\(\int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx\) [2391]

Optimal result

Integrand size = 13, antiderivative size = 67 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=-\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}+\sqrt {3-\frac {1}{\sqrt {x}}} x-\frac {\text {arctanh}\left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}} \]

[Out]

-1/18*arctanh(1/3*(3-1/x^(1/2))^(1/2)*3^(1/2))*3^(1/2)+x*(3-1/x^(1/2))^(1/2)-1/6*x^(1/2)*(3-1/x^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {196, 43, 44, 65, 212} \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\sqrt {3-\frac {1}{\sqrt {x}}} x-\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x} \]

[In]

Int[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

-1/6*(Sqrt[3 - 1/Sqrt[x]]*Sqrt[x]) + Sqrt[3 - 1/Sqrt[x]]*x - ArcTanh[Sqrt[3 - 1/Sqrt[x]]/Sqrt[3]]/(6*Sqrt[3])

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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

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 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

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])

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\sqrt {3-x}}{x^3} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \sqrt {3-\frac {1}{\sqrt {x}}} x+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {3-x} x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}+\sqrt {3-\frac {1}{\sqrt {x}}} x+\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {3-x} x} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}+\sqrt {3-\frac {1}{\sqrt {x}}} x-\frac {1}{6} \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\sqrt {3-\frac {1}{\sqrt {x}}}\right ) \\ & = -\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}+\sqrt {3-\frac {1}{\sqrt {x}}} x-\frac {\tanh ^{-1}\left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{18} \left (-3 \sqrt {3-\frac {1}{\sqrt {x}}} \left (\sqrt {x}-6 x\right )-\sqrt {3} \text {arctanh}\left (\sqrt {1-\frac {1}{3 \sqrt {x}}}\right )\right ) \]

[In]

Integrate[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

(-3*Sqrt[3 - 1/Sqrt[x]]*(Sqrt[x] - 6*x) - Sqrt[3]*ArcTanh[Sqrt[1 - 1/(3*Sqrt[x])]])/18

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.84 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12

method	result	size
meijerg	\(-\frac {i \sqrt {3}\, \sqrt {\operatorname {signum}\left (3 \sqrt {x}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{\frac {1}{4}} \sqrt {3}\, \left (-18 \sqrt {x}+3\right ) \sqrt {-3 \sqrt {x}+1}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (\sqrt {3}\, x^{\frac {1}{4}}\right )}{2}\right )}{9 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (3 \sqrt {x}-1\right )}}\)	\(75\)
derivativedivides	\(-\frac {\sqrt {\frac {3 \sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \left (\ln \left (-\frac {\sqrt {3}}{6}+\sqrt {3}\, \sqrt {x}+\sqrt {3 x -\sqrt {x}}\right ) \sqrt {3}-36 \sqrt {3 x -\sqrt {x}}\, \sqrt {x}+6 \sqrt {3 x -\sqrt {x}}\right )}{36 \sqrt {\left (3 \sqrt {x}-1\right ) \sqrt {x}}}\)	\(91\)
default	\(-\frac {\sqrt {\frac {3 \sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \left (\ln \left (-\frac {\sqrt {3}}{6}+\sqrt {3}\, \sqrt {x}+\sqrt {3 x -\sqrt {x}}\right ) \sqrt {3}-36 \sqrt {3 x -\sqrt {x}}\, \sqrt {x}+6 \sqrt {3 x -\sqrt {x}}\right )}{36 \sqrt {\left (3 \sqrt {x}-1\right ) \sqrt {x}}}\)	\(91\)

[In]

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

[Out]

-1/9*I*3^(1/2)/Pi^(1/2)*signum(3*x^(1/2)-1)^(1/2)/(-signum(3*x^(1/2)-1))^(1/2)*(-1/6*I*Pi^(1/2)*x^(1/4)*3^(1/2
)*(-18*x^(1/2)+3)*(-3*x^(1/2)+1)^(1/2)+1/2*I*Pi^(1/2)*arcsin(3^(1/2)*x^(1/4)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{6} \, {\left (6 \, x - \sqrt {x}\right )} \sqrt {\frac {3 \, x - \sqrt {x}}{x}} + \frac {1}{36} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {x} \sqrt {\frac {3 \, x - \sqrt {x}}{x}} - 6 \, \sqrt {x} + 1\right ) \]

