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

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

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, 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} \begin {gather*} \sin ^{-1}(x) \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[x]

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IntegrateAlgebraic [B]  time = 0.04, size = 20, normalized size = 10.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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fricas [B]  time = 0.83, size = 22, normalized size = 11.00 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

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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giac [B]  time = 0.65, size = 13, normalized size = 6.50 \begin {gather*} 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

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))

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maple [B]  time = 0.00, size = 27, normalized size = 13.50 \begin {gather*} \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.95, size = 2, normalized size = 1.00 \begin {gather*} \arcsin \relax (x) \end {gather*}

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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mupad [B]  time = 0.08, size = 22, normalized size = 11.00 \begin {gather*} -4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*atan(((1 - x)^(1/2) - 1)/((x + 1)^(1/2) - 1))

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sympy [B]  time = 1.04, size = 41, normalized size = 20.50 \begin {gather*} \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 {gather*}

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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