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

Optimal. Leaf size=37 \[ \sqrt{3-x^2}-\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0195136, antiderivative size = 37, 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} \[ \sqrt{3-x^2}-\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - x^2]/x,x]

[Out]

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

Mathematica [A]  time = 0.0068919, size = 33, normalized size = 0.89 \[ \sqrt{3-x^2}-\sqrt{3} \tanh ^{-1}\left (\sqrt{1-\frac{x^2}{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - x^2]/x,x]

[Out]

Sqrt[3 - x^2] - Sqrt[3]*ArcTanh[Sqrt[1 - x^2/3]]

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Maple [A]  time = 0.004, size = 30, normalized size = 0.8 \begin{align*} \sqrt{-{x}^{2}+3}-\sqrt{3}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{-{x}^{2}+3}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42444, size = 55, normalized size = 1.49 \begin{align*} -\sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{-x^{2} + 3}}{{\left | x \right |}} + \frac{6}{{\left | x \right |}}\right ) + \sqrt{-x^{2} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.12917, size = 104, normalized size = 2.81 \begin{align*} \frac{1}{2} \, \sqrt{3} \log \left (-\frac{x^{2} + 2 \, \sqrt{3} \sqrt{-x^{2} + 3} - 6}{x^{2}}\right ) + \sqrt{-x^{2} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.37867, size = 88, normalized size = 2.38 \begin{align*} \begin{cases} i \sqrt{x^{2} - 3} - \sqrt{3} \log{\left (x \right )} + \frac{\sqrt{3} \log{\left (x^{2} \right )}}{2} + \sqrt{3} i \operatorname{asin}{\left (\frac{\sqrt{3}}{x} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{3} > 1 \\\sqrt{3 - x^{2}} + \frac{\sqrt{3} \log{\left (x^{2} \right )}}{2} - \sqrt{3} \log{\left (\sqrt{1 - \frac{x^{2}}{3}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09109, size = 63, normalized size = 1.7 \begin{align*} \frac{1}{2} \, \sqrt{3} \log \left (\frac{\sqrt{3} - \sqrt{-x^{2} + 3}}{\sqrt{3} + \sqrt{-x^{2} + 3}}\right ) + \sqrt{-x^{2} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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