\(\int \frac {\sqrt {9-x^2}}{x} \, dx\) [274]

Optimal result

Integrand size = 15, antiderivative size = 30 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {9-x^2}-3 \text {arctanh}\left (\frac {\sqrt {9-x^2}}{3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 212} \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {9-x^2}-3 \text {arctanh}\left (\frac {\sqrt {9-x^2}}{3}\right ) \]

[In]

Int[Sqrt[9 - x^2]/x,x]

[Out]

Sqrt[9 - x^2] - 3*ArcTanh[Sqrt[9 - x^2]/3]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {9-x}}{x} \, dx,x,x^2\right ) \\ & = \sqrt {9-x^2}+\frac {9}{2} \text {Subst}\left (\int \frac {1}{\sqrt {9-x} x} \, dx,x,x^2\right ) \\ & = \sqrt {9-x^2}-9 \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\sqrt {9-x^2}\right ) \\ & = \sqrt {9-x^2}-3 \text {arctanh}\left (\frac {\sqrt {9-x^2}}{3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {9-x^2}-3 \text {arctanh}\left (\frac {\sqrt {9-x^2}}{3}\right ) \]

[In]

Integrate[Sqrt[9 - x^2]/x,x]

[Out]

Sqrt[9 - x^2] - 3*ArcTanh[Sqrt[9 - x^2]/3]

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
default	\(\sqrt {-x^{2}+9}-3 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {-x^{2}+9}}\right )\)	\(25\)
trager	\(\sqrt {-x^{2}+9}-3 \ln \left (\frac {\sqrt {-x^{2}+9}+3}{x}\right )\)	\(29\)
pseudoelliptic	\(\sqrt {-x^{2}+9}-\frac {3 \ln \left (\sqrt {-x^{2}+9}+3\right )}{2}+\frac {3 \ln \left (\sqrt {-x^{2}+9}-3\right )}{2}\)	\(39\)
meijerg	\(-\frac {3 \left (-2 \left (2-2 \ln \left (2\right )+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{9}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{9}+1}}{2}\right )\right )}{4 \sqrt {\pi }}\)	\(68\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {-x^{2} + 9} + 3 \, \log \left (\frac {\sqrt {-x^{2} + 9} - 3}{x}\right ) \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\begin {cases} i \sqrt {x^{2} - 9} - 3 \log {\left (x \right )} + \frac {3 \log {\left (x^{2} \right )}}{2} + 3 i \operatorname {asin}{\left (\frac {3}{x} \right )} & \text {for}\: \left |{x^{2}}\right | > 9 \\\sqrt {9 - x^{2}} + \frac {3 \log {\left (x^{2} \right )}}{2} - 3 \log {\left (\sqrt {1 - \frac {x^{2}}{9}} + 1 \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*sqrt(x**2 - 9) - 3*log(x) + 3*log(x**2)/2 + 3*I*asin(3/x), Abs(x**2) > 9), (sqrt(9 - x**2) + 3*lo
g(x**2)/2 - 3*log(sqrt(1 - x**2/9) + 1), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {-x^{2} + 9} - 3 \, \log \left (\frac {6 \, \sqrt {-x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]

[In]

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

[Out]

sqrt(-x^2 + 9) - 3*log(6*sqrt(-x^2 + 9)/abs(x) + 18/abs(x))

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=\sqrt {-x^{2} + 9} - \frac {3}{2} \, \log \left (\sqrt {-x^{2} + 9} + 3\right ) + \frac {3}{2} \, \log \left (-\sqrt {-x^{2} + 9} + 3\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {9-x^2}}{x} \, dx=3\,\ln \left (\sqrt {\frac {9}{x^2}-1}-3\,\sqrt {\frac {1}{x^2}}\right )+\sqrt {9-x^2} \]

[In]

int((9 - x^2)^(1/2)/x,x)

[Out]

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