Integrand size = 152, antiderivative size = 21 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=e^{x^2}+(x+\log ((-e+x) (3+\log (x))))^2 \] Output:
exp(x^2)+(ln((3+ln(x))*(x-exp(1)))+x)^2
Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=e^{x^2}+x^2+2 x \log (-((e-x) (3+\log (x))))+\log ^2(-((e-x) (3+\log (x)))) \] Input:
Integrate[(-8*x^2 - 6*x^3 + E*(2*x + 6*x^2) + E^x^2*(6*E*x^2 - 6*x^3) + (- 2*x^2 + 2*E*x^2 - 2*x^3 + E^x^2*(2*E*x^2 - 2*x^3))*Log[x] + (-8*x - 6*x^2 + E*(2 + 6*x) + (-2*x + 2*E*x - 2*x^2)*Log[x])*Log[-3*E + 3*x + (-E + x)*L og[x]])/(3*E*x - 3*x^2 + (E*x - x^2)*Log[x]),x]
Output:
E^x^2 + x^2 + 2*x*Log[-((E - x)*(3 + Log[x]))] + Log[-((E - x)*(3 + Log[x] ))]^2
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.
| \(\displaystyle \int \frac {-6 x^3-8 x^2+e \left (6 x^2+2 x\right )+\left (-6 x^2+\left (-2 x^2+2 e x-2 x\right ) \log (x)-8 x+e (6 x+2)\right ) \log (3 x+(x-e) \log (x)-3 e)+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^3+2 e x^2-2 x^2+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)}{-3 x^2+\left (e x-x^2\right ) \log (x)+3 e x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-6 x^3-8 x^2+e \left (6 x^2+2 x\right )+\left (-6 x^2+\left (-2 x^2+2 e x-2 x\right ) \log (x)-8 x+e (6 x+2)\right ) \log (3 x+(x-e) \log (x)-3 e)+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^3+2 e x^2-2 x^2+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)}{(e-x) x (\log (x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 e^{x^2} x-\frac {2 x^2 \log (x)}{(e-x) (\log (x)+3)}-\frac {6 x^2}{(e-x) (\log (x)+3)}+\frac {2 \left (-3 x^2+x^2 (-\log (x))-4 \left (1-\frac {3 e}{4}\right ) x-(1-e) x \log (x)+e\right ) \log (-((e-x) (\log (x)+3)))}{(e-x) x (\log (x)+3)}-\frac {2 (1-e) x \log (x)}{(e-x) (\log (x)+3)}-\frac {8 x}{(e-x) (\log (x)+3)}+\frac {2 e (3 x+1)}{(e-x) (\log (x)+3)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 (1-e) \int \frac {x}{(e-x) (\log (x)+3)}dx-8 \int \frac {x}{(e-x) (\log (x)+3)}dx+2 e \int \frac {3 x+1}{(e-x) (\log (x)+3)}dx+6 \int \frac {\log (-((e-x) (\log (x)+3)))}{\log (x)+3}dx-6 e \int \frac {\log (-((e-x) (\log (x)+3)))}{(e-x) (\log (x)+3)}dx-2 (4-3 e) \int \frac {\log (-((e-x) (\log (x)+3)))}{(e-x) (\log (x)+3)}dx+2 \int \frac {\log (-((e-x) (\log (x)+3)))}{(e-x) (\log (x)+3)}dx+2 \int \frac {\log (-((e-x) (\log (x)+3)))}{x (\log (x)+3)}dx+2 \int \frac {\log (x) \log (-((e-x) (\log (x)+3)))}{\log (x)+3}dx-2 e \int \frac {\log (x) \log (-((e-x) (\log (x)+3)))}{(e-x) (\log (x)+3)}dx-2 (1-e) \int \frac {\log (x) \log (-((e-x) (\log (x)+3)))}{(e-x) (\log (x)+3)}dx+x^2+e^{x^2}+2 e x+2 (1-e) x+2 e^2 \log (e-x)+2 (1-e) e \log (e-x)\) |
Input:
Int[(-8*x^2 - 6*x^3 + E*(2*x + 6*x^2) + E^x^2*(6*E*x^2 - 6*x^3) + (-2*x^2 + 2*E*x^2 - 2*x^3 + E^x^2*(2*E*x^2 - 2*x^3))*Log[x] + (-8*x - 6*x^2 + E*(2 + 6*x) + (-2*x + 2*E*x - 2*x^2)*Log[x])*Log[-3*E + 3*x + (-E + x)*Log[x]] )/(3*E*x - 3*x^2 + (E*x - x^2)*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(21)=42\).
