3.460 \(\int \frac {\sqrt {9-4 x^2}}{x^3} \, dx\)

Optimal. Leaf size=39 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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, 47, 63, 206} \[ \frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^3,x]

[Out]

-Sqrt[9 - 4*x^2]/(2*x^2) + (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/3

Rule 47

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

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]]

Rubi steps

\begin {align*} \int \frac {\sqrt {9-4 x^2}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {9-4 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{2 x^2}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9-4 x^2}}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {9-4 x^2}\right )\\ &=-\frac {\sqrt {9-4 x^2}}{2 x^2}+\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 0.95 \[ \frac {2}{3} \tanh ^{-1}\left (\sqrt {1-\frac {4 x^2}{9}}\right )-\frac {\sqrt {9-4 x^2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^3,x]

[Out]

-1/2*Sqrt[9 - 4*x^2]/x^2 + (2*ArcTanh[Sqrt[1 - (4*x^2)/9]])/3

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fricas [A]  time = 0.92, size = 38, normalized size = 0.97 \[ -\frac {4 \, x^{2} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) + 3 \, \sqrt {-4 \, x^{2} + 9}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^3,x, algorithm="fricas")

[Out]

-1/6*(4*x^2*log((sqrt(-4*x^2 + 9) - 3)/x) + 3*sqrt(-4*x^2 + 9))/x^2

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giac [A]  time = 1.07, size = 45, normalized size = 1.15 \[ -\frac {\sqrt {-4 \, x^{2} + 9}}{2 \, x^{2}} + \frac {1}{3} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) - \frac {1}{3} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 41, normalized size = 1.05 \[ \frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{3}-\frac {\left (-4 x^{2}+9\right )^{\frac {3}{2}}}{18 x^{2}}-\frac {2 \sqrt {-4 x^{2}+9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2+9)^(1/2)/x^3,x)

[Out]

-1/18/x^2*(-4*x^2+9)^(3/2)-2/9*(-4*x^2+9)^(1/2)+2/3*arctanh(3/(-4*x^2+9)^(1/2))

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maxima [A]  time = 2.99, size = 51, normalized size = 1.31 \[ -\frac {2}{9} \, \sqrt {-4 \, x^{2} + 9} - \frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{18 \, x^{2}} + \frac {2}{3} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^3,x, algorithm="maxima")

[Out]

-2/9*sqrt(-4*x^2 + 9) - 1/18*(-4*x^2 + 9)^(3/2)/x^2 + 2/3*log(6*sqrt(-4*x^2 + 9)/abs(x) + 18/abs(x))

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mupad [B]  time = 4.66, size = 35, normalized size = 0.90 \[ -\frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\frac {3\,\sqrt {\frac {1}{x^2}}}{2}\right )}{3}-\frac {\sqrt {\frac {9}{4}-x^2}}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9 - 4*x^2)^(1/2)/x^3,x)

[Out]

- (2*log((9/(4*x^2) - 1)^(1/2) - (3*(1/x^2)^(1/2))/2))/3 - (9/4 - x^2)^(1/2)/x^2

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sympy [A]  time = 1.73, size = 97, normalized size = 2.49 \[ \begin {cases} \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{3} + \frac {1}{x \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {9}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{3} - \frac {i}{x \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {9 i}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2+9)**(1/2)/x**3,x)

[Out]

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

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