3.12.9 \(\int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\) [1109]

Optimal. Leaf size=20 \[ \sqrt {1-x} \sqrt {1+x}+\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 41, 222} \begin {gather*} \sqrt {1-x} \sqrt {x+1}+\sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/Sqrt[1 + x],x]

[Out]

Sqrt[1 - x]*Sqrt[1 + x] + 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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

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Mathematica [A]
time = 0.06, size = 32, normalized size = 1.60 \begin {gather*} \sqrt {1-x^2}+2 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]/Sqrt[1 + x],x]

[Out]

Sqrt[1 - x^2] + 2*ArcTan[Sqrt[1 + x]/Sqrt[1 - x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.96, size = 89, normalized size = 4.45 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\left (1+x\right )^{\frac {3}{2}}-2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-2 \sqrt {1+x}\right )}{\sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},-\frac {\left (1+x\right )^{\frac {3}{2}}}{\sqrt {1-x}}+2 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]+\frac {2 \sqrt {1+x}}{\sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{I ((1 + x) ^ (3 / 2) - 2 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sqrt[-1 + x] - 2 Sqrt[1 + x]) / Sqrt[-1
+ x], Abs[1 + x] > 2}}, -(1 + x) ^ (3 / 2) / Sqrt[1 - x] + 2 ArcSin[Sqrt[2] Sqrt[1 + x] / 2] + 2 Sqrt[1 + x] /
Sqrt[1 - x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(16)=32\).
time = 0.16, size = 41, normalized size = 2.05

method	result	size
default	\(\sqrt {1-x}\, \sqrt {1+x}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(41\)
risch	\(-\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 12, normalized size = 0.60 \begin {gather*} \sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-x^2 + 1) + arcsin(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
time = 0.30, size = 36, normalized size = 1.80 \begin {gather*} \sqrt {x + 1} \sqrt {-x + 1} - 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-x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.79, size = 99, normalized size = 4.95 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(3/2)/sqrt(x - 1) - 2*I*sqrt(x + 1)/sqrt(x - 1), Abs
(x + 1) > 2), (2*asin(sqrt(2)*sqrt(x + 1)/2) - (x + 1)**(3/2)/sqrt(1 - x) + 2*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]
time = 0.00, size = 36, normalized size = 1.80 \begin {gather*} \sqrt {-x+1} \sqrt {x+1}-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-x)^(1/2)/(1+x)^(1/2),x)

[Out]

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

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Mupad [B]
time = 0.12, size = 12, normalized size = 0.60 \begin {gather*} \mathrm {asin}\left (x\right )+\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2)/(x + 1)^(1/2),x)

[Out]

asin(x) + (1 - x^2)^(1/2)

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