\(\int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx\) [862]

Optimal result

Integrand size = 19, antiderivative size = 10 \[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {728, 116} \[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \]

[In]

Int[1/(Sqrt[1 + x]*Sqrt[x - x^2]),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])

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x}} \, dx \\ & = 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 = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 4.40 \[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=\frac {2 x \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )}{\sqrt {-((-1+x) x)} \sqrt {1+x}} \]

[In]

Integrate[1/(Sqrt[1 + x]*Sqrt[x - x^2]),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. \(42\) vs. \(2(8)=16\).

Time = 1.48 (sec) , antiderivative size = 43, normalized size of antiderivative = 4.30

method	result	size
default	\(\frac {F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2-2 x}\, \sqrt {-\left (-1+x \right ) x}}{\left (1-x \right ) x}\)	\(43\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \sqrt {2-2 x}\, \sqrt {-x}\, F\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-\left (-1+x \right ) x}\, \sqrt {-x^{3}+x}}\)	\(52\)

[In]

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

[Out]

1/(1-x)/x*EllipticF((1+x)^(1/2),1/2*2^(1/2))*(-x)^(1/2)*(2-2*x)^(1/2)*(-(-1+x)*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 \[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) \]

[In]

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

[Out]

-2*I*weierstrassPInverse(4, 0, x)

Sympy [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=\int \frac {1}{\sqrt {- x \left (x - 1\right )} \sqrt {x + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + x} \sqrt {x + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + x} \sqrt {x + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx=\int \frac {1}{\sqrt {x-x^2}\,\sqrt {x+1}} \,d x \]

[In]

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

[Out]

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