3.2.1 \(\int e^{1+x-x^2} (1-2 x) \, dx\) [101]

Optimal. Leaf size=10 \[ e^{1+x-x^2} \]

[Out]

exp(-x^2+x+1)

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2268} \begin {gather*} e^{-x^2+x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 + x - x^2)*(1 - 2*x),x]

[Out]

E^(1 + x - x^2)

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=e^{1+x-x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} e^{1+x-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(1 + x - x^2)

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Maple [A]
time = 0.02, size = 10, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
derivativedivides	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
default	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
norman	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
risch	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
parallelrisch	\({\mathrm e}^{-x^{2}+x +1}\)	\(10\)
parts	\(-\sqrt {\pi }\, {\mathrm e}^{\frac {5}{4}} \erf \left (x -\frac {1}{2}\right ) x +\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {5}{4}} \erf \left (x -\frac {1}{2}\right )}{2}+\frac {{\mathrm e}^{\frac {5}{4}} \left (2 \,\erf \left (x -\frac {1}{2}\right ) x \sqrt {\pi }-\erf \left (x -\frac {1}{2}\right ) \sqrt {\pi }+2 \,{\mathrm e}^{-\left (x -\frac {1}{2}\right )^{2}}\right )}{2}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-x^2+x+1)

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Maxima [A]
time = 0.42, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} + x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x^2 + x + 1)

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Fricas [A]
time = 0.58, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} + x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x^2 + x + 1)

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Sympy [A]
time = 0.04, size = 7, normalized size = 0.70 \begin {gather*} e^{- x^{2} + x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-x**2 + x + 1)

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Giac [A]
time = 0.51, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} + x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x^2 + x + 1)

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Mupad [B]
time = 0.20, size = 11, normalized size = 1.10 \begin {gather*} \mathrm {e}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x - x^2 + 1)*(2*x - 1),x)

[Out]

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

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Chatgpt [F] Failed to verify
time = 1.00, size = 28, normalized size = 2.80 \begin {gather*} -\frac {\left (1-2 x \right ) {\mathrm e}^{-x^{2}+x +1}}{2}-\frac {{\mathrm e}^{-x^{2}+x +1}}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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