Integrand size = 13, antiderivative size = 67 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=-\frac {1}{6} \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}+\sqrt {3-\frac {1}{\sqrt {x}}} x-\frac {\text {arctanh}\left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{\sqrt {3}}\right )}{6 \sqrt {3}} \] Output:
-1/6*(3-1/x^(1/2))^(1/2)*x^(1/2)+(3-1/x^(1/2))^(1/2)*x-1/18*arctanh(1/3*(3 -1/x^(1/2))^(1/2)*3^(1/2))*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{18} \left (-3 \sqrt {3-\frac {1}{\sqrt {x}}} \left (\sqrt {x}-6 x\right )-\sqrt {3} \text {arctanh}\left (\sqrt {1-\frac {1}{3 \sqrt {x}}}\right )\right ) \] Input:
Integrate[Sqrt[3 - 1/Sqrt[x]],x]
Output:
(-3*Sqrt[3 - 1/Sqrt[x]]*(Sqrt[x] - 6*x) - Sqrt[3]*ArcTanh[Sqrt[1 - 1/(3*Sq rt[x])]])/18
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {774, 798, 51, 52, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}d\sqrt {x}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -2 \int \frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{x^{3/2}}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle -2 \left (-\frac {1}{4} \int \frac {1}{\sqrt {3-\frac {1}{\sqrt {x}}} x}d\frac {1}{\sqrt {x}}-\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{2 x}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -2 \left (\frac {1}{4} \left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{3 \sqrt {x}}-\frac {1}{6} \int \frac {1}{\sqrt {3-\frac {1}{\sqrt {x}}} \sqrt {x}}d\frac {1}{\sqrt {x}}\right )-\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{2 x}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -2 \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {1}{3-x}d\sqrt {3-\frac {1}{\sqrt {x}}}+\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{3 \sqrt {x}}\right )-\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{2 x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {1}{4} \left (\frac {\text {arctanh}\left (\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{3 \sqrt {x}}\right )-\frac {\sqrt {3-\frac {1}{\sqrt {x}}}}{2 x}\right )\) |
Input:
Int[Sqrt[3 - 1/Sqrt[x]],x]
Output:
-2*(-1/2*Sqrt[3 - 1/Sqrt[x]]/x + (Sqrt[3 - 1/Sqrt[x]]/(3*Sqrt[x]) + ArcTan h[Sqrt[3 - 1/Sqrt[x]]/Sqrt[3]]/(3*Sqrt[3]))/4)
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] - Simp[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 ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.48 (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 | \(-\frac {\sqrt {\frac {3 \sqrt {x}-1}{\sqrt {x}}}\, \sqrt {x}\, \left (\ln \left (-\frac {\sqrt {3}}{6}+\sqrt {3}\, \sqrt {x}+\sqrt {3 x -\sqrt {x}}\right ) \sqrt {3}-36 \sqrt {3 x -\sqrt {x}}\, \sqrt {x}+6 \sqrt {3 x -\sqrt {x}}\right )}{36 \sqrt {\left (3 \sqrt {x}-1\right ) \sqrt {x}}}\) | \(91\) |
Input:
int((3-1/x^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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)))
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{6} \, {\left (6 \, x - \sqrt {x}\right )} \sqrt {\frac {3 \, x - \sqrt {x}}{x}} + \frac {1}{36} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {x} \sqrt {\frac {3 \, x - \sqrt {x}}{x}} - 6 \, \sqrt {x} + 1\right ) \] Input:
integrate((3-1/x^(1/2))^(1/2),x, algorithm="fricas")
Output:
1/6*(6*x - sqrt(x))*sqrt((3*x - sqrt(x))/x) + 1/36*sqrt(3)*log(2*sqrt(3)*s qrt(x)*sqrt((3*x - sqrt(x))/x) - 6*sqrt(x) + 1)
Result contains complex when optimal does not.
Time = 2.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.46 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\begin {cases} \frac {3 x^{\frac {5}{4}}}{\sqrt {3 \sqrt {x} - 1}} - \frac {3 x^{\frac {3}{4}}}{2 \sqrt {3 \sqrt {x} - 1}} + \frac {\sqrt [4]{x}}{6 \sqrt {3 \sqrt {x} - 1}} - \frac {\sqrt {3} \operatorname {acosh}{\left (\sqrt {3} \sqrt [4]{x} \right )}}{18} & \text {for}\: \left |{\sqrt {x}}\right | > \frac {1}{3} \\- \frac {3 i x^{\frac {5}{4}}}{\sqrt {1 - 3 \sqrt {x}}} + \frac {3 i x^{\frac {3}{4}}}{2 \sqrt {1 - 3 \sqrt {x}}} - \frac {i \sqrt [4]{x}}{6 \sqrt {1 - 3 \sqrt {x}}} + \frac {\sqrt {3} i \operatorname {asin}{\left (\sqrt {3} \sqrt [4]{x} \right )}}{18} & \text {otherwise} \end {cases} \] Input:
integrate((3-1/x**(1/2))**(1/2),x)
Output:
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))
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{36} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {-\frac {1}{\sqrt {x}} + 3}}{\sqrt {3} + \sqrt {-\frac {1}{\sqrt {x}} + 3}}\right ) + \frac {{\left (-\frac {1}{\sqrt {x}} + 3\right )}^{\frac {3}{2}} + 3 \, \sqrt {-\frac {1}{\sqrt {x}} + 3}}{6 \, {\left ({\left (\frac {1}{\sqrt {x}} - 3\right )}^{2} + \frac {6}{\sqrt {x}} - 9\right )}} \] Input:
integrate((3-1/x^(1/2))^(1/2),x, algorithm="maxima")
Output:
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/sqr t(x) - 3)^2 + 6/sqrt(x) - 9)
Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {1}{36} \, {\left (6 \, \sqrt {3 \, x - \sqrt {x}} {\left (6 \, \sqrt {x} - 1\right )} + \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} \sqrt {x} - \sqrt {3 \, x - \sqrt {x}}\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (x\right ) \] Input:
integrate((3-1/x^(1/2))^(1/2),x, algorithm="giac")
Output:
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)
Time = 0.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=\frac {4\,x\,\sqrt {3-\frac {1}{\sqrt {x}}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{2};\ \frac {5}{2};\ 3\,\sqrt {x}\right )}{3\,\sqrt {1-3\,\sqrt {x}}} \] Input:
int((3 - 1/x^(1/2))^(1/2),x)
Output:
(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))
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \sqrt {3-\frac {1}{\sqrt {x}}} \, dx=x^{\frac {3}{4}} \sqrt {3 \sqrt {x}-1}-\frac {x^{\frac {1}{4}} \sqrt {3 \sqrt {x}-1}}{6}-\frac {\sqrt {3}\, \mathrm {log}\left (\sqrt {3 \sqrt {x}-1}+x^{\frac {1}{4}} \sqrt {3}\right )}{18} \] Input:
int((3-1/x^(1/2))^(1/2),x)
Output:
(18*x**(3/4)*sqrt(3*sqrt(x) - 1) - 3*x**(1/4)*sqrt(3*sqrt(x) - 1) - sqrt(3 )*log(sqrt(3*sqrt(x) - 1) + x**(1/4)*sqrt(3)))/18