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

Optimal. Leaf size=30 \[ \sqrt{4-3 x^2}-2 \tanh ^{-1}\left (\frac{1}{2} \sqrt{4-3 x^2}\right ) \]

[Out]

Sqrt[4 - 3*x^2] - 2*ArcTanh[Sqrt[4 - 3*x^2]/2]

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Rubi [A]  time = 0.0165422, 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} \[ \sqrt{4-3 x^2}-2 \tanh ^{-1}\left (\frac{1}{2} \sqrt{4-3 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 - 3*x^2]/x,x]

[Out]

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

Mathematica [A]  time = 0.0052576, size = 30, normalized size = 1. \[ \sqrt{4-3 x^2}-2 \tanh ^{-1}\left (\frac{1}{2} \sqrt{4-3 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[4 - 3*x^2]/x,x]

[Out]

Sqrt[4 - 3*x^2] - 2*ArcTanh[Sqrt[4 - 3*x^2]/2]

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-3\,{x}^{2}+4}-2\,{\it Artanh} \left ( 2\,{\frac{1}{\sqrt{-3\,{x}^{2}+4}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42568, size = 47, normalized size = 1.57 \begin{align*} \sqrt{-3 \, x^{2} + 4} - 2 \, \log \left (\frac{4 \, \sqrt{-3 \, x^{2} + 4}}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-3*x^2 + 4) - 2*log(4*sqrt(-3*x^2 + 4)/abs(x) + 8/abs(x))

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Fricas [A]  time = 1.96684, size = 70, normalized size = 2.33 \begin{align*} \sqrt{-3 \, x^{2} + 4} + 2 \, \log \left (\frac{\sqrt{-3 \, x^{2} + 4} - 2}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.48456, size = 75, normalized size = 2.5 \begin{align*} \begin{cases} i \sqrt{3 x^{2} - 4} - 2 \log{\left (x \right )} + \log{\left (x^{2} \right )} + 2 i \operatorname{asin}{\left (\frac{2 \sqrt{3}}{3 x} \right )} & \text{for}\: \frac{3 \left |{x^{2}}\right |}{4} > 1 \\\sqrt{4 - 3 x^{2}} + \log{\left (x^{2} \right )} - 2 \log{\left (\sqrt{1 - \frac{3 x^{2}}{4}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04958, size = 51, normalized size = 1.7 \begin{align*} \sqrt{-3 \, x^{2} + 4} - \log \left (\sqrt{-3 \, x^{2} + 4} + 2\right ) + \log \left (-\sqrt{-3 \, x^{2} + 4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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