3.2.44 \(\int \frac {12 e^{-\frac {12}{-2+x}}}{4-4 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ e^{\frac {6}{-x+\frac {2+x}{2}}} \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 27, 2209} \begin {gather*} e^{\frac {12}{2-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[12/(E^(12/(-2 + x))*(4 - 4*x + x^2)),x]

[Out]

E^(12/(2 - 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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=12 \int \frac {e^{-\frac {12}{-2+x}}}{4-4 x+x^2} \, dx\\ &=12 \int \frac {e^{-\frac {12}{-2+x}}}{(-2+x)^2} \, dx\\ &=e^{\frac {12}{2-x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.53 \begin {gather*} e^{-\frac {12}{-2+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[12/(E^(12/(-2 + x))*(4 - 4*x + x^2)),x]

[Out]

E^(-12/(-2 + x))

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fricas [A]  time = 0.77, size = 8, normalized size = 0.47 \begin {gather*} e^{\left (-\frac {12}{x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*exp(-6/(x-2))^2/(x^2-4*x+4),x, algorithm="fricas")

[Out]

e^(-12/(x - 2))

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giac [A]  time = 0.42, size = 8, normalized size = 0.47 \begin {gather*} e^{\left (-\frac {12}{x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*exp(-6/(x-2))^2/(x^2-4*x+4),x, algorithm="giac")

[Out]

e^(-12/(x - 2))

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maple [A]  time = 0.09, size = 9, normalized size = 0.53

method	result	size
risch	\({\mathrm e}^{-\frac {12}{x -2}}\)	\(9\)
gosper	\({\mathrm e}^{-\frac {12}{x -2}}\)	\(11\)
derivativedivides	\({\mathrm e}^{-\frac {12}{x -2}}\)	\(11\)
default	\({\mathrm e}^{-\frac {12}{x -2}}\)	\(11\)
norman	\(\frac {x \,{\mathrm e}^{-\frac {12}{x -2}}-2 \,{\mathrm e}^{-\frac {12}{x -2}}}{x -2}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12*exp(-6/(x-2))^2/(x^2-4*x+4),x,method=_RETURNVERBOSE)

[Out]

exp(-12/(x-2))

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maxima [A]  time = 0.54, size = 8, normalized size = 0.47 \begin {gather*} e^{\left (-\frac {12}{x - 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*exp(-6/(x-2))^2/(x^2-4*x+4),x, algorithm="maxima")

[Out]

e^(-12/(x - 2))

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mupad [B]  time = 0.08, size = 8, normalized size = 0.47 \begin {gather*} {\mathrm {e}}^{-\frac {12}{x-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(-12/(x - 2)))/(x^2 - 4*x + 4),x)

[Out]

exp(-12/(x - 2))

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sympy [A]  time = 0.12, size = 7, normalized size = 0.41 \begin {gather*} e^{- \frac {12}{x - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(12*exp(-6/(x-2))**2/(x**2-4*x+4),x)

[Out]

exp(-12/(x - 2))

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