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

Optimal. Leaf size=8 \[ -\sin ^{-1}(1-2 x) \]

[Out]

arcsin(-1+2*x)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 8, 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*} -\sin ^{-1}(1-2 x) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx &=\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] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(8)=16\).
time = 0.03, size = 38, normalized size = 4.75 \begin {gather*} \frac {2 \sqrt {-1+x} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{\sqrt {-((-1+x) x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

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

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.94, size = 19, normalized size = 2.38 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-2 I \text {ArcCosh}\left [\sqrt {x}\right ],\text {Abs}\left [x\right ]>1\right \}\right \},2 \text {ArcSin}\left [\sqrt {x}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(Sqrt[x]*Sqrt[1 - x]),x]')

[Out]

Piecewise[{{-2 I ArcCosh[Sqrt[x]], Abs[x] > 1}}, 2 ArcSin[Sqrt[x]]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(6)=12\).
time = 0.12, size = 27, normalized size = 3.38

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 0.35, size = 14, normalized size = 1.75 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
time = 0.31, size = 14, normalized size = 1.75 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

________________________________________________________________________________________

Sympy [A]
time = 0.46, size = 20, normalized size = 2.50 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 12, normalized size = 1.50 \begin {gather*} -2 \arcsin \left (\sqrt {-x+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arcsin(sqrt(-x + 1))

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 16, normalized size = 2.00 \begin {gather*} -4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(1 - x)^(1/2)),x)

[Out]

-4*atan(((1 - x)^(1/2) - 1)/x^(1/2))

________________________________________________________________________________________