3.27 \(\int \frac{1}{\sqrt{6 x-x^2}} \, dx\)

Optimal. Leaf size=10 \[ -\sin ^{-1}\left (1-\frac{x}{3}\right ) \]

[Out]

-ArcSin[1 - x/3]

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Rubi [A]  time = 0.0062504, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {619, 216} \[ -\sin ^{-1}\left (1-\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[6*x - x^2],x]

[Out]

-ArcSin[1 - x/3]

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]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{6 x-x^2}} \, dx &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,6-2 x\right )\right )\\ &=-\sin ^{-1}\left (1-\frac{x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0096672, size = 14, normalized size = 1.4 \[ -2 \sin ^{-1}\left (\sqrt{1-\frac{x}{6}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[6*x - x^2],x]

[Out]

-2*ArcSin[Sqrt[1 - x/6]]

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Maple [A]  time = 0.065, size = 7, normalized size = 0.7 \begin{align*} \arcsin \left ( -1+{\frac{x}{3}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+6*x)^(1/2),x)

[Out]

arcsin(-1+1/3*x)

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Maxima [A]  time = 3.88425, size = 11, normalized size = 1.1 \begin{align*} -\arcsin \left (-\frac{1}{3} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-1/3*x + 1)

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Fricas [B]  time = 2.27557, size = 42, normalized size = 4.2 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 6 \, x}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan(sqrt(-x^2 + 6*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x^{2} + 6 x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+6*x)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 + 6*x), x)

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Giac [A]  time = 1.61726, size = 8, normalized size = 0.8 \begin{align*} \arcsin \left (\frac{1}{3} \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x)^(1/2),x, algorithm="giac")

[Out]

arcsin(1/3*x - 1)