Integrand size = 131, antiderivative size = 32 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=e^{\left (2-e^{e^{3+x}}-x+x^2\right )^2+\left (3-\frac {2}{\log (3)}\right )^2} \]
Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=e^{13+e^{2 e^{3+x}}-4 x+5 x^2-2 x^3+x^4-2 e^{e^{3+x}} \left (2-x+x^2\right )-\frac {4 (-1+3 \log (3))}{\log ^2(3)}} \]
Integrate[E^((4 - 12*Log[3] + E^(2*E^(3 + x))*Log[3]^2 + E^E^(3 + x)*(-4 + 2*x - 2*x^2)*Log[3]^2 + (13 - 4*x + 5*x^2 - 2*x^3 + x^4)*Log[3]^2)/Log[3] ^2)*(-4 + 2*E^(3 + 2*E^(3 + x) + x) + 10*x - 6*x^2 + 4*x^3 + E^E^(3 + x)*( 2 - 4*x + E^(3 + x)*(-4 + 2*x - 2*x^2))),x]
E^(13 + E^(2*E^(3 + x)) - 4*x + 5*x^2 - 2*x^3 + x^4 - 2*E^E^(3 + x)*(2 - x + x^2) - (4*(-1 + 3*Log[3]))/Log[3]^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.
| \(\displaystyle \int \left (4 x^3-6 x^2+e^{e^{x+3}} \left (e^{x+3} \left (-2 x^2+2 x-4\right )-4 x+2\right )+10 x+2 e^{x+2 e^{x+3}+3}-4\right ) \exp \left (\frac {e^{e^{x+3}} \left (-2 x^2+2 x-4\right ) \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+e^{2 e^{x+3}} \log ^2(3)+4-12 \log (3)}{\log ^2(3)}\right ) \, dx\) |
| \(\Big \downarrow \) 2704 |
| \(\displaystyle \int \left (4 x^3-6 x^2+e^{e^{x+3}} \left (e^{x+3} \left (-2 x^2+2 x-4\right )-4 x+2\right )+10 x+2 e^{x+2 e^{x+3}+3}-4\right ) 3^{-\frac {12}{\log ^2(3)}} \exp \left (\frac {e^{e^{x+3}} \left (-2 x^2+2 x-4\right ) \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+e^{2 e^{x+3}} \log ^2(3)+4}{\log ^2(3)}\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^{-\frac {12}{\log (3)}} \int -2 \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) \left (-2 x^3+3 x^2-5 x-e^{x+2 e^{x+3}+3}-e^{e^{x+3}} \left (-2 x-e^{x+3} \left (x^2-x+2\right )+1\right )+2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 e^{-\frac {12}{\log (3)}} \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) \left (-2 x^3+3 x^2-5 x-e^{x+2 e^{x+3}+3}-e^{e^{x+3}} \left (-2 x-e^{x+3} \left (x^2-x+2\right )+1\right )+2\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -2 e^{-\frac {12}{\log (3)}} \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) \left (-2 x+e^{x+e^{x+3}+3}+1\right ) \left (x^2-x-e^{e^{x+3}}+2\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 e^{-\frac {12}{\log (3)}} \int \left (-\exp \left (x+e^{x+3}+\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+3\right ) \left (-x^2+x+e^{e^{x+3}}-2\right )-\exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) (2 x-1) \left (x^2-x-e^{e^{x+3}}+2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e^{-\frac {12}{\log (3)}} \left (2 \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right )dx-\int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+e^{x+3}\right )dx+2 \int \exp \left (x+e^{x+3}+\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+3\right )dx-\int \exp \left (x+2 e^{x+3}+\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+3\right )dx-5 \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) xdx+2 \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+e^{x+3}\right ) xdx-\int \exp \left (x+e^{x+3}+\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+3\right ) xdx+3 \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) x^2dx+\int \exp \left (x+e^{x+3}+\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}+3\right ) x^2dx-2 \int \exp \left (\frac {-2 e^{e^{x+3}} \log ^2(3) \left (x^2-x+2\right )+e^{2 e^{x+3}} \log ^2(3)+\left (x^4-2 x^3+5 x^2-4 x+13\right ) \log ^2(3)+4}{\log ^2(3)}\right ) x^3dx\right )\) |
Int[E^((4 - 12*Log[3] + E^(2*E^(3 + x))*Log[3]^2 + E^E^(3 + x)*(-4 + 2*x - 2*x^2)*Log[3]^2 + (13 - 4*x + 5*x^2 - 2*x^3 + x^4)*Log[3]^2)/Log[3]^2)*(- 4 + 2*E^(3 + 2*E^(3 + x) + x) + 10*x - 6*x^2 + 4*x^3 + E^E^(3 + x)*(2 - 4* x + E^(3 + x)*(-4 + 2*x - 2*x^2))),x]
3.6.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)* z^(a*b*Log[F]), x] /; FreeQ[{F, a, b}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(29)=58\).
