\(\int e^{-2 e^4+e^{-2 e^4} (-3 e^{3+2 e^4}-3 x+3 x^2)} (-3+6 x) \, dx\) [3019]

Optimal result

Integrand size = 42, antiderivative size = 24 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=2+e^{3 \left (-e^3+e^{-2 e^4} (-1+x) x\right )} \]

[Out]

2+exp(3*x*(-1+x)/exp(exp(4))^2-3*exp(3))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2276, 2268} \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{3 e^{-2 e^4} x^2-3 e^{-2 e^4} x-3 e^3} \]

[In]

Int[E^(-2*E^4 + (-3*E^(3 + 2*E^4) - 3*x + 3*x^2)/E^(2*E^4))*(-3 + 6*x),x]

[Out]

E^(-3*E^3 - (3*x)/E^(2*E^4) + (3*x^2)/E^(2*E^4))

Rule 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2276

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int \exp \left (-e^3 (3+2 e)-3 e^{-2 e^4} x+3 e^{-2 e^4} x^2\right ) (-3+6 x) \, dx \\ & = \exp \left (-3 e^3-3 e^{-2 e^4} x+3 e^{-2 e^4} x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{-3 e^{-2 e^4} \left (e^{3+2 e^4}+x-x^2\right )} \]

[In]

Integrate[E^(-2*E^4 + (-3*E^(3 + 2*E^4) - 3*x + 3*x^2)/E^(2*E^4))*(-3 + 6*x),x]

[Out]

E^((-3*(E^(3 + 2*E^4) + x - x^2))/E^(2*E^4))

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
gosper	\({\mathrm e}^{-3 \left ({\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{4}}-x^{2}+x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}}}\)	\(24\)
parallelrisch	\({\mathrm e}^{-3 \left ({\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{4}}-x^{2}+x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}}}\)	\(24\)
risch	\({\mathrm e}^{3 \left (x^{2}-{\mathrm e}^{2 \,{\mathrm e}^{4}+3}-x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}}}\)	\(25\)
derivativedivides	\({\mathrm e}^{\left (-3 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{4}}+3 x^{2}-3 x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}}}\)	\(26\)
norman	\({\mathrm e}^{\left (-3 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{4}}+3 x^{2}-3 x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}}}\)	\(26\)
default	\(3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (\frac {i {\mathrm e}^{-3 \,{\mathrm e}^{3}} \sqrt {\pi }\, {\mathrm e}^{-\frac {3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}}{4}} \sqrt {3}\, {\mathrm e}^{{\mathrm e}^{4}} \operatorname {erf}\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i {\mathrm e}^{-2 \,{\mathrm e}^{4}} \sqrt {3}\, {\mathrm e}^{{\mathrm e}^{4}}}{2}\right )}{6}+2 \,{\mathrm e}^{-3 \,{\mathrm e}^{3}} \left (\frac {{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}} x^{2}-3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}} x}}{6}-\frac {i \sqrt {\pi }\, {\mathrm e}^{-\frac {3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}}{4}} \sqrt {3}\, {\mathrm e}^{{\mathrm e}^{4}} \operatorname {erf}\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i {\mathrm e}^{-2 \,{\mathrm e}^{4}} \sqrt {3}\, {\mathrm e}^{{\mathrm e}^{4}}}{2}\right )}{12}\right )\right )\)	\(163\)
parts	\(\frac {i {\mathrm e}^{-{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{-3 \,{\mathrm e}^{3}-\frac {3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}}{4}} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}}}{2}\right )}{2}-i {\mathrm e}^{-{\mathrm e}^{4}} \sqrt {\pi }\, {\mathrm e}^{-3 \,{\mathrm e}^{3}-\frac {3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}}{4}} \sqrt {3}\, \operatorname {erf}\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}}}{2}\right ) x +{\mathrm e}^{-3 \,{\mathrm e}^{3}-\frac {3 \,{\mathrm e}^{-2 \,{\mathrm e}^{4}}}{4}} \left (\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}}}{2}\right ) \operatorname {erf}\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}}}{2}\right ) \sqrt {\pi }+{\mathrm e}^{-\left (i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}} x -\frac {i \sqrt {3}\, {\mathrm e}^{-{\mathrm e}^{4}}}{2}\right )^{2}}\right )\)	\(204\)

[In]

int((-3+6*x)*exp((-3*exp(3)*exp(exp(4))^2+3*x^2-3*x)/exp(exp(4))^2)/exp(exp(4))^2,x,method=_RETURNVERBOSE)

[Out]

exp(-3*(exp(3)*exp(exp(4))^2-x^2+x)/exp(exp(4))^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{\left ({\left (3 \, x^{2} - {\left (2 \, e^{4} + 3 \, e^{3}\right )} e^{\left (2 \, e^{4}\right )} - 3 \, x\right )} e^{\left (-2 \, e^{4}\right )} + 2 \, e^{4}\right )} \]

[In]

integrate((-3+6*x)*exp((-3*exp(3)*exp(exp(4))^2+3*x^2-3*x)/exp(exp(4))^2)/exp(exp(4))^2,x, algorithm="fricas")

[Out]

e^((3*x^2 - (2*e^4 + 3*e^3)*e^(2*e^4) - 3*x)*e^(-2*e^4) + 2*e^4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{\frac {3 x^{2} - 3 x - 3 e^{3} e^{2 e^{4}}}{e^{2 e^{4}}}} \]

[In]

integrate((-3+6*x)*exp((-3*exp(3)*exp(exp(4))**2+3*x**2-3*x)/exp(exp(4))**2)/exp(exp(4))**2,x)

[Out]

exp((3*x**2 - 3*x - 3*exp(3)*exp(2*exp(4)))*exp(-2*exp(4)))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{\left (3 \, {\left (x^{2} - x - e^{\left (2 \, e^{4} + 3\right )}\right )} e^{\left (-2 \, e^{4}\right )}\right )} \]

[In]

integrate((-3+6*x)*exp((-3*exp(3)*exp(exp(4))^2+3*x^2-3*x)/exp(exp(4))^2)/exp(exp(4))^2,x, algorithm="maxima")

[Out]

e^(3*(x^2 - x - e^(2*e^4 + 3))*e^(-2*e^4))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx=e^{\left (3 \, x^{2} e^{\left (-2 \, e^{4}\right )} - 3 \, x e^{\left (-2 \, e^{4}\right )} - 3 \, e^{3}\right )} \]

[In]

integrate((-3+6*x)*exp((-3*exp(3)*exp(exp(4))^2+3*x^2-3*x)/exp(exp(4))^2)/exp(exp(4))^2,x, algorithm="giac")

[Out]

e^(3*x^2*e^(-2*e^4) - 3*x*e^(-2*e^4) - 3*e^3)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int e^{-2 e^4+e^{-2 e^4} \left (-3 e^{3+2 e^4}-3 x+3 x^2\right )} (-3+6 x) \, dx={\mathrm {e}}^{-3\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-3\,x\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}}\,{\mathrm {e}}^{3\,x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}} \]

[In]

int(exp(-2*exp(4))*exp(-exp(-2*exp(4))*(3*x - 3*x^2 + 3*exp(2*exp(4))*exp(3)))*(6*x - 3),x)

[Out]

exp(-3*exp(3))*exp(-3*x*exp(-2*exp(4)))*exp(3*x^2*exp(-2*exp(4)))