3.1.69 \(\int \frac {x}{\sqrt {4 x-x^2}} \, dx\) [69]

Optimal. Leaf size=26 \[ -\sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]

[Out]

2*arcsin(-1+1/2*x)-(-x^2+4*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 633, 222} \begin {gather*} -2 \text {ArcSin}\left (1-\frac {x}{2}\right )-\sqrt {4 x-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4*x - x^2],x]

[Out]

-Sqrt[4*x - x^2] - 2*ArcSin[1 - x/2]

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]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {4 x-x^2}} \, dx &=-\sqrt {4 x-x^2}+2 \int \frac {1}{\sqrt {4 x-x^2}} \, dx\\ &=-\sqrt {4 x-x^2}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,4-2 x\right )\\ &=-\sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 43, normalized size = 1.65 \begin {gather*} \frac {(-4+x) x+4 \sqrt {-4+x} \sqrt {x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-4+x}{x}}}\right )}{\sqrt {-((-4+x) x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4*x - x^2],x]

[Out]

((-4 + x)*x + 4*Sqrt[-4 + x]*Sqrt[x]*ArcTanh[1/Sqrt[(-4 + x)/x]])/Sqrt[-((-4 + x)*x)]

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Maple [A]
time = 0.42, size = 23, normalized size = 0.88

method	result	size
default	\(2 \arcsin \left (\frac {x}{2}-1\right )-\sqrt {-x^{2}+4 x}\)	\(23\)
risch	\(\frac {x \left (x -4\right )}{\sqrt {-x \left (x -4\right )}}+2 \arcsin \left (\frac {x}{2}-1\right )\)	\(23\)
meijerg	\(\frac {4 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }}\)	\(36\)
trager	\(-\sqrt {-x^{2}+4 x}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+4 x}\right )\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*x-1)-(-x^2+4*x)^(1/2)

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Maxima [A]
time = 0.54, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 2 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4*x) - 2*arcsin(-1/2*x + 1)

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Fricas [A]
time = 1.64, size = 32, normalized size = 1.23 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- x \left (x - 4\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(-x*(x - 4)), x)

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Giac [A]
time = 0.66, size = 22, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 4 \, x} + 2 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4*x) + 2*arcsin(1/2*x - 1)

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Mupad [B]
time = 0.16, size = 22, normalized size = 0.85 \begin {gather*} 2\,\mathrm {asin}\left (\frac {x}{2}-1\right )-\sqrt {4\,x-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x - x^2)^(1/2),x)

[Out]

2*asin(x/2 - 1) - (4*x - x^2)^(1/2)

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