3.20.61 \(\int \frac {-2+e^{e^4} (4-4 x+x^2)}{8-8 x+2 x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{2} x \left (e^{e^4}+\frac {x}{-2 x+x^2}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {27, 12, 1850} \begin {gather*} \frac {e^{e^4} x}{2}-\frac {1}{2-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^E^4*(4 - 4*x + x^2))/(8 - 8*x + 2*x^2),x]

[Out]

-(2 - x)^(-1) + (E^E^4*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 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+e^{e^4} \left (4-4 x+x^2\right )}{2 (-2+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {-2+e^{e^4} \left (4-4 x+x^2\right )}{(-2+x)^2} \, dx\\ &=\frac {1}{2} \int \left (e^{e^4}-\frac {2}{(-2+x)^2}\right ) \, dx\\ &=-\frac {1}{2-x}+\frac {e^{e^4} x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{2} \left (\frac {2}{-2+x}+e^{e^4} (-2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^E^4*(4 - 4*x + x^2))/(8 - 8*x + 2*x^2),x]

[Out]

(2/(-2 + x) + E^E^4*(-2 + x))/2

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fricas [A]  time = 0.99, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\left (x^{2} - 2 \, x\right )} e^{\left (e^{4}\right )} + 2}{2 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+4)*exp(exp(4))-2)/(2*x^2-8*x+8),x, algorithm="fricas")

[Out]

1/2*((x^2 - 2*x)*e^(e^4) + 2)/(x - 2)

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giac [A]  time = 0.36, size = 12, normalized size = 0.55 \begin {gather*} \frac {1}{2} \, x e^{\left (e^{4}\right )} + \frac {1}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+4)*exp(exp(4))-2)/(2*x^2-8*x+8),x, algorithm="giac")

[Out]

1/2*x*e^(e^4) + 1/(x - 2)

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maple [A]  time = 0.37, size = 13, normalized size = 0.59

method	result	size
default	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2}+\frac {1}{x -2}\)	\(13\)
risch	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2}+\frac {1}{x -2}\)	\(13\)
gosper	\(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2-4 \,{\mathrm e}^{{\mathrm e}^{4}}}{2 x -4}\)	\(22\)
norman	\(\frac {\frac {x^{2} {\mathrm e}^{{\mathrm e}^{4}}}{2}+1-2 \,{\mathrm e}^{{\mathrm e}^{4}}}{x -2}\)	\(22\)
meijerg	\(-\frac {x}{4 \left (1-\frac {x}{2}\right )}-{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {x \left (-\frac {3 x}{2}+6\right )}{6 \left (1-\frac {x}{2}\right )}-2 \ln \left (1-\frac {x}{2}\right )\right )-2 \,{\mathrm e}^{{\mathrm e}^{4}} \left (\frac {x}{2-x}+\ln \left (1-\frac {x}{2}\right )\right )+\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2-x}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-4*x+4)*exp(exp(4))-2)/(2*x^2-8*x+8),x,method=_RETURNVERBOSE)

[Out]

1/2*x*exp(exp(4))+1/(x-2)

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maxima [A]  time = 0.68, size = 12, normalized size = 0.55 \begin {gather*} \frac {1}{2} \, x e^{\left (e^{4}\right )} + \frac {1}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-4*x+4)*exp(exp(4))-2)/(2*x^2-8*x+8),x, algorithm="maxima")

[Out]

1/2*x*e^(e^4) + 1/(x - 2)

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mupad [B]  time = 1.14, size = 12, normalized size = 0.55 \begin {gather*} \frac {1}{x-2}+\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^4}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(4))*(x^2 - 4*x + 4) - 2)/(2*x^2 - 8*x + 8),x)

[Out]

1/(x - 2) + (x*exp(exp(4)))/2

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sympy [A]  time = 0.09, size = 12, normalized size = 0.55 \begin {gather*} \frac {x e^{e^{4}}}{2} + \frac {1}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-4*x+4)*exp(exp(4))-2)/(2*x**2-8*x+8),x)

[Out]

x*exp(exp(4))/2 + 1/(x - 2)

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