3.89.57 \(\int (14+e^{-2 x+x^2} (-1+2 x-2 x^2)) \, dx\)

Optimal. Leaf size=13 \[ \left (14-e^{(-2+x) x}\right ) x \]

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Rubi [B]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 2.23, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2288} \begin {gather*} 14 x-\frac {e^{x^2-2 x} \left (x-x^2\right )}{1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[14 + E^(-2*x + x^2)*(-1 + 2*x - 2*x^2),x]

[Out]

14*x - (E^(-2*x + x^2)*(x - x^2))/(1 - x)

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=14 x+\int e^{-2 x+x^2} \left (-1+2 x-2 x^2\right ) \, dx\\ &=14 x-\frac {e^{-2 x+x^2} \left (x-x^2\right )}{1-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.92 \begin {gather*} -\left (\left (-14+e^{(-2+x) x}\right ) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[14 + E^(-2*x + x^2)*(-1 + 2*x - 2*x^2),x]

[Out]

-((-14 + E^((-2 + x)*x))*x)

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fricas [A]  time = 0.53, size = 15, normalized size = 1.15 \begin {gather*} -x e^{\left (x^{2} - 2 \, x\right )} + 14 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="fricas")

[Out]

-x*e^(x^2 - 2*x) + 14*x

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giac [A]  time = 0.13, size = 15, normalized size = 1.15 \begin {gather*} -x e^{\left (x^{2} - 2 \, x\right )} + 14 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="giac")

[Out]

-x*e^(x^2 - 2*x) + 14*x

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maple [A]  time = 0.04, size = 14, normalized size = 1.08

method	result	size
risch	\(14 x -x \,{\mathrm e}^{\left (x -2\right ) x}\)	\(14\)
default	\(14 x -x \,{\mathrm e}^{x^{2}-2 x}\)	\(16\)
norman	\(14 x -x \,{\mathrm e}^{x^{2}-2 x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x,method=_RETURNVERBOSE)

[Out]

14*x-x*exp((x-2)*x)

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maxima [C]  time = 0.42, size = 120, normalized size = 9.23 \begin {gather*} \frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x - i\right ) e^{\left (-1\right )} + {\left (\frac {{\left (x - 1\right )}^{3} \Gamma \left (\frac {3}{2}, -{\left (x - 1\right )}^{2}\right )}{\left (-{\left (x - 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (x - 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x - 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 1\right )}^{2}}} - 2 \, e^{\left ({\left (x - 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} + {\left (\frac {\sqrt {\pi } {\left (x - 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x - 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 1\right )}^{2}}} + e^{\left ({\left (x - 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} + 14 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^2+2*x-1)*exp(x^2-2*x)+14,x, algorithm="maxima")

[Out]

1/2*I*sqrt(pi)*erf(I*x - I)*e^(-1) + ((x - 1)^3*gamma(3/2, -(x - 1)^2)/(-(x - 1)^2)^(3/2) - sqrt(pi)*(x - 1)*(
erf(sqrt(-(x - 1)^2)) - 1)/sqrt(-(x - 1)^2) - 2*e^((x - 1)^2))*e^(-1) + (sqrt(pi)*(x - 1)*(erf(sqrt(-(x - 1)^2
)) - 1)/sqrt(-(x - 1)^2) + e^((x - 1)^2))*e^(-1) + 14*x

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mupad [B]  time = 5.08, size = 13, normalized size = 1.00 \begin {gather*} -x\,\left ({\mathrm {e}}^{x^2-2\,x}-14\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(14 - exp(x^2 - 2*x)*(2*x^2 - 2*x + 1),x)

[Out]

-x*(exp(x^2 - 2*x) - 14)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.92 \begin {gather*} - x e^{x^{2} - 2 x} + 14 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2+2*x-1)*exp(x**2-2*x)+14,x)

[Out]

-x*exp(x**2 - 2*x) + 14*x

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