))/(18*x**2*exp((-x**2+x)*ln(x)-x**3+x**2)**2-12*x*ln(21/5)*exp((-x**2+x)* ln(x)-x**3+x**2)+2*ln(21/5)**2),x)
Output:
-x/3 + (-2*x*log(21) + 2*x*log(5) - log(21) + log(5))/(18*x*exp(-x**3 + x* *2 + (-x**2 + x)*log(x)) - 6*log(21) + 6*log(5))
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=\frac {{\left (2 \, x^{2} + x\right )} e^{\left (x^{2} + x \log \left (x\right )\right )}}{2 \, {\left ({\left (\log \left (7\right ) - \log \left (5\right ) + \log \left (3\right )\right )} e^{\left (x^{3} + x^{2} \log \left (x\right )\right )} - 3 \, x e^{\left (x^{2} + x \log \left (x\right )\right )}\right )}} \] Input:
integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log (x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18* x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^3+ x^2)+2*log(21/5)^2),x, algorithm="maxima")
Output:
1/2*(2*x^2 + x)*e^(x^2 + x*log(x))/((log(7) - log(5) + log(3))*e^(x^3 + x^ 2*log(x)) - 3*x*e^(x^2 + x*log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.38 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.06 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=-\frac {6 \, x^{2} e^{\left (-x^{3} - x^{2} \log \left (x\right ) + x^{2} + x \log \left (x\right )\right )} + \log \left (21\right ) - \log \left (5\right )}{6 \, {\left (3 \, x e^{\left (-x^{3} - x^{2} \log \left (x\right ) + x^{2} + x \log \left (x\right )\right )} - \log \left (21\right ) + \log \left (5\right )\right )}} \] Input:
integrate((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log (x)+(-6*x^4-x^3+3*x^2+5*x+1)*log(21/5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18* x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^3+ x^2)+2*log(21/5)^2),x, algorithm="giac")
Output:
-1/6*(6*x^2*e^(-x^3 - x^2*log(x) + x^2 + x*log(x)) + log(21) - log(5))/(3* x*e^(-x^3 - x^2*log(x) + x^2 + x*log(x)) - log(21) + log(5))
Time = 4.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=\frac {x^{x^2}\,\ln \left (\frac {21}{5}\right )+6\,x^x\,x^2\,{\mathrm {e}}^{x^2-x^3}}{6\,\left (x^{x^2}\,\ln \left (\frac {21}{5}\right )-3\,x\,x^x\,{\mathrm {e}}^{x^2-x^3}\right )} \] Input:
int((exp(log(x)*(x - x^2) + x^2 - x^3)*(log(21/5)*(5*x + 3*x^2 - x^3 - 6*x ^4 + 1) + log(21/5)*log(x)*(x - 4*x^3)) - 6*x^2*exp(2*log(x)*(x - x^2) + 2 *x^2 - 2*x^3))/(18*x^2*exp(2*log(x)*(x - x^2) + 2*x^2 - 2*x^3) + 2*log(21/ 5)^2 - 12*x*exp(log(x)*(x - x^2) + x^2 - x^3)*log(21/5)),x)
Output:
(x^(x^2)*log(21/5) + 6*x^x*x^2*exp(x^2 - x^3))/(6*(x^(x^2)*log(21/5) - 3*x *x^x*exp(x^2 - x^3)))
Time = 0.92 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-6 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2+e^{x^2-x^3+\left (x-x^2\right ) \log (x)} \left (\left (1+5 x+3 x^2-x^3-6 x^4\right ) \log \left (\frac {21}{5}\right )+\left (x-4 x^3\right ) \log \left (\frac {21}{5}\right ) \log (x)\right )}{18 e^{2 x^2-2 x^3+2 \left (x-x^2\right ) \log (x)} x^2-12 e^{x^2-x^3+\left (x-x^2\right ) \log (x)} x \log \left (\frac {21}{5}\right )+2 \log ^2\left (\frac {21}{5}\right )} \, dx=\frac {x^{x^{2}} e^{x^{3}} \mathrm {log}\left (\frac {21}{5}\right )+6 x^{x} e^{x^{2}} x^{2}}{6 x^{x^{2}} e^{x^{3}} \mathrm {log}\left (\frac {21}{5}\right )-18 x^{x} e^{x^{2}} x} \] Input:
int((-6*x^2*exp((-x^2+x)*log(x)-x^3+x^2)^2+((-4*x^3+x)*log(21/5)*log(x)+(- 6*x^4-x^3+3*x^2+5*x+1)*log(21/5))*exp((-x^2+x)*log(x)-x^3+x^2))/(18*x^2*ex p((-x^2+x)*log(x)-x^3+x^2)^2-12*x*log(21/5)*exp((-x^2+x)*log(x)-x^3+x^2)+2 *log(21/5)^2),x)
Output:
(x**(x**2)*e**(x**3)*log(21/5) + 6*x**x*e**(x**2)*x**2)/(6*(x**(x**2)*e**( x**3)*log(21/5) - 3*x**x*e**(x**2)*x))