Time = 29.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67
| method | result | size |
| parallelrisch | \(-{\mathrm e}^{2}+x^{2}+2 \ln \left (\left (x -{\mathrm e}\right ) \ln \left (x \right )-3 \,{\mathrm e}+3 x \right ) x +\ln \left (\left (x -{\mathrm e}\right ) \ln \left (x \right )-3 \,{\mathrm e}+3 x \right )^{2}+{\mathrm e}^{x^{2}}\) | \(56\) |
| risch | \(2 \ln \left (3+\ln \left (x \right )\right ) x +{\mathrm e}^{x^{2}}+x^{2}-i \pi x \,\operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )-i \pi \ln \left (3+\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )-i \pi \ln \left (x -{\mathrm e}\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )+2 i \pi x +\left (2 x +2 \ln \left (3+\ln \left (x \right )\right )\right ) \ln \left ({\mathrm e}-x \right )-2 i \pi \ln \left (3+\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}-2 i \pi \ln \left (x -{\mathrm e}\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}-2 i \pi x \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi \ln \left (3+\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi \ln \left (3+\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi x \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{3}+i \pi \ln \left (x -{\mathrm e}\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{3}+i \pi \ln \left (3+\ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{3}+i \pi x \,\operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi \ln \left (x -{\mathrm e}\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+i \pi \ln \left (x -{\mathrm e}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}-x \right )\right ) \operatorname {csgn}\left (i \left (3+\ln \left (x \right )\right ) \left ({\mathrm e}-x \right )\right )^{2}+\ln \left (3+\ln \left (x \right )\right )^{2}+\ln \left ({\mathrm e}-x \right )^{2}+2 i \pi \ln \left (x -{\mathrm e}\right )+2 i \pi \ln \left (3+\ln \left (x \right )\right )\) | \(544\) |
Input:
int((((2*x*exp(1)-2*x^2-2*x)*ln(x)+(6*x+2)*exp(1)-6*x^2-8*x)*ln((x-exp(1)) *ln(x)-3*exp(1)+3*x)+((2*x^2*exp(1)-2*x^3)*exp(x^2)+2*x^2*exp(1)-2*x^3-2*x ^2)*ln(x)+(6*x^2*exp(1)-6*x^3)*exp(x^2)+(6*x^2+2*x)*exp(1)-6*x^3-8*x^2)/(( x*exp(1)-x^2)*ln(x)+3*x*exp(1)-3*x^2),x,method=_RETURNVERBOSE)
Output:
-exp(1)^2+x^2+2*ln((x-exp(1))*ln(x)-3*exp(1)+3*x)*x+ln((x-exp(1))*ln(x)-3* exp(1)+3*x)^2+exp(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=x^{2} + 2 \, x \log \left ({\left (x - e\right )} \log \left (x\right ) + 3 \, x - 3 \, e\right ) + \log \left ({\left (x - e\right )} \log \left (x\right ) + 3 \, x - 3 \, e\right )^{2} + e^{\left (x^{2}\right )} \] Input:
