Integrand size = 15, antiderivative size = 200 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=2 \sqrt {x}+\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 \sqrt [6]{x}\right )\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt {5} \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \]
6/5*ln(1-x^(1/6))-3/10*ln(2+x^(1/6)+2*x^(1/3)+x^(1/6)*5^(1/2))*(-5^(1/2)+1 )-3/10*ln(2+x^(1/6)+2*x^(1/3)-x^(1/6)*5^(1/2))*(5^(1/2)+1)+2*x^(1/2)+3/5*a rctan((1+4*x^(1/6)-5^(1/2))/(10+2*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-3/5 *arctan(1/20*(1+4*x^(1/6)+5^(1/2))*(50+10*5^(1/2))^(1/2))*(10+2*5^(1/2))^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.63 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=2 \sqrt {x}+\frac {6}{5} \log \left (-1+\sqrt [6]{x}\right )-\frac {6}{5} \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt [6]{x}-\text {$\#$1}\right )-2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
2*Sqrt[x] + (6*Log[-1 + x^(1/6)])/5 - (6*RootSum[1 + #1 + #1^2 + #1^3 + #1 ^4 & , (-Log[x^(1/6) - #1] - 2*Log[x^(1/6) - #1]*#1 + 2*Log[x^(1/6) - #1]* #1^2 + Log[x^(1/6) - #1]*#1^3)/(1 + 2*#1 + 3*#1^2 + 4*#1^3) & ])/5
Time = 0.40 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {2027, 864, 25, 843, 823, 16, 27, 1142, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt {x}-\frac {1}{\sqrt [3]{x}}} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {\sqrt [3]{x}}{x^{5/6}-1}dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle 6 \int -\frac {x^{7/6}}{1-x^{5/6}}d\sqrt [6]{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -6 \int \frac {x^{7/6}}{1-x^{5/6}}d\sqrt [6]{x}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 6 \left (\frac {\sqrt {x}}{3}-\int \frac {\sqrt [3]{x}}{1-x^{5/6}}d\sqrt [6]{x}\right )\) |
| \(\Big \downarrow \) 823 |
| \(\displaystyle 6 \left (-\frac {1}{5} \int \frac {1}{1-\sqrt [6]{x}}d\sqrt [6]{x}-\frac {2}{5} \int -\frac {-\left (\left (1+\sqrt {5}\right ) \sqrt [6]{x}\right )+\sqrt {5}+1}{2 \left (2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2\right )}d\sqrt [6]{x}-\frac {2}{5} \int -\frac {-\left (\left (1-\sqrt {5}\right ) \sqrt [6]{x}\right )-\sqrt {5}+1}{2 \left (2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2\right )}d\sqrt [6]{x}+\frac {\sqrt {x}}{3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 6 \left (-\frac {2}{5} \int -\frac {-\left (\left (1+\sqrt {5}\right ) \sqrt [6]{x}\right )+\sqrt {5}+1}{2 \left (2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2\right )}d\sqrt [6]{x}-\frac {2}{5} \int -\frac {-\left (\left (1-\sqrt {5}\right ) \sqrt [6]{x}\right )-\sqrt {5}+1}{2 \left (2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2\right )}d\sqrt [6]{x}+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \left (\frac {1}{5} \int \frac {-\left (\left (1+\sqrt {5}\right ) \sqrt [6]{x}\right )+\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}+\frac {1}{5} \int \frac {-\left (\left (1-\sqrt {5}\right ) \sqrt [6]{x}\right )-\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 6 \left (\frac {1}{5} \left (\sqrt {5} \int \frac {1}{2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}-\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}\right )+\frac {1}{5} \left (-\sqrt {5} \int \frac {1}{2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}+\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}\right )+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 6 \left (\frac {1}{5} \left (-2 \sqrt {5} \int \frac {1}{-\sqrt [3]{x}-2 \left (5+\sqrt {5}\right )}d\left (4 \sqrt [6]{x}-\sqrt {5}+1\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}-\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}\right )+\frac {1}{5} \left (2 \sqrt {5} \int \frac {1}{-\sqrt [3]{x}-2 \left (5-\sqrt {5}\right )}d\left (4 \sqrt [6]{x}+\sqrt {5}+1\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}+\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}\right )+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 6 \left (\frac {1}{5} \left (\sqrt {\frac {10}{5+\sqrt {5}}} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}-\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}\right )+\frac {1}{5} \left (-\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 \sqrt [6]{x}+\sqrt {5}+1}{2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2}d\sqrt [6]{x}-\sqrt {\frac {10}{5-\sqrt {5}}} \arctan \left (\frac {4 \sqrt [6]{x}+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )\right )+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 6 \left (\frac {1}{5} \left (\sqrt {\frac {10}{5+\sqrt {5}}} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2\right )\right )+\frac {1}{5} \left (-\sqrt {\frac {10}{5-\sqrt {5}}} \arctan \left (\frac {4 \sqrt [6]{x}+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2\right )\right )+\frac {\sqrt {x}}{3}+\frac {1}{5} \log \left (1-\sqrt [6]{x}\right )\right )\) |