Time = 1.43 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\ln \left (3\right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{3+x}}+\left (-2 x^{2}+2 x -4\right ) \ln \left (3\right )^{2} {\mathrm e}^{{\mathrm e}^{3+x}}+\left (x^{4}-2 x^{3}+5 x^{2}-4 x +13\right ) \ln \left (3\right )^{2}-12 \ln \left (3\right )+4}{\ln \left (3\right )^{2}}}\) | \(68\) |
| risch | \({\mathrm e}^{\frac {x^{4} \ln \left (3\right )^{2}-2 \ln \left (3\right )^{2} {\mathrm e}^{{\mathrm e}^{3+x}} x^{2}-2 x^{3} \ln \left (3\right )^{2}+2 \ln \left (3\right )^{2} {\mathrm e}^{{\mathrm e}^{3+x}} x +5 x^{2} \ln \left (3\right )^{2}-4 \ln \left (3\right )^{2} {\mathrm e}^{{\mathrm e}^{3+x}}+\ln \left (3\right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{3+x}}-4 x \ln \left (3\right )^{2}+13 \ln \left (3\right )^{2}-12 \ln \left (3\right )+4}{\ln \left (3\right )^{2}}}\) | \(101\) |
int((2*exp(3+x)*exp(exp(3+x))^2+((-2*x^2+2*x-4)*exp(3+x)-4*x+2)*exp(exp(3+ x))+4*x^3-6*x^2+10*x-4)*exp((ln(3)^2*exp(exp(3+x))^2+(-2*x^2+2*x-4)*ln(3)^ 2*exp(exp(3+x))+(x^4-2*x^3+5*x^2-4*x+13)*ln(3)^2-12*ln(3)+4)/ln(3)^2),x,me thod=_RETURNVERBOSE)
exp((ln(3)^2*exp(exp(3+x))^2+(-2*x^2+2*x-4)*ln(3)^2*exp(exp(3+x))+(x^4-2*x ^3+5*x^2-4*x+13)*ln(3)^2-12*ln(3)+4)/ln(3)^2)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=e^{\left (-\frac {2 \, {\left (x^{2} - x + 2\right )} e^{\left (e^{\left (x + 3\right )}\right )} \log \left (3\right )^{2} - {\left (x^{4} - 2 \, x^{3} + 5 \, x^{2} - 4 \, x + 13\right )} \log \left (3\right )^{2} - e^{\left (2 \, e^{\left (x + 3\right )}\right )} \log \left (3\right )^{2} + 12 \, \log \left (3\right ) - 4}{\log \left (3\right )^{2}}\right )} \]
integrate((2*exp(3+x)*exp(exp(3+x))^2+((-2*x^2+2*x-4)*exp(3+x)-4*x+2)*exp( exp(3+x))+4*x^3-6*x^2+10*x-4)*exp((log(3)^2*exp(exp(3+x))^2+(-2*x^2+2*x-4) *log(3)^2*exp(exp(3+x))+(x^4-2*x^3+5*x^2-4*x+13)*log(3)^2-12*log(3)+4)/log (3)^2),x, algorithm=\
e^(-(2*(x^2 - x + 2)*e^(e^(x + 3))*log(3)^2 - (x^4 - 2*x^3 + 5*x^2 - 4*x + 13)*log(3)^2 - e^(2*e^(x + 3))*log(3)^2 + 12*log(3) - 4)/log(3)^2)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=e^{\frac {\left (- 2 x^{2} + 2 x - 4\right ) e^{e^{x + 3}} \log {\left (3 \right )}^{2} + \left (x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 13\right ) \log {\left (3 \right )}^{2} + e^{2 e^{x + 3}} \log {\left (3 \right )}^{2} - 12 \log {\left (3 \right )} + 4}{\log {\left (3 \right )}^{2}}} \]
integrate((2*exp(3+x)*exp(exp(3+x))**2+((-2*x**2+2*x-4)*exp(3+x)-4*x+2)*ex p(exp(3+x))+4*x**3-6*x**2+10*x-4)*exp((ln(3)**2*exp(exp(3+x))**2+(-2*x**2+ 2*x-4)*ln(3)**2*exp(exp(3+x))+(x**4-2*x**3+5*x**2-4*x+13)*ln(3)**2-12*ln(3 )+4)/ln(3)**2),x)
