\(\int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx\) [648]

Optimal result

Integrand size = 15, antiderivative size = 8 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-\arcsin (1-2 x) \]

[Out]

arcsin(-1+2*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {55, 633, 222} \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-\arcsin (1-2 x) \]

[In]

Int[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

-ArcSin[1 - 2*x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(8)=16\).

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 5.00 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-\frac {2 \sqrt {-1+x} \sqrt {x} \log \left (\sqrt {-1+x}-\sqrt {x}\right )}{\sqrt {-((-1+x) x)}} \]

[In]

Integrate[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

(-2*Sqrt[-1 + x]*Sqrt[x]*Log[Sqrt[-1 + x] - Sqrt[x]])/Sqrt[-((-1 + x)*x)]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
meijerg	\(2 \arcsin \left (\sqrt {x}\right )\)	\(7\)
default	\(\frac {\sqrt {x \left (1-x \right )}\, \arcsin \left (-1+2 x \right )}{\sqrt {x}\, \sqrt {1-x}}\)	\(27\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \]

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=\begin {cases} - 2 i \operatorname {acosh}{\left (\sqrt {x} \right )} & \text {for}\: \left |{x}\right | > 1 \\2 \operatorname {asin}{\left (\sqrt {x} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \]

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=2 \, \arcsin \left (\sqrt {x}\right ) \]

[In]

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

[Out]

2*arcsin(sqrt(x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx=-4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x}}\right ) \]

[In]

int(1/(x^(1/2)*(1 - x)^(1/2)),x)

[Out]

-4*atan(((1 - x)^(1/2) - 1)/x^(1/2))