6*(Sqrt[x]/3 + Log[1 - x^(1/6)]/5 + (Sqrt[10/(5 + Sqrt[5])]*ArcTan[(1 - Sq rt[5] + 4*x^(1/6))/Sqrt[2*(5 + Sqrt[5])]] - ((1 + Sqrt[5])*Log[2 + (1 - Sq rt[5])*x^(1/6) + 2*x^(1/3)])/4)/5 + (-(Sqrt[10/(5 - Sqrt[5])]*ArcTan[(1 + Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5 - Sqrt[5])]]) - ((1 - Sqrt[5])*Log[2 + (1 + Sqrt[5])*x^(1/6) + 2*x^(1/3)])/4)/5)
3.3.20.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2* k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r^(m + 1)/(a*n*s^m) Int[1/(r - s*x), x] - 2*((-r)^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 1)/2}], x]] /; FreeQ[{a, b }, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Time = 0.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.62
| method | result | size |
| meijerg | \(-\frac {6 \left (-1\right )^{\frac {2}{5}} \left (\frac {5 \sqrt {x}\, \left (-1\right )^{\frac {3}{5}}}{3}+\left (-1\right )^{\frac {3}{5}} \left (\ln \left (1-x^{\frac {1}{6}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )+2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}{1-\cos \left (\frac {2 \pi }{5}\right ) x^{\frac {1}{6}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}+x^{\frac {1}{3}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}{1+\cos \left (\frac {\pi }{5}\right ) x^{\frac {1}{6}}}\right )\right )\right )}{5}\) | \(124\) |
| derivativedivides | \(2 \sqrt {x}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}+\frac {3 \left (\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (\sqrt {5}+1\right )}{10}-\frac {12 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}\) | \(172\) |
| default | \(2 \sqrt {x}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}+\frac {3 \left (\sqrt {5}-1\right ) \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right )}{10}+\frac {12 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \ln \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (\sqrt {5}+1\right )}{10}-\frac {12 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}\) | \(172\) |
-6/5*(-1)^(2/5)*(5/3*x^(1/2)*(-1)^(3/5)+(-1)^(3/5)*(ln(1-x^(1/6))-cos(1/5* Pi)*ln(1-2*cos(2/5*Pi)*x^(1/6)+x^(1/3))+2*sin(1/5*Pi)*arctan(sin(2/5*Pi)*x ^(1/6)/(1-cos(2/5*Pi)*x^(1/6)))+cos(2/5*Pi)*ln(1+2*cos(1/5*Pi)*x^(1/6)+x^( 1/3))-2*sin(2/5*Pi)*arctan(sin(1/5*Pi)*x^(1/6)/(1+cos(1/5*Pi)*x^(1/6)))))
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (133) = 266\).
Time = 0.91 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.19 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx =\text {Too large to display} \]
1/10*(3*sqrt(5) - sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqr t(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(9/4*(sqrt(2)*sqrt(sqrt( 5) - 5) + sqrt(5) + 1)^2 + 9/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 3*sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2) *sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1 ) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqr t(5) - 5) + 18*sqrt(5) - 90)*(sqrt(5) - 1) + 72*x^(1/6) + 36) + 1/10*(3*sq rt(5) + sqrt(-27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt (2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt( sqrt(5) - 5) + 18*sqrt(5) - 90) - 3)*log(9/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 - 3*sqrt( -27/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 9/2*(sqrt(2)*sqrt(sqrt (5) - 5) + sqrt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*( sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*sqrt(2)*sqrt(sqrt(5) - 5) + 18*sqrt(5) - 90)*(sqrt(5) - 1) + 72*x^(1/6) + 36) - 3/10*(sqrt(2)*sqrt(s qrt(5) - 5) + sqrt(5) + 1)*log(-9/4*(sqrt(2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 36*x^(1/6)) + 3/10*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)*lo...
\[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\int \frac {\sqrt [3]{x}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac {2}{3}} + \sqrt [3]{x} + \sqrt {x} + 1\right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (133) = 266\).
Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.36 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=-\frac {6}{5} \, \left (-1\right )^{\frac {3}{5}} \log \left (\left (-1\right )^{\frac {1}{5}} + x^{\frac {1}{6}}\right ) - \frac {6 \, \sqrt {5} \left (-1\right )^{\frac {3}{5}} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {2 \, \sqrt {5} - 10}} + \frac {6 \, \sqrt {5} \left (-1\right )^{\frac {3}{5}} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} - 10}} + 2 \, \sqrt {x} + \frac {6 \, \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {2}{5}} + \left (-1\right )^{\frac {2}{5}}\right )}} - \frac {6 \, \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {2}{5}} - \left (-1\right )^{\frac {2}{5}}\right )}} \]
-6/5*(-1)^(3/5)*log((-1)^(1/5) + x^(1/6)) - 6/5*sqrt(5)*(-1)^(3/5)*log((sq rt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4*x^(1/6 ))/(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4* x^(1/6)))/sqrt(2*sqrt(5) - 10) + 6/5*sqrt(5)*(-1)^(3/5)*log((sqrt(5)*(-1)^ (1/5) - (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6))/(sqrt(5 )*(-1)^(1/5) + (-1)^(1/5)*sqrt(-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6))) /sqrt(-2*sqrt(5) - 10) + 2*sqrt(x) + 6/5*log(-x^(1/6)*(sqrt(5)*(-1)^(1/5) + (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(2/5) + (-1)^(2/5) ) - 6/5*log(x^(1/6)*(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x ^(1/3))/(sqrt(5)*(-1)^(2/5) - (-1)^(2/5))
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\frac {3}{5} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, x^{\frac {1}{6}} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {3}{5} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, x^{\frac {1}{6}} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {3}{10} \, \sqrt {5} \log \left (\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{10} \, \sqrt {5} \log \left (-\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} - \frac {3}{10} \, \log \left (x^{\frac {2}{3}} + \sqrt {x} + x^{\frac {1}{3}} + x^{\frac {1}{6}} + 1\right ) + \frac {6}{5} \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \]
3/5*sqrt(-2*sqrt(5) + 10)*arctan(-(sqrt(5) - 4*x^(1/6) - 1)/sqrt(2*sqrt(5) + 10)) - 3/5*sqrt(2*sqrt(5) + 10)*arctan((sqrt(5) + 4*x^(1/6) + 1)/sqrt(- 2*sqrt(5) + 10)) + 3/10*sqrt(5)*log(1/2*x^(1/6)*(sqrt(5) + 1) + x^(1/3) + 1) - 3/10*sqrt(5)*log(-1/2*x^(1/6)*(sqrt(5) - 1) + x^(1/3) + 1) + 2*sqrt(x ) - 3/10*log(x^(2/3) + sqrt(x) + x^(1/3) + x^(1/6) + 1) + 6/5*log(abs(x^(1 /6) - 1))
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.12 \[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=\frac {6\,\ln \left (1296\,x^{1/6}-1296\right )}{5}-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )+\ln \left (750\,x^{1/6}\,{\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )+2\,\sqrt {x} \]
(6*log(1296*x^(1/6) - 1296))/5 - log(- 750*x^(1/6)*((3*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/10 - (3*5^(1/2))/10 + 3/10)^3 - 1296)*((3*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/10 - (3*5^(1/2))/10 + 3/10) + log(750*x^(1/6)*((3*2^(1/2)*(- 5^ (1/2) - 5)^(1/2))/10 + (3*5^(1/2))/10 - 3/10)^3 - 1296)*((3*2^(1/2)*(- 5^( 1/2) - 5)^(1/2))/10 + (3*5^(1/2))/10 - 3/10) - log(- 750*x^(1/6)*((3*5^(1/ 2))/10 - (3*2^(1/2)*(5^(1/2) - 5)^(1/2))/10 + 3/10)^3 - 1296)*((3*5^(1/2)) /10 - (3*2^(1/2)*(5^(1/2) - 5)^(1/2))/10 + 3/10) - log(- 750*x^(1/6)*((3*5 ^(1/2))/10 + (3*2^(1/2)*(5^(1/2) - 5)^(1/2))/10 + 3/10)^3 - 1296)*((3*5^(1 /2))/10 + (3*2^(1/2)*(5^(1/2) - 5)^(1/2))/10 + 3/10) + 2*x^(1/2)
\[ \int \frac {1}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx=2 \sqrt {x}-\left (\int -\frac {x^{\frac {1}{6}}}{\sqrt {x}\, x -x^{\frac {2}{3}}}d x \right ) \]