3.30.29 \(\int \frac {-1+e^x (1-x)}{81 x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {1-e^x-2 x}{81 x} \]

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 2197} \begin {gather*} \frac {1}{81 x}-\frac {e^x}{81 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^x*(1 - x))/(81*x^2),x]

[Out]

1/(81*x) - E^x/(81*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {-1+e^x (1-x)}{x^2} \, dx\\ &=\frac {1}{81} \int \left (-\frac {1}{x^2}-\frac {e^x (-1+x)}{x^2}\right ) \, dx\\ &=\frac {1}{81 x}-\frac {1}{81} \int \frac {e^x (-1+x)}{x^2} \, dx\\ &=\frac {1}{81 x}-\frac {e^x}{81 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} -\frac {-1+e^x}{81 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x*(1 - x))/(81*x^2),x]

[Out]

-1/81*(-1 + E^x)/x

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fricas [A]  time = 0.78, size = 9, normalized size = 0.53 \begin {gather*} -\frac {e^{x} - 1}{81 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((-x+1)*exp(x)-1)/x^2,x, algorithm="fricas")

[Out]

-1/81*(e^x - 1)/x

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giac [A]  time = 0.18, size = 9, normalized size = 0.53 \begin {gather*} -\frac {e^{x} - 1}{81 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((-x+1)*exp(x)-1)/x^2,x, algorithm="giac")

[Out]

-1/81*(e^x - 1)/x

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maple [A]  time = 0.02, size = 11, normalized size = 0.65

method	result	size
norman	\(\frac {\frac {1}{81}-\frac {{\mathrm e}^{x}}{81}}{x}\)	\(11\)
default	\(-\frac {{\mathrm e}^{x}}{81 x}+\frac {1}{81 x}\)	\(14\)
risch	\(-\frac {{\mathrm e}^{x}}{81 x}+\frac {1}{81 x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*((1-x)*exp(x)-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

(1/81-1/81*exp(x))/x

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maxima [C]  time = 0.71, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{81 \, x} - \frac {1}{81} \, {\rm Ei}\relax (x) + \frac {1}{81} \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((-x+1)*exp(x)-1)/x^2,x, algorithm="maxima")

[Out]

1/81/x - 1/81*Ei(x) + 1/81*gamma(-1, -x)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.53 \begin {gather*} -\frac {{\mathrm {e}}^x-1}{81\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(x)*(x - 1))/81 + 1/81)/x^2,x)

[Out]

-(exp(x) - 1)/(81*x)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.59 \begin {gather*} - \frac {e^{x}}{81 x} + \frac {1}{81 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((-x+1)*exp(x)-1)/x**2,x)

[Out]

-exp(x)/(81*x) + 1/(81*x)

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