\(\int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx\) [2390]

Optimal result

Integrand size = 15, antiderivative size = 188 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=2 \sqrt {x}+\frac {3}{5} \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1-\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {10+2 \sqrt {5}}}\right )-\frac {3}{5} \sqrt {10+2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}+4 \sqrt [6]{x}}{\sqrt {10-2 \sqrt {5}}}\right )+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2+\left (1-\sqrt {5}\right ) \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2+\left (1+\sqrt {5}\right ) \sqrt [6]{x}+2 \sqrt [3]{x}\right ) \]

[Out]

6/5*ln(1-x^(1/6))-3/10*ln(2+2*x^(1/3)+x^(1/6)*(5^(1/2)+1))*(-5^(1/2)+1)-3/10*ln(2+2*x^(1/3)+x^(1/6)*(-5^(1/2)+
1))*(5^(1/2)+1)+2*x^(1/2)+3/5*arctan((1+4*x^(1/6)-5^(1/2))/(10+2*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)-3/5*arct
an((1+4*x^(1/6)+5^(1/2))/(10-2*5^(1/2))^(1/2))*(10+2*5^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {348, 327, 300, 648, 632, 210, 642, 31} \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=\frac {3}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 \sqrt [6]{x}-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 \sqrt [6]{x}+\sqrt {5}+1\right )\right )+2 \sqrt {x}+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac {3}{10} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac {3}{10} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt {5} \sqrt [6]{x}+\sqrt [6]{x}+2\right ) \]

[In]

Int[x^(1/3)/(-1 + x^(5/6)),x]

[Out]

2*Sqrt[x] + (3*Sqrt[2*(5 - Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5 + Sqrt[5])]])/5 - (3*Sqrt[2*(5
+ Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3*(1
+ Sqrt[5])*Log[2 + x^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 - Sqrt[5])*Log[2 + x^(1/6) + Sqrt[5]*x^(
1/6) + 2*x^(1/3)])/10

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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])

Rule 300

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] - Dist[2*((-r)^(m + 1)/(a*n*s^m
)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1
] && NegQ[a/b]

Rule 327

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] - Dist[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]

Rule 348

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[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]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = 6 \text {Subst}\left (\int \frac {x^7}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}+6 \text {Subst}\left (\int \frac {x^2}{-1+x^5} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}-\frac {6}{5} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {\frac {1}{4} \left (-1-\sqrt {5}\right )+\frac {1}{4} \left (1+\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {12}{5} \text {Subst}\left (\int \frac {\frac {1}{4} \left (-1+\sqrt {5}\right )+\frac {1}{4} \left (1-\sqrt {5}\right ) x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}+\frac {6}{5} \log \left (1-\sqrt [6]{x}\right )+\frac {3 \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )}{\sqrt {5}}-\frac {3 \text {Subst}\left (\int \frac {1}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )}{\sqrt {5}}-\frac {1}{10} \left (3 \left (1-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1+\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac {1}{10} \left (3 \left (1+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1-\sqrt {5}\right )+2 x}{1+\frac {1}{2} \left (1-\sqrt {5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right ) \\ & = 2 \sqrt {x}+\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 )-\frac {6 \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5-\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right )+2 \sqrt [6]{x}\right )}{\sqrt {5}}+\frac {6 \text {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-5+\sqrt {5}\right )-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right )+2 \sqrt [6]{x}\right )}{\sqrt {5}} \\ & = 2 \sqrt {x}+6 \sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \tan ^{-1}\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 )} \tan ^{-1}\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 ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=2 \sqrt {x}+\frac {6}{5} \log \left (-1+\sqrt [6]{x}\right )-\frac {6}{5} \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt [6]{x}-\text {$\#$1}\right )-2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt [6]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[x^(1/3)/(-1 + x^(5/6)),x]

[Out]

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

Maple [A] (verified)

