3.9.61 \(\int \frac {1-x}{\sqrt {1-x^2}} \, dx\) [861]

Optimal. Leaf size=14 \[ \sqrt {1-x^2}+\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, 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 = {655, 222} \begin {gather*} \text {ArcSin}(x)+\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)/Sqrt[1 - x^2],x]

[Out]

Sqrt[1 - x^2] + ArcSin[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]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(32\) vs. \(2(14)=28\).
time = 0.06, size = 32, normalized size = 2.29 \begin {gather*} \sqrt {1-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)/Sqrt[1 - x^2],x]

[Out]

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

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Maple [A]
time = 0.43, size = 13, normalized size = 0.93

method	result	size
default	\(\arcsin \left (x \right )+\sqrt {-x^{2}+1}\)	\(13\)
risch	\(-\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+\arcsin \left (x \right )\)	\(20\)
meijerg	\(\arcsin \left (x \right )+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\)	\(29\)
trager	\(\sqrt {-x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 12, normalized size = 0.86 \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)/(-x^2+1)^(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. 28 vs. \(2 (12) = 24\).
time = 3.61, size = 28, normalized size = 2.00 \begin {gather*} \sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.71 \begin {gather*} \sqrt {1 - x^{2}} + \operatorname {asin}{\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(1 - x**2) + asin(x)

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Giac [A]
time = 1.96, size = 12, normalized size = 0.86 \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)/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

sqrt(-x^2 + 1) + arcsin(x)

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Mupad [B]
time = 0.11, size = 12, normalized size = 0.86 \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(-(x - 1)/(1 - x^2)^(1/2),x)

[Out]

asin(x) + (1 - x^2)^(1/2)

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