3.1110 \(\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\)

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

[Out]

ArcSin[x]

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Rubi [A]  time = 0.0014997, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {41, 216} \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*Sqrt[1 + x]),x]

[Out]

ArcSin[x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx &=\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0042368, size = 2, normalized size = 1. \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[x]

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Maple [B]  time = 0.002, size = 27, normalized size = 13.5 \begin{align*}{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53142, size = 3, normalized size = 1.5 \begin{align*} \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x)

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Fricas [B]  time = 1.81425, size = 61, normalized size = 30.5 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [B]  time = 1.09657, size = 41, normalized size = 20.5 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.05936, size = 18, normalized size = 9. \begin{align*} 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(x + 1))