### 3.274 $$\int \frac{\sqrt{9-x^2}}{x} \, dx$$

Optimal. Leaf size=30 $\sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right )$

[Out]

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

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Rubi [A]  time = 0.0149736, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {266, 50, 63, 206} $\sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[9 - x^2]/x,x]

[Out]

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

Rule 266

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]]

Rule 50

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 63

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.0051874, size = 30, normalized size = 1. $\sqrt{9-x^2}-3 \tanh ^{-1}\left (\frac{\sqrt{9-x^2}}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[9 - x^2]/x,x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-{x}^{2}+9}-3\,{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-{x}^{2}+9}}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42481, size = 47, normalized size = 1.57 \begin{align*} \sqrt{-x^{2} + 9} - 3 \, \log \left (\frac{6 \, \sqrt{-x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]  time = 1.87756, size = 65, normalized size = 2.17 \begin{align*} \sqrt{-x^{2} + 9} + 3 \, \log \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.30047, size = 73, normalized size = 2.43 \begin{align*} \begin{cases} - \frac{x}{\sqrt{-1 + \frac{9}{x^{2}}}} - 3 \operatorname{acosh}{\left (\frac{3}{x} \right )} + \frac{9}{x \sqrt{-1 + \frac{9}{x^{2}}}} & \text{for}\: \frac{9}{\left |{x^{2}}\right |} > 1 \\\frac{i x}{\sqrt{1 - \frac{9}{x^{2}}}} + 3 i \operatorname{asin}{\left (\frac{3}{x} \right )} - \frac{9 i}{x \sqrt{1 - \frac{9}{x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06007, size = 54, normalized size = 1.8 \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)