3.9.57 \(\int \frac {\sqrt {1-x}}{\sqrt {1-x^2}} \, dx\) [857]

Optimal. Leaf size=9 \[ 2 \sqrt {1+x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {26, 32} \begin {gather*} 2 \sqrt {x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/Sqrt[1 - x^2],x]

[Out]

2*Sqrt[1 + x]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(9)=18\).
time = 0.03, size = 22, normalized size = 2.44 \begin {gather*} \frac {2 \sqrt {1-x^2}}{\sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]/Sqrt[1 - x^2],x]

[Out]

(2*Sqrt[1 - x^2])/Sqrt[1 - x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(23\) vs. \(2(7)=14\).
time = 0.56, size = 24, normalized size = 2.67

method	result	size
gosper	\(\frac {2 \left (1+x \right ) \sqrt {1-x}}{\sqrt {-x^{2}+1}}\)	\(22\)
default	\(-\frac {2 \sqrt {1-x}\, \sqrt {-x^{2}+1}}{-1+x}\)	\(24\)
risch	\(-\frac {2 \sqrt {\frac {\left (1-x \right ) \left (-x^{2}+1\right )}{\left (-1+x \right )^{2}}}\, \left (-1+x \right ) \sqrt {1+x}}{\sqrt {1-x}\, \sqrt {-x^{2}+1}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 7, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.35, size = 23, normalized size = 2.56 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1} \sqrt {-x + 1}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - x}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - x)/sqrt(-(x - 1)*(x + 1)), x)

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Giac [A]
time = 2.64, size = 13, normalized size = 1.44 \begin {gather*} -2 \, \sqrt {2} + 2 \, \sqrt {x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(2) + 2*sqrt(x + 1)

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Mupad [B]
time = 3.64, size = 18, normalized size = 2.00 \begin {gather*} \frac {2\,\sqrt {1-x^2}}{\sqrt {1-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2)/(1 - x^2)^(1/2),x)

[Out]

(2*(1 - x^2)^(1/2))/(1 - x)^(1/2)

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