Integrand size = 83, antiderivative size = 21 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \]
Time = 5.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (1+x)} x^2 \]
Integrate[E^(58*x + 60*x^2 - 2*E^E^(-2*x)*(x + x^2))*(E^(2*x)*(2*x + 60*x^ 2 + 120*x^3) + E^E^(-2*x)*(4*x^3 + 4*x^4 + E^(2*x)*(-2*x^2 - 4*x^3))),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.
\(\displaystyle \int e^{60 x^2-2 e^{e^{-2 x}} \left (x^2+x\right )+58 x} \left (e^{2 x} \left (120 x^3+60 x^2+2 x\right )+e^{e^{-2 x}} \left (4 x^4+4 x^3+e^{2 x} \left (-4 x^3-2 x^2\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \left (e^{2 x} \left (120 x^3+60 x^2+2 x\right )+e^{e^{-2 x}} \left (4 x^4+4 x^3+e^{2 x} \left (-4 x^3-2 x^2\right )\right )\right ) \exp \left (-2 x \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 \left (2 x^2-2 e^{2 x} x+2 x-e^{2 x}\right ) x^2 \exp \left (e^{-2 x}-2 x \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right )\right )+2 \left (60 x^2+30 x+1\right ) x \exp \left (2 x-2 x \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \exp \left (e^{-2 x}-2 x \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right )\right ) x^4dx+4 \int \exp \left (e^{-2 x}-2 x \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right )\right ) x^3dx-4 \int \exp \left (-2 \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right ) x+2 x+e^{-2 x}\right ) x^3dx-2 \int \exp \left (-2 \left (e^{e^{-2 x}} x-30 x+e^{e^{-2 x}}-29\right ) x+2 x+e^{-2 x}\right ) x^2dx+120 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (x+1)} x^3dx+60 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (x+1)} x^2dx+2 \int e^{-2 \left (-30+e^{e^{-2 x}}\right ) x (x+1)} xdx\) |
Int[E^(58*x + 60*x^2 - 2*E^E^(-2*x)*(x + x^2))*(E^(2*x)*(2*x + 60*x^2 + 12 0*x^3) + E^E^(-2*x)*(4*x^3 + 4*x^4 + E^(2*x)*(-2*x^2 - 4*x^3))),x]
3.28.15.3.1 Defintions of rubi rules used
Time = 0.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
risch | \(x^{2} {\mathrm e}^{-2 x \left (1+x \right ) \left ({\mathrm e}^{{\mathrm e}^{-2 x}}-30\right )}\) | \(19\) |
parallelrisch | \(x^{2} {\mathrm e}^{-2 \left (x^{2}+x \right ) {\mathrm e}^{{\mathrm e}^{-2 x}}+60 x^{2}+60 x}\) | \(28\) |
int((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+60*x^2 +2*x)*exp(x)^2)/exp(x)^2/exp((x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2,x,meth od=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{\left (60 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x\right )} \]
integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+ 60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2, x, algorithm=\
Time = 9.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{60 x^{2} + 60 x - 2 \left (x^{2} + x\right ) e^{e^{- 2 x}}} \]
integrate((((-4*x**3-2*x**2)*exp(x)**2+4*x**4+4*x**3)*exp(1/exp(x)**2)+(12 0*x**3+60*x**2+2*x)*exp(x)**2)/exp(x)**2/exp((x**2+x)*exp(1/exp(x)**2)-30* x**2-30*x)**2,x)
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^{2} e^{\left (-2 \, x^{2} e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x^{2} - 2 \, x e^{\left (e^{\left (-2 \, x\right )}\right )} + 60 \, x\right )} \]
integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+ 60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2, x, algorithm=\
\[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=\int { 2 \, {\left ({\left (60 \, x^{3} + 30 \, x^{2} + x\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{4} + 2 \, x^{3} - {\left (2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (e^{\left (-2 \, x\right )}\right )}\right )} e^{\left (60 \, x^{2} - 2 \, {\left (x^{2} + x\right )} e^{\left (e^{\left (-2 \, x\right )}\right )} + 58 \, x\right )} \,d x } \]
integrate((((-4*x^3-2*x^2)*exp(x)^2+4*x^4+4*x^3)*exp(1/exp(x)^2)+(120*x^3+ 60*x^2+2*x)*exp(x)^2)/exp(x)^2/exp((x^2+x)*exp(1/exp(x)^2)-30*x^2-30*x)^2, x, algorithm=\
integrate(2*((60*x^3 + 30*x^2 + x)*e^(2*x) + (2*x^4 + 2*x^3 - (2*x^3 + x^2 )*e^(2*x))*e^(e^(-2*x)))*e^(60*x^2 - 2*(x^2 + x)*e^(e^(-2*x)) + 58*x), x)
Time = 12.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int e^{58 x+60 x^2-2 e^{e^{-2 x}} \left (x+x^2\right )} \left (e^{2 x} \left (2 x+60 x^2+120 x^3\right )+e^{e^{-2 x}} \left (4 x^3+4 x^4+e^{2 x} \left (-2 x^2-4 x^3\right )\right )\right ) \, dx=x^2\,{\mathrm {e}}^{60\,x}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}}}\,{\mathrm {e}}^{60\,x^2}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}}} \]