3.1555 \(\int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx\)

Optimal. Leaf size=10 \[ -\sin ^{-1}\left (1-\frac {x}{2}\right ) \]

[Out]

arcsin(-1+1/2*x)

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Rubi [A]  time = 0.01, antiderivative size = 10, 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 = {53, 619, 216} \[ -\sin ^{-1}\left (1-\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[4 - x]*Sqrt[x]),x]

[Out]

-ArcSin[1 - x/2]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + 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]

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.40 \[ -2 \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[4 - x]*Sqrt[x]),x]

[Out]

-2*ArcSin[Sqrt[1 - x/4]]

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fricas [B]  time = 0.43, size = 14, normalized size = 1.40 \[ -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 4)/sqrt(x))

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giac [A]  time = 1.10, size = 8, normalized size = 0.80 \[ 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*sqrt(x))

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maple [B]  time = 0.00, size = 27, normalized size = 2.70 \[ \frac {\sqrt {\left (-x +4\right ) x}\, \arcsin \left (\frac {x}{2}-1\right )}{\sqrt {-x +4}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-x)^(1/2)/x^(1/2),x)

[Out]

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

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maxima [B]  time = 2.99, size = 14, normalized size = 1.40 \[ -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 4)/sqrt(x))

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mupad [B]  time = 0.29, size = 16, normalized size = 1.60 \[ -4\,\mathrm {atan}\left (\frac {\sqrt {4-x}-2}{\sqrt {x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(4 - x)^(1/2)),x)

[Out]

-4*atan(((4 - x)^(1/2) - 2)/x^(1/2))

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sympy [A]  time = 0.99, size = 26, normalized size = 2.60 \[ \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {x}}{2} \right )} & \text {for}\: \frac {\left |{x}\right |}{4} > 1 \\2 \operatorname {asin}{\left (\frac {\sqrt {x}}{2} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(x)/2), Abs(x)/4 > 1), (2*asin(sqrt(x)/2), True))

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