3.9.59 \(\int \frac {\sqrt {1-x^2}}{\sqrt {1+x}} \, dx\) [859]

Optimal. Leaf size=13 \[ -\frac {2}{3} (1-x)^{3/2} \]

[Out]

-2/3*(1-x)^(3/2)

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {26, 32} \begin {gather*} -\frac {2}{3} (1-x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[1 + x],x]

[Out]

(-2*(1 - x)^(3/2))/3

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^2}}{\sqrt {1+x}} \, dx &=\int \sqrt {1-x} \, dx\\ &=-\frac {2}{3} (1-x)^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.69 \begin {gather*} -\frac {2 \left (1-x^2\right )^{3/2}}{3 (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - x^2)^(3/2))/(3*(1 + x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(19\) vs. \(2(9)=18\).
time = 0.60, size = 20, normalized size = 1.54

method	result	size
gosper	\(\frac {2 \left (-1+x \right ) \sqrt {-x^{2}+1}}{3 \sqrt {1+x}}\)	\(20\)
default	\(\frac {2 \left (-1+x \right ) \sqrt {-x^{2}+1}}{3 \sqrt {1+x}}\)	\(20\)
risch	\(-\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (-1+x \right )^{2}}{3 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(-1+x)*(-x^2+1)^(1/2)/(1+x)^(1/2)

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.92 \begin {gather*} \frac {2}{3} \, {\left (x - 1\right )} \sqrt {-x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x - 1)*sqrt(-x + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (9) = 18\).
time = 0.34, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1} {\left (x - 1\right )}}{3 \, \sqrt {x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-x^2 + 1)*(x - 1)/sqrt(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.30, size = 15, normalized size = 1.15 \begin {gather*} -\frac {2}{3} \, {\left (-x + 1\right )}^{\frac {3}{2}} + \frac {4}{3} \, \sqrt {2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-x + 1)^(3/2) + 4/3*sqrt(2)

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Mupad [B]
time = 3.52, size = 20, normalized size = 1.54 \begin {gather*} \frac {\left (\frac {2\,x}{3}-\frac {2}{3}\right )\,\sqrt {1-x^2}}{\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(x + 1)^(1/2),x)

[Out]

(((2*x)/3 - 2/3)*(1 - x^2)^(1/2))/(x + 1)^(1/2)

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