integrate((((2*exp(1)*x-2*x^2-2*x)*log(x)+(6*x+2)*exp(1)-6*x^2-8*x)*log((x -exp(1))*log(x)-3*exp(1)+3*x)+((2*x^2*exp(1)-2*x^3)*exp(x^2)+2*x^2*exp(1)- 2*x^3-2*x^2)*log(x)+(6*x^2*exp(1)-6*x^3)*exp(x^2)+(6*x^2+2*x)*exp(1)-6*x^3 -8*x^2)/((exp(1)*x-x^2)*log(x)+3*exp(1)*x-3*x^2),x, algorithm="fricas")
Output:
x^2 + 2*x*log((x - e)*log(x) + 3*x - 3*e) + log((x - e)*log(x) + 3*x - 3*e )^2 + e^(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.47 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=x^{2} + 2 x \log {\left (3 x + \left (x - e\right ) \log {\left (x \right )} - 3 e \right )} + e^{x^{2}} + \log {\left (3 x + \left (x - e\right ) \log {\left (x \right )} - 3 e \right )}^{2} \] Input:
integrate((((2*exp(1)*x-2*x**2-2*x)*ln(x)+(6*x+2)*exp(1)-6*x**2-8*x)*ln((x -exp(1))*ln(x)-3*exp(1)+3*x)+((2*x**2*exp(1)-2*x**3)*exp(x**2)+2*x**2*exp( 1)-2*x**3-2*x**2)*ln(x)+(6*x**2*exp(1)-6*x**3)*exp(x**2)+(6*x**2+2*x)*exp( 1)-6*x**3-8*x**2)/((exp(1)*x-x**2)*ln(x)+3*exp(1)*x-3*x**2),x)
Output:
x**2 + 2*x*log(3*x + (x - E)*log(x) - 3*E) + exp(x**2) + log(3*x + (x - E) *log(x) - 3*E)**2
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=x^{2} + 2 \, {\left (x + \log \left (\log \left (x\right ) + 3\right )\right )} \log \left (x - e\right ) + \log \left (x - e\right )^{2} + 2 \, x \log \left (\log \left (x\right ) + 3\right ) + \log \left (\log \left (x\right ) + 3\right )^{2} + e^{\left (x^{2}\right )} \] Input:
integrate((((2*exp(1)*x-2*x^2-2*x)*log(x)+(6*x+2)*exp(1)-6*x^2-8*x)*log((x -exp(1))*log(x)-3*exp(1)+3*x)+((2*x^2*exp(1)-2*x^3)*exp(x^2)+2*x^2*exp(1)- 2*x^3-2*x^2)*log(x)+(6*x^2*exp(1)-6*x^3)*exp(x^2)+(6*x^2+2*x)*exp(1)-6*x^3 -8*x^2)/((exp(1)*x-x^2)*log(x)+3*exp(1)*x-3*x^2),x, algorithm="maxima")
Output:
x^2 + 2*(x + log(log(x) + 3))*log(x - e) + log(x - e)^2 + 2*x*log(log(x) + 3) + log(log(x) + 3)^2 + e^(x^2)
\[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=\int { \frac {2 \, {\left (3 \, x^{3} + 4 \, x^{2} - {\left (3 \, x^{2} + x\right )} e + 3 \, {\left (x^{3} - x^{2} e\right )} e^{\left (x^{2}\right )} + {\left (3 \, x^{2} - {\left (3 \, x + 1\right )} e + {\left (x^{2} - x e + x\right )} \log \left (x\right ) + 4 \, x\right )} \log \left ({\left (x - e\right )} \log \left (x\right ) + 3 \, x - 3 \, e\right ) + {\left (x^{3} - x^{2} e + x^{2} + {\left (x^{3} - x^{2} e\right )} e^{\left (x^{2}\right )}\right )} \log \left (x\right )\right )}}{3 \, x^{2} - 3 \, x e + {\left (x^{2} - x e\right )} \log \left (x\right )} \,d x } \] Input:
integrate((((2*exp(1)*x-2*x^2-2*x)*log(x)+(6*x+2)*exp(1)-6*x^2-8*x)*log((x -exp(1))*log(x)-3*exp(1)+3*x)+((2*x^2*exp(1)-2*x^3)*exp(x^2)+2*x^2*exp(1)- 2*x^3-2*x^2)*log(x)+(6*x^2*exp(1)-6*x^3)*exp(x^2)+(6*x^2+2*x)*exp(1)-6*x^3 -8*x^2)/((exp(1)*x-x^2)*log(x)+3*exp(1)*x-3*x^2),x, algorithm="giac")
