\(\int \frac {\sqrt {9-x^2}}{x^2} \, dx\) [118]

Optimal result

Integrand size = 15, antiderivative size = 25 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=-\frac {\sqrt {9-x^2}}{x}-\arcsin \left (\frac {x}{3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {283, 222} \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=-\arcsin \left (\frac {x}{3}\right )-\frac {\sqrt {9-x^2}}{x} \]

[In]

Int[Sqrt[9 - x^2]/x^2,x]

[Out]

-(Sqrt[9 - x^2]/x) - ArcSin[x/3]

Rule 222

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 283

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=-\frac {\sqrt {9-x^2}}{x}+2 \arctan \left (\frac {\sqrt {9-x^2}}{3+x}\right ) \]

[In]

Integrate[Sqrt[9 - x^2]/x^2,x]

[Out]

-(Sqrt[9 - x^2]/x) + 2*ArcTan[Sqrt[9 - x^2]/(3 + x)]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {x^{2}-9}{x \sqrt {-x^{2}+9}}-\arcsin \left (\frac {x}{3}\right )\)	\(26\)
pseudoelliptic	\(\frac {\arctan \left (\frac {\sqrt {-x^{2}+9}}{x}\right ) x -\sqrt {-x^{2}+9}}{x}\)	\(33\)
default	\(-\frac {\left (-x^{2}+9\right )^{\frac {3}{2}}}{9 x}-\frac {x \sqrt {-x^{2}+9}}{9}-\arcsin \left (\frac {x}{3}\right )\)	\(34\)
meijerg	\(-\frac {i \left (-\frac {12 i \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{9}+1}}{x}-4 i \sqrt {\pi }\, \arcsin \left (\frac {x}{3}\right )\right )}{4 \sqrt {\pi }}\)	\(36\)
trager	\(-\frac {\sqrt {-x^{2}+9}}{x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+9}\right )\)	\(42\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=\frac {2 \, x \arctan \left (\frac {\sqrt {-x^{2} + 9} - 3}{x}\right ) - \sqrt {-x^{2} + 9}}{x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=- \operatorname {asin}{\left (\frac {x}{3} \right )} - \frac {\sqrt {9 - x^{2}}}{x} \]

[In]

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

[Out]

-asin(x/3) - sqrt(9 - x**2)/x

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=-\frac {\sqrt {-x^{2} + 9}}{x} - \arcsin \left (\frac {1}{3} \, x\right ) \]

[In]

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

[Out]

-sqrt(-x^2 + 9)/x - arcsin(1/3*x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=\frac {x}{2 \, {\left (\sqrt {-x^{2} + 9} - 3\right )}} - \frac {\sqrt {-x^{2} + 9} - 3}{2 \, x} - \arcsin \left (\frac {1}{3} \, x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {9-x^2}}{x^2} \, dx=-\mathrm {asin}\left (\frac {x}{3}\right )-\frac {\sqrt {9-x^2}}{x} \]

[In]

int((9 - x^2)^(1/2)/x^2,x)

[Out]

- asin(x/3) - (9 - x^2)^(1/2)/x