3.236 \(\int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx\)

Optimal. Leaf size=8 \[ -\sin ^{-1}(2-x) \]

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Rubi [A]  time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {619, 216} \[ -\sin ^{-1}(2-x) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 4*x - x^2],x]

[Out]

-ArcSin[2 - x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,4-2 x\right )\right )\\ &=-\sin ^{-1}(2-x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 8, normalized size = 1.00 \[ -\sin ^{-1}(2-x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-3 + 4*x - x^2],x]

[Out]

-ArcSin[2 - x]

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IntegrateAlgebraic [B]  time = 0.09, size = 23, normalized size = 2.88 \[ -2 \tan ^{-1}\left (\frac {\sqrt {-x^2+4 x-3}}{x-1}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[-3 + 4*x - x^2],x]

[Out]

-2*ArcTan[Sqrt[-3 + 4*x - x^2]/(-1 + x)]

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fricas [B]  time = 0.98, size = 29, normalized size = 3.62 \[ -\arctan \left (\frac {\sqrt {-x^{2} + 4 \, x - 3} {\left (x - 2\right )}}{x^{2} - 4 \, x + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4*x-3)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(-x^2 + 4*x - 3)*(x - 2)/(x^2 - 4*x + 3))

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giac [A]  time = 0.63, size = 4, normalized size = 0.50 \[ \arcsin \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4*x-3)^(1/2),x, algorithm="giac")

[Out]

arcsin(x - 2)

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maple [A]  time = 0.32, size = 5, normalized size = 0.62

method	result	size
default	\(\arcsin \left (-2+x \right )\)	\(5\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+4 x -3}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+4*x-3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsin(-2+x)

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maxima [A]  time = 1.17, size = 8, normalized size = 1.00 \[ -\arcsin \left (-x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4*x-3)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-x + 2)

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mupad [B]  time = 0.18, size = 4, normalized size = 0.50 \[ \mathrm {asin}\left (x-2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x - x^2 - 3)^(1/2),x)

[Out]

asin(x - 2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{2} + 4 x - 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+4*x-3)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 + 4*x - 3), x)

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