\(\int \frac {\sqrt {3-x^2}}{x} \, dx\) [150]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\sqrt {3-x^2}-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 212} \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\sqrt {3-x^2}-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right ) \]

[In]

Int[Sqrt[3 - x^2]/x,x]

[Out]

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

Rule 52

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 65

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 212

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

Rule 272

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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {3-x}}{x} \, dx,x,x^2\right ) \\ & = \sqrt {3-x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3-x} x} \, dx,x,x^2\right ) \\ & = \sqrt {3-x^2}-3 \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\sqrt {3-x^2}\right ) \\ & = \sqrt {3-x^2}-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3-x^2}}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\sqrt {3-x^2}-\sqrt {3} \text {arctanh}\left (\sqrt {1-\frac {x^2}{3}}\right ) \]

[In]

Integrate[Sqrt[3 - x^2]/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81

method	result	size
default	\(\sqrt {-x^{2}+3}-\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}}{\sqrt {-x^{2}+3}}\right )\)	\(30\)
pseudoelliptic	\(-\operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+3}\, \sqrt {3}}{3}\right ) \sqrt {3}+\sqrt {-x^{2}+3}\)	\(31\)
trager	\(\sqrt {-x^{2}+3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\sqrt {-x^{2}+3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x}\right )\)	\(41\)
meijerg	\(-\frac {\sqrt {3}\, \left (-2 \left (2-2 \ln \left (2\right )+2 \ln \left (x \right )-\ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{3}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{3}+1}}{2}\right )\right )}{4 \sqrt {\pi }}\)	\(71\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\frac {1}{2} \, \sqrt {3} \log \left (-\frac {x^{2} + 2 \, \sqrt {3} \sqrt {-x^{2} + 3} - 6}{x^{2}}\right ) + \sqrt {-x^{2} + 3} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\begin {cases} i \sqrt {x^{2} - 3} - \sqrt {3} \log {\left (x \right )} + \frac {\sqrt {3} \log {\left (x^{2} \right )}}{2} + \sqrt {3} i \operatorname {asin}{\left (\frac {\sqrt {3}}{x} \right )} & \text {for}\: \left |{x^{2}}\right | > 3 \\\sqrt {3 - x^{2}} + \frac {\sqrt {3} \log {\left (x^{2} \right )}}{2} - \sqrt {3} \log {\left (\sqrt {1 - \frac {x^{2}}{3}} + 1 \right )} & \text {otherwise} \end {cases} \]

[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)
, (sqrt(3 - x**2) + sqrt(3)*log(x**2)/2 - sqrt(3)*log(sqrt(1 - x**2/3) + 1), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=-\sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {-x^{2} + 3}}{{\left | x \right |}} + \frac {6}{{\left | x \right |}}\right ) + \sqrt {-x^{2} + 3} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\frac {1}{2} \, \sqrt {3} \log \left (\frac {\sqrt {3} - \sqrt {-x^{2} + 3}}{\sqrt {3} + \sqrt {-x^{2} + 3}}\right ) + \sqrt {-x^{2} + 3} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {3-x^2}}{x} \, dx=\sqrt {3}\,\ln \left (\sqrt {\frac {3}{x^2}-1}-\sqrt {3}\,\sqrt {\frac {1}{x^2}}\right )+\sqrt {3-x^2} \]

[In]

int((3 - x^2)^(1/2)/x,x)

[Out]

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