Integrand size = 151, antiderivative size = 28 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=-1+x \left (2+e^{x+\frac {\log ^2\left (\left (-2+e^x-x\right ) x\right )}{x}}+x\right ) \]
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x \left (2+e^{x+\frac {\log ^2\left (-x \left (2-e^x+x\right )\right )}{x}}+x\right ) \]
Integrate[(-4*x - 6*x^2 - 2*x^3 + E^x*(2*x + 2*x^2) + E^((x^2 + Log[-2*x + E^x*x - x^2]^2)/x)*(-2*x - 3*x^2 - x^3 + E^x*(x + x^2) + (-4 - 4*x + E^x* (2 + 2*x))*Log[-2*x + E^x*x - x^2] + (2 - E^x + x)*Log[-2*x + E^x*x - x^2] ^2))/(-2*x + E^x*x - 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.
\(\displaystyle \int \frac {-2 x^3-6 x^2+e^x \left (2 x^2+2 x\right )+e^{\frac {x^2+\log ^2\left (-x^2+e^x x-2 x\right )}{x}} \left (-x^3-3 x^2+e^x \left (x^2+x\right )+\left (x-e^x+2\right ) \log ^2\left (-x^2+e^x x-2 x\right )+\left (-4 x+e^x (2 x+2)-4\right ) \log \left (-x^2+e^x x-2 x\right )-2 x\right )-4 x}{-x^2+e^x x-2 x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{x+\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}} \left (-x^3+e^x x^2-3 x^2+e^x x-2 x+x \log ^2\left (-x \left (x-e^x+2\right )\right )-e^x \log ^2\left (-x \left (x-e^x+2\right )\right )+2 \log ^2\left (-x \left (x-e^x+2\right )\right )+2 e^x x \log \left (-x \left (x-e^x+2\right )\right )-4 x \log \left (-x \left (x-e^x+2\right )\right )+2 e^x \log \left (-x \left (x-e^x+2\right )\right )-4 \log \left (-x \left (x-e^x+2\right )\right )\right )}{\left (-x+e^x-2\right ) x}+2 (x+1)\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (2 (x+1)+\frac {e^{x+\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}} \left (-x (x+1) \left (x-e^x+2\right )+\left (x-e^x+2\right ) \log ^2\left (-x \left (x-e^x+2\right )\right )+2 \left (e^x-2\right ) (x+1) \log \left (-x \left (x-e^x+2\right )\right )\right )}{\left (-x+e^x-2\right ) x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )}{-x+e^x-2}dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log \left (-x^2+e^x x-2 x\right )}{x}dx+2 \int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} x \log \left (-x^2+e^x x-2 x\right )}{-x+e^x-2}dx-\int \frac {e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} \log ^2\left (-x^2+e^x x-2 x\right )}{x}dx+\int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x}dx+\int e^{\frac {\log ^2\left (-x \left (x-e^x+2\right )\right )}{x}+x} xdx+(x+1)^2\) |
Int[(-4*x - 6*x^2 - 2*x^3 + E^x*(2*x + 2*x^2) + E^((x^2 + Log[-2*x + E^x*x - x^2]^2)/x)*(-2*x - 3*x^2 - x^3 + E^x*(x + x^2) + (-4 - 4*x + E^x*(2 + 2 *x))*Log[-2*x + E^x*x - x^2] + (2 - E^x + x)*Log[-2*x + E^x*x - x^2]^2))/( -2*x + E^x*x - x^2),x]
3.9.21.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.95 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(-4+x^{2}+{\mathrm e}^{\frac {{\ln \left (x \left ({\mathrm e}^{x}-2-x \right )\right )}^{2}+x^{2}}{x}} x +2 x\) | \(32\) |
risch | \(\text {Expression too large to display}\) | \(812\) |
int((((x-exp(x)+2)*ln(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*ln(exp(x) *x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((ln(exp(x)*x-x^2-2*x)^2+x^2) /x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x,method=_RETUR NVERBOSE)
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x^{2} + x e^{\left (\frac {x^{2} + \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2}}{x}\right )} + 2 \, x \]
integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg orithm=\
Timed out. \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=\text {Timed out} \]
integrate((((x-exp(x)+2)*ln(exp(x)*x-x**2-2*x)**2+((2+2*x)*exp(x)-4*x-4)*l n(exp(x)*x-x**2-2*x)+(x**2+x)*exp(x)-x**3-3*x**2-2*x)*exp((ln(exp(x)*x-x** 2-2*x)**2+x**2)/x)+(2*x**2+2*x)*exp(x)-2*x**3-6*x**2-4*x)/(exp(x)*x-x**2-2 *x),x)
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=x^{2} + x e^{\left (x + \frac {\log \left (x\right )^{2}}{x} + \frac {2 \, \log \left (x\right ) \log \left (-x + e^{x} - 2\right )}{x} + \frac {\log \left (-x + e^{x} - 2\right )^{2}}{x}\right )} + 2 \, x \]
integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg orithm=\
\[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=\int { \frac {2 \, x^{3} + 6 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{x} + {\left (x^{3} - {\left (x - e^{x} + 2\right )} \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2} + 3 \, x^{2} - {\left (x^{2} + x\right )} e^{x} - 2 \, {\left ({\left (x + 1\right )} e^{x} - 2 \, x - 2\right )} \log \left (-x^{2} + x e^{x} - 2 \, x\right ) + 2 \, x\right )} e^{\left (\frac {x^{2} + \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2}}{x}\right )} + 4 \, x}{x^{2} - x e^{x} + 2 \, x} \,d x } \]
integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2+2*x)*exp(x)-4*x-4)*lo g(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2* x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, alg orithm=\
integrate((2*x^3 + 6*x^2 - 2*(x^2 + x)*e^x + (x^3 - (x - e^x + 2)*log(-x^2 + x*e^x - 2*x)^2 + 3*x^2 - (x^2 + x)*e^x - 2*((x + 1)*e^x - 2*x - 2)*log( -x^2 + x*e^x - 2*x) + 2*x)*e^((x^2 + log(-x^2 + x*e^x - 2*x)^2)/x) + 4*x)/ (x^2 - x*e^x + 2*x), x)
Time = 12.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-4 x-6 x^2-2 x^3+e^x \left (2 x+2 x^2\right )+e^{\frac {x^2+\log ^2\left (-2 x+e^x x-x^2\right )}{x}} \left (-2 x-3 x^2-x^3+e^x \left (x+x^2\right )+\left (-4-4 x+e^x (2+2 x)\right ) \log \left (-2 x+e^x x-x^2\right )+\left (2-e^x+x\right ) \log ^2\left (-2 x+e^x x-x^2\right )\right )}{-2 x+e^x x-x^2} \, dx=2\,x+x^2+x\,{\mathrm {e}}^{\frac {{\ln \left (x\,{\mathrm {e}}^x-2\,x-x^2\right )}^2}{x}}\,{\mathrm {e}}^x \]