[In]

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

[Out]

1/6*(6*x - sqrt(x))*sqrt((3*x - sqrt(x))/x) + 1/36*sqrt(3)*log(2*sqrt(3)*sqrt(x)*sqrt((3*x - sqrt(x))/x) - 6*s
qrt(x) + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.46 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\begin {cases} \frac {3 x^{\frac {5}{4}}}{\sqrt {3 \sqrt {x} - 1}} - \frac {3 x^{\frac {3}{4}}}{2 \sqrt {3 \sqrt {x} - 1}} + \frac {\sqrt [4]{x}}{6 \sqrt {3 \sqrt {x} - 1}} - \frac {\sqrt {3} \operatorname {acosh}{\left (\sqrt {3} \sqrt [4]{x} \right )}}{18} & \text {for}\: \left |{\sqrt {x}}\right | > \frac {1}{3} \\- \frac {3 i x^{\frac {5}{4}}}{\sqrt {1 - 3 \sqrt {x}}} + \frac {3 i x^{\frac {3}{4}}}{2 \sqrt {1 - 3 \sqrt {x}}} - \frac {i \sqrt [4]{x}}{6 \sqrt {1 - 3 \sqrt {x}}} + \frac {\sqrt {3} i \operatorname {asin}{\left (\sqrt {3} \sqrt [4]{x} \right )}}{18} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*x**(5/4)/sqrt(3*sqrt(x) - 1) - 3*x**(3/4)/(2*sqrt(3*sqrt(x) - 1)) + x**(1/4)/(6*sqrt(3*sqrt(x) -
1)) - sqrt(3)*acosh(sqrt(3)*x**(1/4))/18, Abs(sqrt(x)) > 1/3), (-3*I*x**(5/4)/sqrt(1 - 3*sqrt(x)) + 3*I*x**(3/
4)/(2*sqrt(1 - 3*sqrt(x))) - I*x**(1/4)/(6*sqrt(1 - 3*sqrt(x))) + sqrt(3)*I*asin(sqrt(3)*x**(1/4))/18, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{36} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {-\frac {1}{\sqrt {x}} + 3}}{\sqrt {3} + \sqrt {-\frac {1}{\sqrt {x}} + 3}}\right ) + \frac {{\left (-\frac {1}{\sqrt {x}} + 3\right )}^{\frac {3}{2}} + 3 \, \sqrt {-\frac {1}{\sqrt {x}} + 3}}{6 \, {\left ({\left (\frac {1}{\sqrt {x}} - 3\right )}^{2} + \frac {6}{\sqrt {x}} - 9\right )}} \]

[In]

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

[Out]

1/36*sqrt(3)*log(-(sqrt(3) - sqrt(-1/sqrt(x) + 3))/(sqrt(3) + sqrt(-1/sqrt(x) + 3))) + 1/6*((-1/sqrt(x) + 3)^(
3/2) + 3*sqrt(-1/sqrt(x) + 3))/((1/sqrt(x) - 3)^2 + 6/sqrt(x) - 9)

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{36} \, {\left (6 \, \sqrt {3 \, x - \sqrt {x}} {\left (6 \, \sqrt {x} - 1\right )} + \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} \sqrt {x} - \sqrt {3 \, x - \sqrt {x}}\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

1/36*(6*sqrt(3*x - sqrt(x))*(6*sqrt(x) - 1) + sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*sqrt(x) - sqrt(3*x - sqrt(x)
)) + 1)))*sgn(x)

Mupad [B] (verification not implemented)

Time = 6.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {4\,x\,\sqrt {3-\frac {1}{\sqrt {x}}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{2};\ \frac {5}{2};\ 3\,\sqrt {x}\right )}{3\,\sqrt {1-3\,\sqrt {x}}} \]

[In]

int((3 - 1/x^(1/2))^(1/2),x)

[Out]

(4*x*(3 - 1/x^(1/2))^(1/2)*hypergeom([-1/2, 3/2], 5/2, 3*x^(1/2)))/(3*(1 - 3*x^(1/2))^(1/2))