Integrand size = 68, antiderivative size = 26 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=e^{-2 x} \left (1+e+2 x-e^{-e^4} (1+x)\right )^2 \]
Time = 5.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=e^{-2 \left (e^4+x\right )} \left (-1+e^{1+e^4}-x+e^{e^4} (1+2 x)\right )^2 \]
Integrate[E^(-2*E^4 - 2*x)*(-2*x - 2*x^2 + E^(2*E^4)*(2 - 2*E^2 - 8*E*x - 8*x^2) + E^E^4*(-2 + 4*x + 8*x^2 + E*(2 + 4*x))),x]
Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(26)=52\).
Time = 0.50 (sec) , antiderivative size = 192, normalized size of antiderivative = 7.38, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2626, 2009}
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.
\(\displaystyle \int e^{-2 x-2 e^4} \left (-2 x^2+e^{2 e^4} \left (-8 x^2-8 e x-2 e^2+2\right )+e^{e^4} \left (8 x^2+4 x+e (4 x+2)-2\right )-2 x\right ) \, dx\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \int \left (-2 e^{-2 x-2 e^4} x^2-2 e^{-2 x} \left (4 x^2+4 e x+e^2-1\right )+2 e^{-2 x-e^4} \left (4 x^2+2 (1+e) x+e-1\right )-2 e^{-2 x-2 e^4} x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{-2 x-2 e^4} x^2-4 e^{-2 x-e^4} x^2+4 e^{-2 x} x^2+4 e^{1-2 x} x+2 e^{-2 x-2 e^4} x-4 e^{-2 x-e^4} x+4 e^{-2 x} x-2 (1+e) e^{-2 x-e^4} x+2 e^{1-2 x}+e^{-2 x-2 e^4}+(1-e) e^{-2 x-e^4}-2 e^{-2 x-e^4}+2 e^{-2 x}-\left (1-e^2\right ) e^{-2 x}-(1+e) e^{-2 x-e^4}\) |
Int[E^(-2*E^4 - 2*x)*(-2*x - 2*x^2 + E^(2*E^4)*(2 - 2*E^2 - 8*E*x - 8*x^2) + E^E^4*(-2 + 4*x + 8*x^2 + E*(2 + 4*x))),x]
2*E^(1 - 2*x) + E^(-2*E^4 - 2*x) - 2*E^(-E^4 - 2*x) + (1 - E)*E^(-E^4 - 2* x) + 2/E^(2*x) - E^(-E^4 - 2*x)*(1 + E) - (1 - E^2)/E^(2*x) + 4*E^(1 - 2*x )*x + 2*E^(-2*E^4 - 2*x)*x - 4*E^(-E^4 - 2*x)*x + (4*x)/E^(2*x) - 2*E^(-E^ 4 - 2*x)*(1 + E)*x + E^(-2*E^4 - 2*x)*x^2 - 4*E^(-E^4 - 2*x)*x^2 + (4*x^2) /E^(2*x)
3.30.25.3.1 Defintions of rubi rules used
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
gosper | \(\left ({\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+2 x \,{\mathrm e}^{{\mathrm e}^{4}}+{\mathrm e}^{{\mathrm e}^{4}}-x -1\right )^{2} {\mathrm e}^{-2 x} {\mathrm e}^{-2 \,{\mathrm e}^{4}}\) | \(33\) |
risch | \(\left ({\mathrm e}^{2 \,{\mathrm e}^{4}+2}+4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -2 x \,{\mathrm e}^{{\mathrm e}^{4}+1}-4 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}+1}-6 x \,{\mathrm e}^{{\mathrm e}^{4}}+x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+2 x +1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}-2 x}\) | \(102\) |
parallelrisch | \({\mathrm e}^{-2 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e} x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}-2 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{4}} x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -4 x^{2} {\mathrm e}^{{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-6 x \,{\mathrm e}^{{\mathrm e}^{4}}+x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+2 x \right ) {\mathrm e}^{-2 x}\) | \(105\) |
norman | \(\left (\left ({\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}\right ) {\mathrm e}^{-{\mathrm e}^{4}}+\left (4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-4 \,{\mathrm e}^{{\mathrm e}^{4}}+1\right ) {\mathrm e}^{-{\mathrm e}^{4}} x^{2}+2 \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}-{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}-3 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-{\mathrm e}^{4}} x \right ) {\mathrm e}^{-2 x} {\mathrm e}^{-{\mathrm e}^{4}}\) | \(117\) |
meijerg | \(-{\mathrm e}^{2} \left (1-{\mathrm e}^{-2 x}\right )+{\mathrm e}^{1-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )+1-{\mathrm e}^{-2 x}-{\mathrm e}^{-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )+\frac {\left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (1-\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{2}\right )}{4}+\frac {\left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+8 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (2-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{3}\right )}{8}\) | \(134\) |