Output:
integrate(2*(3*x^3 + 4*x^2 - (3*x^2 + x)*e + 3*(x^3 - x^2*e)*e^(x^2) + (3* x^2 - (3*x + 1)*e + (x^2 - x*e + x)*log(x) + 4*x)*log((x - e)*log(x) + 3*x - 3*e) + (x^3 - x^2*e + x^2 + (x^3 - x^2*e)*e^(x^2))*log(x))/(3*x^2 - 3*x *e + (x^2 - x*e)*log(x)), x)
Time = 3.52 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.43 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx={\mathrm {e}}^{x^2}+{\ln \left (3\,x-3\,\mathrm {e}+\ln \left (x\right )\,\left (x-\mathrm {e}\right )\right )}^2+x^2-\frac {\ln \left (3\,x-3\,\mathrm {e}+\ln \left (x\right )\,\left (x-\mathrm {e}\right )\right )\,\left (2\,x^2\,\mathrm {e}-2\,x^3\right )}{x\,\left (x-\mathrm {e}\right )} \] Input:
int((exp(1)*(2*x + 6*x^2) + log(x)*(2*x^2*exp(1) + exp(x^2)*(2*x^2*exp(1) - 2*x^3) - 2*x^2 - 2*x^3) + exp(x^2)*(6*x^2*exp(1) - 6*x^3) - 8*x^2 - 6*x^ 3 - log(3*x - 3*exp(1) + log(x)*(x - exp(1)))*(8*x + 6*x^2 + log(x)*(2*x - 2*x*exp(1) + 2*x^2) - exp(1)*(6*x + 2)))/(log(x)*(x*exp(1) - x^2) + 3*x*e xp(1) - 3*x^2),x)
Output:
exp(x^2) + log(3*x - 3*exp(1) + log(x)*(x - exp(1)))^2 + x^2 - (log(3*x - 3*exp(1) + log(x)*(x - exp(1)))*(2*x^2*exp(1) - 2*x^3))/(x*(x - exp(1)))
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.05 \[ \int \frac {-8 x^2-6 x^3+e \left (2 x+6 x^2\right )+e^{x^2} \left (6 e x^2-6 x^3\right )+\left (-2 x^2+2 e x^2-2 x^3+e^{x^2} \left (2 e x^2-2 x^3\right )\right ) \log (x)+\left (-8 x-6 x^2+e (2+6 x)+\left (-2 x+2 e x-2 x^2\right ) \log (x)\right ) \log (-3 e+3 x+(-e+x) \log (x))}{3 e x-3 x^2+\left (e x-x^2\right ) \log (x)} \, dx=e^{x^{2}}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+3\right ) e +\mathrm {log}\left (-\mathrm {log}\left (x \right ) e +\mathrm {log}\left (x \right ) x -3 e +3 x \right )^{2}+2 \,\mathrm {log}\left (-\mathrm {log}\left (x \right ) e +\mathrm {log}\left (x \right ) x -3 e +3 x \right ) e +2 \,\mathrm {log}\left (-\mathrm {log}\left (x \right ) e +\mathrm {log}\left (x \right ) x -3 e +3 x \right ) x -2 \,\mathrm {log}\left (e -x \right ) e +x^{2} \] Input:
int((((2*exp(1)*x-2*x^2-2*x)*log(x)+(6*x+2)*exp(1)-6*x^2-8*x)*log((x-exp(1 ))*log(x)-3*exp(1)+3*x)+((2*x^2*exp(1)-2*x^3)*exp(x^2)+2*x^2*exp(1)-2*x^3- 2*x^2)*log(x)+(6*x^2*exp(1)-6*x^3)*exp(x^2)+(6*x^2+2*x)*exp(1)-6*x^3-8*x^2 )/((exp(1)*x-x^2)*log(x)+3*exp(1)*x-3*x^2),x)
Output:
e**(x**2) - 2*log(log(x) + 3)*e + log( - log(x)*e + log(x)*x - 3*e + 3*x)* *2 + 2*log( - log(x)*e + log(x)*x - 3*e + 3*x)*e + 2*log( - log(x)*e + log (x)*x - 3*e + 3*x)*x - 2*log(e - x)*e + x**2