\(\int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx\) [869]

Optimal result

Integrand size = 22, antiderivative size = 24 \[ \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx=-\frac {2 \sqrt {-x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {x}} \]

[Out]

-2*EllipticE((-x)^(1/2),I)*(-x)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {112, 111} \[ \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx=-\frac {2 \sqrt {-x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {x}} \]

[In]

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

[Out]

(-2*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[x]

Rule 111

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

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(-b)*x]/Sqrt[b*
x], Int[Sqrt[e + f*x]/(Sqrt[(-b)*x]*Sqrt[c + d*x]), x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &
& GtQ[c, 0] && GtQ[e, 0] && LtQ[-b/d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x} \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {1+x}} \, dx}{\sqrt {x}} \\ & = -\frac {2 \sqrt {-x} E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {x}} \\ \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 = 40, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx=-\frac {2}{3} \sqrt {x} \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )\right ) \]

[In]

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

[Out]

(-2*Sqrt[x]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x*Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/3

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {2 \sqrt {2}\, \sqrt {-x}\, E\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}}\)	\(25\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \left (\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x^{3}+x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 E\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-x^{3}+x}}\right )}{\sqrt {x}\, \sqrt {1-x}\, \sqrt {1+x}}\)	\(119\)

[In]

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

[Out]

2*2^(1/2)*(-x)^(1/2)*EllipticE((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 = 16, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) - 2 i \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \]

[In]

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

[Out]

-2*I*weierstrassPInverse(4, 0, x) - 2*I*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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