\(\int e^{e^x+3 x-6 x^2} (-1-3 x-e^x x+12 x^2) \, dx\) [7565]

Optimal result

Integrand size = 31, antiderivative size = 19 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=3-e^{e^x+3 x-6 x^2} x \]

[Out]

3-exp(exp(x)-6*x^2+3*x)*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2326} \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-\frac {e^{-6 x^2+3 x+e^x} \left (-12 x^2+e^x x+3 x\right )}{-12 x+e^x+3} \]

[In]

Int[E^(E^x + 3*x - 6*x^2)*(-1 - 3*x - E^x*x + 12*x^2),x]

[Out]

-((E^(E^x + 3*x - 6*x^2)*(3*x + E^x*x - 12*x^2))/(3 + E^x - 12*x))

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{e^x+3 x-6 x^2} \left (3 x+e^x x-12 x^2\right )}{3+e^x-12 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-e^{e^x+3 x-6 x^2} x \]

[In]

Integrate[E^(E^x + 3*x - 6*x^2)*(-1 - 3*x - E^x*x + 12*x^2),x]

[Out]

-(E^(E^x + 3*x - 6*x^2)*x)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
norman	\(-{\mathrm e}^{{\mathrm e}^{x}-6 x^{2}+3 x} x\)	\(16\)
risch	\(-{\mathrm e}^{{\mathrm e}^{x}-6 x^{2}+3 x} x\)	\(16\)
parallelrisch	\(-{\mathrm e}^{{\mathrm e}^{x}-6 x^{2}+3 x} x\)	\(16\)

[In]

int((-exp(x)*x+12*x^2-3*x-1)*exp(exp(x)-6*x^2+3*x),x,method=_RETURNVERBOSE)

[Out]

-exp(exp(x)-6*x^2+3*x)*x

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-x e^{\left (-6 \, x^{2} + 3 \, x + e^{x}\right )} \]

[In]

integrate((-exp(x)*x+12*x^2-3*x-1)*exp(exp(x)-6*x^2+3*x),x, algorithm="fricas")

[Out]

-x*e^(-6*x^2 + 3*x + e^x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=- x e^{- 6 x^{2} + 3 x + e^{x}} \]

[In]

integrate((-exp(x)*x+12*x**2-3*x-1)*exp(exp(x)-6*x**2+3*x),x)

[Out]

-x*exp(-6*x**2 + 3*x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-x e^{\left (-6 \, x^{2} + 3 \, x + e^{x}\right )} \]

[In]

integrate((-exp(x)*x+12*x^2-3*x-1)*exp(exp(x)-6*x^2+3*x),x, algorithm="maxima")

[Out]

-x*e^(-6*x^2 + 3*x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-x e^{\left (-6 \, x^{2} + 3 \, x + e^{x}\right )} \]

[In]

integrate((-exp(x)*x+12*x^2-3*x-1)*exp(exp(x)-6*x^2+3*x),x, algorithm="giac")

[Out]

-x*e^(-6*x^2 + 3*x + e^x)

Mupad [B] (verification not implemented)

Time = 12.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int e^{e^x+3 x-6 x^2} \left (-1-3 x-e^x x+12 x^2\right ) \, dx=-x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-6\,x^2} \]

[In]

int(-exp(3*x + exp(x) - 6*x^2)*(3*x + x*exp(x) - 12*x^2 + 1),x)

[Out]

-x*exp(3*x)*exp(exp(x))*exp(-6*x^2)