3.118 \(\int \frac {\sqrt {9-x^2}}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right ) \]

[Out]

-arcsin(1/3*x)-(-x^2+9)^(1/2)/x

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - x^2]/x^2,x]

[Out]

-(Sqrt[9 - x^2]/x) - ArcSin[x/3]

Rule 216

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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {9-x^2}}{x^2} \, dx &=-\frac {\sqrt {9-x^2}}{x}-\int \frac {1}{\sqrt {9-x^2}} \, dx\\ &=-\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {\sqrt {9-x^2}}{x}-\sin ^{-1}\left (\frac {x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - x^2]/x^2,x]

[Out]

-(Sqrt[9 - x^2]/x) - ArcSin[x/3]

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fricas [A]  time = 0.40, size = 35, normalized size = 1.40 \[ \frac {2 \, x \arctan \left (\frac {\sqrt {-x^{2} + 9} - 3}{x}\right ) - \sqrt {-x^{2} + 9}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+9)^(1/2)/x^2,x, algorithm="fricas")

[Out]

(2*x*arctan((sqrt(-x^2 + 9) - 3)/x) - sqrt(-x^2 + 9))/x

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giac [A]  time = 0.98, size = 39, normalized size = 1.56 \[ \frac {x}{2 \, {\left (\sqrt {-x^{2} + 9} - 3\right )}} - \frac {\sqrt {-x^{2} + 9} - 3}{2 \, x} - \arcsin \left (\frac {1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+9)^(1/2)/x^2,x, algorithm="giac")

[Out]

1/2*x/(sqrt(-x^2 + 9) - 3) - 1/2*(sqrt(-x^2 + 9) - 3)/x - arcsin(1/3*x)

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maple [A]  time = 0.01, size = 34, normalized size = 1.36 \[ -\frac {\sqrt {-x^{2}+9}\, x}{9}-\arcsin \left (\frac {x}{3}\right )-\frac {\left (-x^{2}+9\right )^{\frac {3}{2}}}{9 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+9)^(1/2)/x^2,x)

[Out]

-1/9/x*(-x^2+9)^(3/2)-1/9*x*(-x^2+9)^(1/2)-arcsin(1/3*x)

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maxima [A]  time = 0.98, size = 21, normalized size = 0.84 \[ -\frac {\sqrt {-x^{2} + 9}}{x} - \arcsin \left (\frac {1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+9)^(1/2)/x^2,x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 9)/x - arcsin(1/3*x)

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mupad [B]  time = 0.04, size = 21, normalized size = 0.84 \[ -\mathrm {asin}\left (\frac {x}{3}\right )-\frac {\sqrt {9-x^2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9 - x^2)^(1/2)/x^2,x)

[Out]

- asin(x/3) - (9 - x^2)^(1/2)/x

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sympy [A]  time = 0.26, size = 15, normalized size = 0.60 \[ - \operatorname {asin}{\left (\frac {x}{3} \right )} - \frac {\sqrt {9 - x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+9)**(1/2)/x**2,x)

[Out]

-asin(x/3) - sqrt(9 - x**2)/x

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