3.35.10 \(\int e^{-x} (7-7 e^x-7 x) \, dx\)

Optimal. Leaf size=20 \[ 4-7 \left (x-e^{-x} x\right )-\log ^2(5) \]

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Rubi [A]  time = 0.08, antiderivative size = 12, normalized size of antiderivative = 0.60, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6741, 12, 6742, 2194, 2176} \begin {gather*} 7 e^{-x} x-7 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7 - 7*E^x - 7*x)/E^x,x]

[Out]

-7*x + (7*x)/E^x

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]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 7 e^{-x} \left (1-e^x-x\right ) \, dx\\ &=7 \int e^{-x} \left (1-e^x-x\right ) \, dx\\ &=7 \int \left (-1+e^{-x}-e^{-x} x\right ) \, dx\\ &=-7 x+7 \int e^{-x} \, dx-7 \int e^{-x} x \, dx\\ &=-7 e^{-x}-7 x+7 e^{-x} x-7 \int e^{-x} \, dx\\ &=-7 x+7 e^{-x} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.60 \begin {gather*} -7 \left (x-e^{-x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7 - 7*E^x - 7*x)/E^x,x]

[Out]

-7*(x - x/E^x)

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fricas [A]  time = 0.63, size = 14, normalized size = 0.70 \begin {gather*} -7 \, {\left (x e^{x} - x\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*exp(x)-7*x+7)/exp(x),x, algorithm="fricas")

[Out]

-7*(x*e^x - x)*e^(-x)

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giac [A]  time = 0.15, size = 11, normalized size = 0.55 \begin {gather*} 7 \, x e^{\left (-x\right )} - 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*exp(x)-7*x+7)/exp(x),x, algorithm="giac")

[Out]

7*x*e^(-x) - 7*x

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maple [A]  time = 0.01, size = 12, normalized size = 0.60

method	result	size
default	\(7 x \,{\mathrm e}^{-x}-7 x\)	\(12\)
risch	\(7 x \,{\mathrm e}^{-x}-7 x\)	\(12\)
norman	\(\left (7 x -7 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{-x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-7*exp(x)-7*x+7)/exp(x),x,method=_RETURNVERBOSE)

[Out]

7*x/exp(x)-7*x

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maxima [A]  time = 0.61, size = 19, normalized size = 0.95 \begin {gather*} 7 \, {\left (x + 1\right )} e^{\left (-x\right )} - 7 \, x - 7 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*exp(x)-7*x+7)/exp(x),x, algorithm="maxima")

[Out]

7*(x + 1)*e^(-x) - 7*x - 7*e^(-x)

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mupad [B]  time = 0.05, size = 9, normalized size = 0.45 \begin {gather*} 7\,x\,\left ({\mathrm {e}}^{-x}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(7*x + 7*exp(x) - 7),x)

[Out]

7*x*(exp(-x) - 1)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.40 \begin {gather*} - 7 x + 7 x e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*exp(x)-7*x+7)/exp(x),x)

[Out]

-7*x + 7*x*exp(-x)

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