Time = 3.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.66

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	\(2 \sqrt {x}+\frac {3 \left (\sqrt {5}-1\right ) \ln \left (\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+2\right )}{10}+\frac {12 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \left (\sqrt {5}+1\right ) \ln \left (-\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+2\right )}{10}-\frac {12 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}\)	\(172\)
default	\(2 \sqrt {x}+\frac {3 \left (\sqrt {5}-1\right ) \ln \left (\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+2\right )}{10}+\frac {12 \left (-\sqrt {5}+1-\frac {\left (\sqrt {5}-1\right ) \left (\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}+\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {3 \left (\sqrt {5}+1\right ) \ln \left (-\sqrt {5}\, x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}}+2\right )}{10}-\frac {12 \left (-\sqrt {5}-1-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}+1\right )}{4}\right ) \arctan \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {6 \ln \left (x^{\frac {1}{6}}-1\right )}{5}\)	\(172\)

[In]

int(x^(1/3)/(-1+x^(5/6)),x,method=_RETURNVERBOSE)

[Out]

-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)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (130) = 260\).

Time = 0.98 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.39 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=\frac {1}{10} \, {\left (3 \, \sqrt {5} - \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (\frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 3 \, \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} {\left (\sqrt {5} - 1\right )} + 72 \, x^{\frac {1}{6}} + 36\right ) + \frac {1}{10} \, {\left (3 \, \sqrt {5} + \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} - 3\right )} \log \left (\frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} - 3 \, \sqrt {-\frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + \frac {9}{2} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} - 3\right )} {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} - \frac {27}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 18 \, \sqrt {2} \sqrt {\sqrt {5} - 5} + 18 \, \sqrt {5} - 90} {\left (\sqrt {5} - 1\right )} + 72 \, x^{\frac {1}{6}} + 36\right ) - \frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )} \log \left (-\frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} + \sqrt {5} + 1\right )}^{2} + 36 \, x^{\frac {1}{6}}\right ) + \frac {3}{10} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )} \log \left (-\frac {9}{4} \, {\left (\sqrt {2} \sqrt {\sqrt {5} - 5} - \sqrt {5} - 1\right )}^{2} + 36 \, x^{\frac {1}{6}}\right ) + 2 \, \sqrt {x} + \frac {6}{5} \, \log \left (x^{\frac {1}{6}} - 1\right ) \]

[In]

integrate(x^(1/3)/(-1+x^(5/6)),x, algorithm="fricas")

[Out]

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) + sq
rt(5) - 3)*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 27/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 18*s
qrt(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*(sq
rt(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*(sqr
t(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) + 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*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*(sq
rt(2)*sqrt(sqrt(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)*log(-9/4*(sqrt(2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 36*x^(1/6)
) + 2*sqrt(x) + 6/5*log(x^(1/6) - 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (167) = 334\).

Time = 16.00 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=2 \sqrt {x} + \frac {6 \log {\left (\sqrt [6]{x} - 1 \right )}}{5} - \frac {3 \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} + \frac {3 \sqrt {5} \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \sqrt {5} \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \sqrt {10} \sqrt {5 - \sqrt {5}} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {5 - \sqrt {5}}} + \frac {\sqrt {2}}{2 \sqrt {5 - \sqrt {5}}} + \frac {\sqrt {10}}{2 \sqrt {5 - \sqrt {5}}} \right )}}{10} - \frac {3 \sqrt {2} \sqrt {5 - \sqrt {5}} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {5 - \sqrt {5}}} + \frac {\sqrt {2}}{2 \sqrt {5 - \sqrt {5}}} + \frac {\sqrt {10}}{2 \sqrt {5 - \sqrt {5}}} \right )}}{10} - \frac {3 \sqrt {2} \sqrt {\sqrt {5} + 5} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {\sqrt {5} + 5}} - \frac {\sqrt {10}}{2 \sqrt {\sqrt {5} + 5}} + \frac {\sqrt {2}}{2 \sqrt {\sqrt {5} + 5}} \right )}}{10} + \frac {3 \sqrt {10} \sqrt {\sqrt {5} + 5} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {\sqrt {5} + 5}} - \frac {\sqrt {10}}{2 \sqrt {\sqrt {5} + 5}} + \frac {\sqrt {2}}{2 \sqrt {\sqrt {5} + 5}} \right )}}{10} \]

[In]

integrate(x**(1/3)/(-1+x**(5/6)),x)

[Out]

