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

Optimal. Leaf size=20 \[ \frac {1}{2} \left (5-e^7-x-3 e^x x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 2176, 2194} \begin {gather*} -\frac {x}{2}+\frac {3 e^x}{2}-\frac {3}{2} e^x (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^x*(-3 - 3*x))/2,x]

[Out]

(3*E^x)/2 - x/2 - (3*E^x*(1 + x))/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{2} \int \left (-1+e^x (-3-3 x)\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{2} \int e^x (-3-3 x) \, dx\\ &=-\frac {x}{2}-\frac {3}{2} e^x (1+x)+\frac {3 \int e^x \, dx}{2}\\ &=\frac {3 e^x}{2}-\frac {x}{2}-\frac {3}{2} e^x (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{2} \left (-x-3 e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x*(-3 - 3*x))/2,x]

[Out]

(-x - 3*E^x*x)/2

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fricas [A]  time = 0.65, size = 9, normalized size = 0.45 \begin {gather*} -\frac {3}{2} \, x e^{x} - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x*e^x - 1/2*x

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giac [A]  time = 0.14, size = 9, normalized size = 0.45 \begin {gather*} -\frac {3}{2} \, x e^{x} - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x*e^x - 1/2*x

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maple [A]  time = 0.01, size = 10, normalized size = 0.50

method	result	size
default	\(-\frac {x}{2}-\frac {3 \,{\mathrm e}^{x} x}{2}\)	\(10\)
norman	\(-\frac {x}{2}-\frac {3 \,{\mathrm e}^{x} x}{2}\)	\(10\)
risch	\(-\frac {x}{2}-\frac {3 \,{\mathrm e}^{x} x}{2}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 15, normalized size = 0.75 \begin {gather*} -\frac {3}{2} \, {\left (x - 1\right )} e^{x} - \frac {1}{2} \, x - \frac {3}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x - 1)*e^x - 1/2*x - 3/2*e^x

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mupad [B]  time = 0.05, size = 9, normalized size = 0.45 \begin {gather*} -\frac {x\,\left (3\,{\mathrm {e}}^x+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (exp(x)*(3*x + 3))/2 - 1/2,x)

[Out]

-(x*(3*exp(x) + 1))/2

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sympy [A]  time = 0.09, size = 12, normalized size = 0.60 \begin {gather*} - \frac {3 x e^{x}}{2} - \frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x*exp(x)/2 - x/2

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