3.27.35 \(\int (3+e^{\frac {1}{2} (-1+2 x)} (-1-x)) \, dx\)

Optimal. Leaf size=13 \[ \left (3-e^{-\frac {1}{2}+x}\right ) x \]

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Rubi [B]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 2.38, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2176, 2194} \begin {gather*} 3 x+e^{\frac {1}{2} (2 x-1)}-e^{\frac {1}{2} (2 x-1)} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3 + E^((-1 + 2*x)/2)*(-1 - x),x]

[Out]

E^((-1 + 2*x)/2) + 3*x - E^((-1 + 2*x)/2)*(1 + x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3 x+\int e^{\frac {1}{2} (-1+2 x)} (-1-x) \, dx\\ &=3 x-e^{\frac {1}{2} (-1+2 x)} (1+x)+\int e^{\frac {1}{2} (-1+2 x)} \, dx\\ &=e^{\frac {1}{2} (-1+2 x)}+3 x-e^{\frac {1}{2} (-1+2 x)} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 14, normalized size = 1.08 \begin {gather*} 3 x-e^{-\frac {1}{2}+x} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x - E^(-1/2 + x)*x

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fricas [A]  time = 0.58, size = 11, normalized size = 0.85 \begin {gather*} -x e^{\left (x - \frac {1}{2}\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*e^(x - 1/2) + 3*x

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giac [A]  time = 0.19, size = 11, normalized size = 0.85 \begin {gather*} -x e^{\left (x - \frac {1}{2}\right )} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*e^(x - 1/2) + 3*x

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maple [A]  time = 0.02, size = 12, normalized size = 0.92

method	result	size
norman	\(3 x -{\mathrm e}^{x -\frac {1}{2}} x\)	\(12\)
risch	\(3 x -{\mathrm e}^{x -\frac {1}{2}} x\)	\(12\)
default	\(3 x -{\mathrm e}^{x -\frac {1}{2}} \left (x -\frac {1}{2}\right )-\frac {{\mathrm e}^{x -\frac {1}{2}}}{2}\)	\(20\)
derivativedivides	\(3 x -\frac {3}{2}-{\mathrm e}^{x -\frac {1}{2}} \left (x -\frac {1}{2}\right )-\frac {{\mathrm e}^{x -\frac {1}{2}}}{2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x-1)*exp(x-1/2)+3,x,method=_RETURNVERBOSE)

[Out]

3*x-exp(x-1/2)*x

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maxima [B]  time = 0.64, size = 25, normalized size = 1.92 \begin {gather*} -{\left (x e^{\frac {1}{2}} - e^{\frac {1}{2}}\right )} e^{\left (x - 1\right )} + 3 \, x - e^{\left (x - \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e^(1/2) - e^(1/2))*e^(x - 1) + 3*x - e^(x - 1/2)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.69 \begin {gather*} -x\,\left ({\mathrm {e}}^{x-\frac {1}{2}}-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3 - exp(x - 1/2)*(x + 1),x)

[Out]

-x*(exp(x - 1/2) - 3)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.77 \begin {gather*} - x e^{x - \frac {1}{2}} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(x - 1/2) + 3*x

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