3.1.27 \(\int \frac {1}{\sqrt {6 x-x^2}} \, dx\) [27]

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

[Out]

arcsin(-1+1/3*x)

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = {633, 222} \begin {gather*} -\text {ArcSin}\left (1-\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[6*x - x^2],x]

[Out]

-ArcSin[1 - x/3]

Rule 222

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 633

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 {6 x-x^2}} \, dx &=-\left (\frac {1}{6} \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(10)=20\).
time = 0.03, size = 38, normalized size = 3.80 \begin {gather*} \frac {2 \sqrt {-6+x} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-6+x}}\right )}{\sqrt {-((-6+x) x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[6*x - x^2],x]

[Out]

(2*Sqrt[-6 + x]*Sqrt[x]*ArcTanh[Sqrt[x]/Sqrt[-6 + x]])/Sqrt[-((-6 + x)*x)]

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Maple [A]
time = 0.38, size = 7, normalized size = 0.70

method	result	size
default	\(\arcsin \left (-1+\frac {x}{3}\right )\)	\(7\)
meijerg	\(2 \arcsin \left (\frac {\sqrt {6}\, \sqrt {x}}{6}\right )\)	\(12\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+6 x}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(-1+1/3*x)

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Maxima [A]
time = 0.50, size = 8, normalized size = 0.80 \begin {gather*} -\arcsin \left (-\frac {1}{3} \, x + 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
time = 1.66, size = 18, normalized size = 1.80 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} + 6 \, x}}{x}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} + 6 x}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (6) = 12\).
time = 2.84, size = 25, normalized size = 2.50 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 6 \, x} {\left (x - 3\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 6*x)*(x - 3) + 9/2*arcsin(1/3*x - 1)

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Mupad [B]
time = 0.11, size = 6, normalized size = 0.60 \begin {gather*} \mathrm {asin}\left (\frac {x}{3}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(6*x - x^2)^(1/2),x)

[Out]

asin(x/3 - 1)

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