### 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]

-(Sqrt[9 - x^2]/x) - ArcSin[x/3]

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Rubi [A]  time = 0.0042819, antiderivative size = 25, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0073502, size = 25, normalized size = 1. $-\frac{\sqrt{9-x^2}}{x}-\sin ^{-1}\left (\frac{x}{3}\right )$

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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Maple [A]  time = 0.003, size = 34, normalized size = 1.4 \begin{align*} -{\frac{1}{9\,x} \left ( -{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{9}\sqrt{-{x}^{2}+9}}-\arcsin \left ({\frac{x}{3}} \right ) \end{align*}

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 = 1.40731, size = 28, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-x^{2} + 9}}{x} - \arcsin \left (\frac{1}{3} \, x\right ) \end{align*}

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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Fricas [A]  time = 1.79534, size = 77, normalized size = 3.08 \begin{align*} \frac{2 \, x \arctan \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) - \sqrt{-x^{2} + 9}}{x} \end{align*}

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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Sympy [A]  time = 0.226875, size = 15, normalized size = 0.6 \begin{align*} - \operatorname{asin}{\left (\frac{x}{3} \right )} - \frac{\sqrt{9 - x^{2}}}{x} \end{align*}

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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Giac [A]  time = 1.06067, size = 53, normalized size = 2.12 \begin{align*} \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 ) \end{align*}

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)