3.10.33 \(\int (1-x^2) \sqrt {\frac {1}{2-x^2}} \, dx\) [933]

Optimal. Leaf size=18 \[ \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 18, 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 = {1972, 391} \begin {gather*} \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)*Sqrt[(2 - x^2)^(-1)],x]

[Out]

x/(2*Sqrt[(2 - x^2)^(-1)])

Rule 391

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*x*((a + b*x^n)^(p + 1)/a), x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rule 1972

Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(a + b*x^n)
^(p*q)], Int[u*(a + b*x^n)^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)*Sqrt[(2 - x^2)^(-1)],x]

[Out]

x/(2*Sqrt[(2 - x^2)^(-1)])

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Maple [A]
time = 0.23, size = 20, normalized size = 1.11

method	result	size
gosper	\(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\)	\(20\)
default	\(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\)	\(20\)
trager	\(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\)	\(20\)
risch	\(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\)	\(20\)
meijerg	\(\sqrt {\frac {1}{-x^{2}+2}}\, \sqrt {-x^{2}+2}\, \arcsin \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {i \sqrt {\frac {1}{-x^{2}+2}}\, \sqrt {-x^{2}+2}\, \left (\frac {i \sqrt {\pi }\, x \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\sqrt {\pi }}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x^2 - 1)*sqrt(-1/(x^2 - 2)), x)

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Fricas [A]
time = 0.36, size = 20, normalized size = 1.11 \begin {gather*} -\frac {1}{2} \, {\left (x^{3} - 2 \, x\right )} \sqrt {-\frac {1}{x^{2} - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^3 - 2*x)*sqrt(-1/(x^2 - 2))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 0.41, size = 29, normalized size = 1.61 \begin {gather*} - \frac {x^{3} \sqrt {- \frac {1}{x^{2} - 2}}}{2} + x \sqrt {- \frac {1}{x^{2} - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.88, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 2} x \mathrm {sgn}\left (x^{2} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 2)*x*sgn(x^2 - 2)

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Mupad [B]
time = 3.50, size = 19, normalized size = 1.06 \begin {gather*} -\frac {x\,\left (x^2-2\right )\,\sqrt {-\frac {1}{x^2-2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)*(-1/(x^2 - 2))^(1/2),x)

[Out]

-(x*(x^2 - 2)*(-1/(x^2 - 2))^(1/2))/2

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