\(\int (-1+e^x-2 x-2 e^{2-x^2} x) \, dx\) [7711]

Optimal result

Integrand size = 20, antiderivative size = 22 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=-2+e^x+e^{2-x^2}-x-x^2 \]

[Out]

-2-x+exp(x)+exp(-x^2+2)-x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2225, 2240} \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=-x^2+e^{2-x^2}-x+e^x \]

[In]

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

[Out]

E^x + E^(2 - x^2) - x - x^2

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = -x-x^2-2 \int e^{2-x^2} x \, dx+\int e^x \, dx \\ & = e^x+e^{2-x^2}-x-x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=e^x+e^{2-x^2}-x-x^2 \]

[In]

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

[Out]

E^x + E^(2 - x^2) - x - x^2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
default	\(-x^{2}-x +{\mathrm e}^{-x^{2}+2}+{\mathrm e}^{x}\)	\(20\)
norman	\(-x^{2}-x +{\mathrm e}^{-x^{2}+2}+{\mathrm e}^{x}\)	\(20\)
risch	\(-x^{2}-x +{\mathrm e}^{-x^{2}+2}+{\mathrm e}^{x}\)	\(20\)
parallelrisch	\(-x^{2}-x +{\mathrm e}^{-x^{2}+2}+{\mathrm e}^{x}\)	\(20\)
parts	\(-x^{2}-x +{\mathrm e}^{-x^{2}+2}+{\mathrm e}^{x}\)	\(20\)

[In]

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

[Out]

-x^2-x+exp(-x^2+2)+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=-x^{2} - x + e^{\left (-x^{2} + 2\right )} + e^{x} \]

[In]

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

[Out]

-x^2 - x + e^(-x^2 + 2) + e^x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=- x^{2} - x + e^{x} + e^{2 - x^{2}} \]

[In]

integrate(exp(x)-2*x*exp(-x**2+2)-2*x-1,x)

[Out]

-x**2 - x + exp(x) + exp(2 - x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=-x^{2} - x + e^{\left (-x^{2} + 2\right )} + e^{x} \]

[In]

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

[Out]

-x^2 - x + e^(-x^2 + 2) + e^x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx=-x^{2} - x + e^{\left (-x^{2} + 2\right )} + e^{x} \]

[In]

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

[Out]

-x^2 - x + e^(-x^2 + 2) + e^x

Mupad [B] (verification not implemented)

Time = 13.70 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (-1+e^x-2 x-2 e^{2-x^2} x\right ) \, dx={\mathrm {e}}^x-x+{\mathrm {e}}^{2-x^2}-x^2 \]

[In]

int(exp(x) - 2*x - 2*x*exp(2 - x^2) - 1,x)

[Out]

exp(x) - x + exp(2 - x^2) - x^2