3.12.10 \(\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\) [1110]

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

[Out]

arcsin(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, 222} \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 222

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 [B] Leaf count is larger than twice the leaf count of optimal. \(14\) vs. \(2(2)=4\).
time = 0.02, size = 14, normalized size = 7.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/Sqrt[1 - x^2]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.31, size = 35, normalized size = 17.50 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-2 I \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ],\text {Abs}\left [1+x\right ]>2\right \}\right \},2 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(1/2)*(1 + x)^(1/2)),x]')

[Out]

Piecewise[{{-2 I ArcCosh[Sqrt[2] Sqrt[1 + x] / 2], Abs[1 + x] > 2}}, 2 ArcSin[Sqrt[2] Sqrt[1 + x] / 2]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(2)=4\).
time = 0.14, size = 27, normalized size = 13.50

method	result	size
default	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.37, size = 2, normalized size = 1.00 \begin {gather*} \arcsin \left (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="maxima")

[Out]

arcsin(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (2) = 4\).
time = 0.30, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 39, normalized size = 19.50 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \left |{x + 1}\right | > 2 \\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), (2*asin(sqrt(2)*sqrt(x + 1)/2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (2) = 4\).
time = 0.00, size = 19, normalized size = 9.50 \begin {gather*} -2 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\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)

[Out]

-2*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

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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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