default | \({\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (2 \,{\mathrm e}^{-2 x} x +{\mathrm e}^{-2 x}+{\mathrm e}^{-2 x} x^{2}+{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{-2 x} {\mathrm e}^{2 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+8 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x^{2}}{2}-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x^{2}}{2}-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+{\mathrm e}^{-2 x} {\mathrm e}^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}-{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+4 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )\right )\) | \(192\) |
int(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+8*x^2+ 4*x-2)*exp(exp(4))-2*x^2-2*x)/exp(x)^2/exp(exp(4))^2,x,method=_RETURNVERBO SE)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e + 4 \, x + e^{2} + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + {\left (x + 1\right )} e + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} \]
integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+ 8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/exp(x)^2/exp(exp(4))^2,x, algorithm=\
(4*x^2 + 2*(2*x + 1)*e + 4*x + e^2 + 1)*e^(-2*x) - 2*(2*x^2 + (x + 1)*e + 3*x + 1)*e^(-2*x - e^4) + (x^2 + 2*x + 1)*e^(-2*x - 2*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=\frac {\left (- 4 x^{2} e^{e^{4}} + x^{2} + 4 x^{2} e^{2 e^{4}} - 6 x e^{e^{4}} - 2 e x e^{e^{4}} + 2 x + 4 x e^{2 e^{4}} + 4 e x e^{2 e^{4}} - 2 e e^{e^{4}} - 2 e^{e^{4}} + 1 + e^{2 e^{4}} + 2 e e^{2 e^{4}} + e^{2} e^{2 e^{4}}\right ) e^{- 2 x}}{e^{2 e^{4}}} \]
integrate(((-2*exp(1)**2-8*x*exp(1)-8*x**2+2)*exp(exp(4))**2+((4*x+2)*exp( 1)+8*x**2+4*x-2)*exp(exp(4))-2*x**2-2*x)/exp(x)**2/exp(exp(4))**2,x)
(-4*x**2*exp(exp(4)) + x**2 + 4*x**2*exp(2*exp(4)) - 6*x*exp(exp(4)) - 2*E *x*exp(exp(4)) + 2*x + 4*x*exp(2*exp(4)) + 4*E*x*exp(2*exp(4)) - 2*E*exp(e xp(4)) - 2*exp(exp(4)) + 1 + exp(2*exp(4)) + 2*E*exp(2*exp(4)) + exp(2)*ex p(2*exp(4)))*exp(-2*x)*exp(-2*exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.04 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (2 \, x e + e\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x e + e\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} - e^{\left (-2 \, x\right )} - e^{\left (-2 \, x - e^{4} + 1\right )} + e^{\left (-2 \, x - e^{4}\right )} + e^{\left (-2 \, x + 2\right )} \]
integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+ 8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/exp(x)^2/exp(exp(4))^2,x, algorithm=\
2*(2*x^2 + 2*x + 1)*e^(-2*x) + 2*(2*x*e + e)*e^(-2*x) - 2*(2*x^2 + 2*x + 1 )*e^(-2*x - e^4) - (2*x*e + e)*e^(-2*x - e^4) - (2*x + 1)*e^(-2*x - e^4) + 1/2*(2*x^2 + 2*x + 1)*e^(-2*x - 2*e^4) + 1/2*(2*x + 1)*e^(-2*x - 2*e^4) - e^(-2*x) - e^(-2*x - e^4 + 1) + e^(-2*x - e^4) + e^(-2*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\left (4 \, x^{2} + 4 \, x + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (x + 1\right )} e^{\left (-2 \, x - e^{4} + 1\right )} - 2 \, {\left (2 \, x^{2} + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + 2 \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x + 1\right )} + e^{\left (-2 \, x + 2\right )} \]
integrate(((-2*exp(1)^2-8*x*exp(1)-8*x^2+2)*exp(exp(4))^2+((4*x+2)*exp(1)+ 8*x^2+4*x-2)*exp(exp(4))-2*x^2-2*x)/exp(x)^2/exp(exp(4))^2,x, algorithm=\
(4*x^2 + 4*x + 1)*e^(-2*x) - 2*(x + 1)*e^(-2*x - e^4 + 1) - 2*(2*x^2 + 3*x + 1)*e^(-2*x - e^4) + (x^2 + 2*x + 1)*e^(-2*x - 2*e^4) + 2*(2*x + 1)*e^(- 2*x + 1) + e^(-2*x + 2)
Time = 12.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.19 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}+{\mathrm {e}}^{2\,{\mathrm {e}}^4+2}-2\,{\mathrm {e}}^{{\mathrm {e}}^4}+1\right )+x^2\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,{\left (2\,{\mathrm {e}}^{{\mathrm {e}}^4}-1\right )}^2+x\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left (4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}-6\,{\mathrm {e}}^{{\mathrm {e}}^4}+2\right ) \]
int(-exp(-2*exp(4))*exp(-2*x)*(2*x + exp(2*exp(4))*(2*exp(2) + 8*x*exp(1) + 8*x^2 - 2) - exp(exp(4))*(4*x + 8*x^2 + exp(1)*(4*x + 2) - 2) + 2*x^2),x )