exp(((-2*x**2 + 2*x - 4)*exp(exp(x + 3))*log(3)**2 + (x**4 - 2*x**3 + 5*x* *2 - 4*x + 13)*log(3)**2 + exp(2*exp(x + 3))*log(3)**2 - 12*log(3) + 4)/lo g(3)**2)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.62 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=e^{\left (x^{4} - 2 \, x^{3} - 2 \, x^{2} e^{\left (e^{\left (x + 3\right )}\right )} + 5 \, x^{2} + 2 \, x e^{\left (e^{\left (x + 3\right )}\right )} - 4 \, x - \frac {12}{\log \left (3\right )} + \frac {4}{\log \left (3\right )^{2}} + e^{\left (2 \, e^{\left (x + 3\right )}\right )} - 4 \, e^{\left (e^{\left (x + 3\right )}\right )} + 13\right )} \]
integrate((2*exp(3+x)*exp(exp(3+x))^2+((-2*x^2+2*x-4)*exp(3+x)-4*x+2)*exp( exp(3+x))+4*x^3-6*x^2+10*x-4)*exp((log(3)^2*exp(exp(3+x))^2+(-2*x^2+2*x-4) *log(3)^2*exp(exp(3+x))+(x^4-2*x^3+5*x^2-4*x+13)*log(3)^2-12*log(3)+4)/log (3)^2),x, algorithm=\
e^(x^4 - 2*x^3 - 2*x^2*e^(e^(x + 3)) + 5*x^2 + 2*x*e^(e^(x + 3)) - 4*x - 1 2/log(3) + 4/log(3)^2 + e^(2*e^(x + 3)) - 4*e^(e^(x + 3)) + 13)
\[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx=\int { 2 \, {\left (2 \, x^{3} - 3 \, x^{2} - {\left ({\left (x^{2} - x + 2\right )} e^{\left (x + 3\right )} + 2 \, x - 1\right )} e^{\left (e^{\left (x + 3\right )}\right )} + 5 \, x + e^{\left (x + 2 \, e^{\left (x + 3\right )} + 3\right )} - 2\right )} e^{\left (-\frac {2 \, {\left (x^{2} - x + 2\right )} e^{\left (e^{\left (x + 3\right )}\right )} \log \left (3\right )^{2} - {\left (x^{4} - 2 \, x^{3} + 5 \, x^{2} - 4 \, x + 13\right )} \log \left (3\right )^{2} - e^{\left (2 \, e^{\left (x + 3\right )}\right )} \log \left (3\right )^{2} + 12 \, \log \left (3\right ) - 4}{\log \left (3\right )^{2}}\right )} \,d x } \]
integrate((2*exp(3+x)*exp(exp(3+x))^2+((-2*x^2+2*x-4)*exp(3+x)-4*x+2)*exp( exp(3+x))+4*x^3-6*x^2+10*x-4)*exp((log(3)^2*exp(exp(3+x))^2+(-2*x^2+2*x-4) *log(3)^2*exp(exp(3+x))+(x^4-2*x^3+5*x^2-4*x+13)*log(3)^2-12*log(3)+4)/log (3)^2),x, algorithm=\
integrate(2*(2*x^3 - 3*x^2 - ((x^2 - x + 2)*e^(x + 3) + 2*x - 1)*e^(e^(x + 3)) + 5*x + e^(x + 2*e^(x + 3) + 3) - 2)*e^(-(2*(x^2 - x + 2)*e^(e^(x + 3 ))*log(3)^2 - (x^4 - 2*x^3 + 5*x^2 - 4*x + 13)*log(3)^2 - e^(2*e^(x + 3))* log(3)^2 + 12*log(3) - 4)/log(3)^2), x)
Time = 9.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.38 \[ \int e^{\frac {4-12 \log (3)+e^{2 e^{3+x}} \log ^2(3)+e^{e^{3+x}} \left (-4+2 x-2 x^2\right ) \log ^2(3)+\left (13-4 x+5 x^2-2 x^3+x^4\right ) \log ^2(3)}{\log ^2(3)}} \left (-4+2 e^{3+2 e^{3+x}+x}+10 x-6 x^2+4 x^3+e^{e^{3+x}} \left (2-4 x+e^{3+x} \left (-4+2 x-2 x^2\right )\right )\right ) \, dx={\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{13}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^3\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4}{{\ln \left (3\right )}^2}}\,{\mathrm {e}}^{-\frac {12}{\ln \left (3\right )}}\,{\mathrm {e}}^{-2\,x^3}\,{\mathrm {e}}^{5\,x^2}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{{\mathrm {e}}^3\,{\mathrm {e}}^x}} \]
int(exp((exp(2*exp(x + 3))*log(3)^2 - 12*log(3) + log(3)^2*(5*x^2 - 4*x - 2*x^3 + x^4 + 13) - exp(exp(x + 3))*log(3)^2*(2*x^2 - 2*x + 4) + 4)/log(3) ^2)*(10*x - exp(exp(x + 3))*(4*x + exp(x + 3)*(2*x^2 - 2*x + 4) - 2) + 2*e xp(2*exp(x + 3))*exp(x + 3) - 6*x^2 + 4*x^3 - 4),x)