3.134 \(\int \frac{\sqrt{-4+9 x^2}}{x} \, dx\)

Optimal. Leaf size=30 \[ \sqrt{9 x^2-4}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{9 x^2-4}\right ) \]

[Out]

Sqrt[-4 + 9*x^2] - 2*ArcTan[Sqrt[-4 + 9*x^2]/2]

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Rubi [A]  time = 0.015983, 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, 203} \[ \sqrt{9 x^2-4}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{9 x^2-4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-4 + 9*x^2]/x,x]

[Out]

Sqrt[-4 + 9*x^2] - 2*ArcTan[Sqrt[-4 + 9*x^2]/2]

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 203

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

Rubi steps

\begin{align*} \int \frac{\sqrt{-4+9 x^2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-4+9 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt{-4+9 x^2}-2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-4+9 x}} \, dx,x,x^2\right )\\ &=\sqrt{-4+9 x^2}-\frac{4}{9} \operatorname{Subst}\left (\int \frac{1}{\frac{4}{9}+\frac{x^2}{9}} \, dx,x,\sqrt{-4+9 x^2}\right )\\ &=\sqrt{-4+9 x^2}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{-4+9 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0055626, size = 30, normalized size = 1. \[ \sqrt{9 x^2-4}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{9 x^2-4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-4 + 9*x^2]/x,x]

[Out]

Sqrt[-4 + 9*x^2] - 2*ArcTan[Sqrt[-4 + 9*x^2]/2]

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} \sqrt{9\,{x}^{2}-4}+2\,\arctan \left ( 2\,{\frac{1}{\sqrt{9\,{x}^{2}-4}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2-4)^(1/2)/x,x)

[Out]

(9*x^2-4)^(1/2)+2*arctan(2/(9*x^2-4)^(1/2))

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Maxima [A]  time = 1.40573, size = 26, normalized size = 0.87 \begin{align*} \sqrt{9 \, x^{2} - 4} + 2 \, \arcsin \left (\frac{2}{3 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x, algorithm="maxima")

[Out]

sqrt(9*x^2 - 4) + 2*arcsin(2/3/abs(x))

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Fricas [A]  time = 1.97538, size = 78, normalized size = 2.6 \begin{align*} \sqrt{9 \, x^{2} - 4} - 4 \, \arctan \left (-\frac{3}{2} \, x + \frac{1}{2} \, \sqrt{9 \, x^{2} - 4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x, algorithm="fricas")

[Out]

sqrt(9*x^2 - 4) - 4*arctan(-3/2*x + 1/2*sqrt(9*x^2 - 4))

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Sympy [A]  time = 1.29256, size = 92, normalized size = 3.07 \begin{align*} \begin{cases} - \frac{3 i x}{\sqrt{-1 + \frac{4}{9 x^{2}}}} - 2 i \operatorname{acosh}{\left (\frac{2}{3 x} \right )} + \frac{4 i}{3 x \sqrt{-1 + \frac{4}{9 x^{2}}}} & \text{for}\: \frac{4}{9 \left |{x^{2}}\right |} > 1 \\\frac{3 x}{\sqrt{1 - \frac{4}{9 x^{2}}}} + 2 \operatorname{asin}{\left (\frac{2}{3 x} \right )} - \frac{4}{3 x \sqrt{1 - \frac{4}{9 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2-4)**(1/2)/x,x)

[Out]

Piecewise((-3*I*x/sqrt(-1 + 4/(9*x**2)) - 2*I*acosh(2/(3*x)) + 4*I/(3*x*sqrt(-1 + 4/(9*x**2))), 4/(9*Abs(x**2)
) > 1), (3*x/sqrt(1 - 4/(9*x**2)) + 2*asin(2/(3*x)) - 4/(3*x*sqrt(1 - 4/(9*x**2))), True))

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Giac [A]  time = 1.06677, size = 32, normalized size = 1.07 \begin{align*} \sqrt{9 \, x^{2} - 4} - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{9 \, x^{2} - 4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x, algorithm="giac")

[Out]

sqrt(9*x^2 - 4) - 2*arctan(1/2*sqrt(9*x^2 - 4))