3.16.55 \(\int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx\) [1555]

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 = {55, 633, 222} \begin {gather*} -\text {ArcSin}\left (1-\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[4 - x]*Sqrt[x]),x]

[Out]

-ArcSin[1 - x/2]

Rule 55

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 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 {4-x} \sqrt {x}} \, dx &=\int \frac {1}{\sqrt {4 x-x^2}} \, dx\\ &=-\left (\frac {1}{4} \text {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 [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 {-4+x} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-4+x}}\right )}{\sqrt {-((-4+x) x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[4 - x]*Sqrt[x]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(6)=12\).
time = 0.15, size = 27, normalized size = 2.70

method	result	size
meijerg	\(2 \arcsin \left (\frac {\sqrt {x}}{2}\right )\)	\(9\)
default	\(\frac {\sqrt {\left (4-x \right ) x}\, \arcsin \left (-1+\frac {x}{2}\right )}{\sqrt {4-x}\, \sqrt {x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 0.51, size = 14, normalized size = 1.40 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 1.16, size = 14, normalized size = 1.40 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 24, normalized size = 2.40 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {x}}{2} \right )} & \text {for}\: \left |{x}\right | > 4 \\2 \operatorname {asin}{\left (\frac {\sqrt {x}}{2} \right )} & \text {otherwise} \end {cases} \end {gather*}

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), (2*asin(sqrt(x)/2), True))

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Giac [A]
time = 1.39, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {x}\right ) \end {gather*}

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

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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