\(\int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx\) [861]

Optimal result

Integrand size = 22, antiderivative size = 10 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \]

[Out]

2*EllipticF(x^(1/2),I)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {116} \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \]

[In]

Int[1/(Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (GtQ[-b/d, 0] || LtQ[-b/f, 0])

Rubi steps \begin{align*} \text {integral}& = 2 F\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 4.40 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=\frac {2 x \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )}{\sqrt {-((-1+x) x)} \sqrt {1+x}} \]

[In]

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

[Out]

(2*x*Sqrt[1 - x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, x^2])/(Sqrt[-((-1 + x)*x)]*Sqrt[1 + x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).

Time = 1.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.40

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-x}\, F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}}\)	\(24\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \sqrt {2-2 x}\, \sqrt {-x}\, F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, \sqrt {1-x}\, \sqrt {-x^{3}+x}}\)	\(54\)

[In]

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

[Out]

2^(1/2)*(-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))/x^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) \]

[In]

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

[Out]

-2*I*weierstrassPInverse(4, 0, x)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (7) = 14\).

Time = 10.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 6.60 \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]

[In]

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

[Out]

I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), x**(-2))/(4*pi**(3/2)) - I*meijerg(
((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), exp_polar(-2*I*pi)/x**2)/(4*pi**(3/2))

Maxima [F]

\[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=\int { \frac {1}{\sqrt {x + 1} \sqrt {x} \sqrt {-x + 1}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=\int { \frac {1}{\sqrt {x + 1} \sqrt {x} \sqrt {-x + 1}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {1-x}\,\sqrt {x+1}} \,d x \]

[In]

int(1/(x^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)),x)

[Out]

int(1/(x^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)), x)