2*sqrt(x) + 6*log(x**(1/6) - 1)/5 - 3*log(8*x**(1/6) + 8*sqrt(5)*x**(1/6) + 16*x**(1/3) + 16)/10 + 3*sqrt(5)*l
og(8*x**(1/6) + 8*sqrt(5)*x**(1/6) + 16*x**(1/3) + 16)/10 - 3*sqrt(5)*log(-8*sqrt(5)*x**(1/6) + 8*x**(1/6) + 1
6*x**(1/3) + 16)/10 - 3*log(-8*sqrt(5)*x**(1/6) + 8*x**(1/6) + 16*x**(1/3) + 16)/10 - 3*sqrt(10)*sqrt(5 - sqrt
(5))*atan(2*sqrt(2)*x**(1/6)/sqrt(5 - sqrt(5)) + sqrt(2)/(2*sqrt(5 - sqrt(5))) + sqrt(10)/(2*sqrt(5 - sqrt(5))
))/10 - 3*sqrt(2)*sqrt(5 - sqrt(5))*atan(2*sqrt(2)*x**(1/6)/sqrt(5 - sqrt(5)) + sqrt(2)/(2*sqrt(5 - sqrt(5)))
+ sqrt(10)/(2*sqrt(5 - sqrt(5))))/10 - 3*sqrt(2)*sqrt(sqrt(5) + 5)*atan(2*sqrt(2)*x**(1/6)/sqrt(sqrt(5) + 5) -
sqrt(10)/(2*sqrt(sqrt(5) + 5)) + sqrt(2)/(2*sqrt(sqrt(5) + 5)))/10 + 3*sqrt(10)*sqrt(sqrt(5) + 5)*atan(2*sqrt
(2)*x**(1/6)/sqrt(sqrt(5) + 5) - sqrt(10)/(2*sqrt(sqrt(5) + 5)) + sqrt(2)/(2*sqrt(sqrt(5) + 5)))/10

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (130) = 260\).

Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=-\frac {6}{5} \, \left (-1\right )^{\frac {3}{5}} \log \left (\left (-1\right )^{\frac {1}{5}} + x^{\frac {1}{6}}\right ) - \frac {6 \, \sqrt {5} \left (-1\right )^{\frac {3}{5}} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} - 10} + \left (-1\right )^{\frac {1}{5}} - 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {2 \, \sqrt {5} - 10}} + \frac {6 \, \sqrt {5} \left (-1\right )^{\frac {3}{5}} \log \left (\frac {\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}{\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} - 10} - \left (-1\right )^{\frac {1}{5}} + 4 \, x^{\frac {1}{6}}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} - 10}} + 2 \, \sqrt {x} + \frac {6 \, \log \left (-x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} + \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {2}{5}} + \left (-1\right )^{\frac {2}{5}}\right )}} - \frac {6 \, \log \left (x^{\frac {1}{6}} {\left (\sqrt {5} \left (-1\right )^{\frac {1}{5}} - \left (-1\right )^{\frac {1}{5}}\right )} + 2 \, \left (-1\right )^{\frac {2}{5}} + 2 \, x^{\frac {1}{3}}\right )}{5 \, {\left (\sqrt {5} \left (-1\right )^{\frac {2}{5}} - \left (-1\right )^{\frac {2}{5}}\right )}} \]

[In]

integrate(x^(1/3)/(-1+x^(5/6)),x, algorithm="maxima")

[Out]

-6/5*(-1)^(3/5)*log((-1)^(1/5) + x^(1/6)) - 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) + 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))

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=\frac {3}{5} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, x^{\frac {1}{6}} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) - \frac {3}{5} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, x^{\frac {1}{6}} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {3}{10} \, \sqrt {5} \log \left (\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} + 1\right )} + x^{\frac {1}{3}} + 1\right ) - \frac {3}{10} \, \sqrt {5} \log \left (-\frac {1}{2} \, x^{\frac {1}{6}} {\left (\sqrt {5} - 1\right )} + x^{\frac {1}{3}} + 1\right ) + 2 \, \sqrt {x} - \frac {3}{10} \, \log \left (x^{\frac {2}{3}} + \sqrt {x} + x^{\frac {1}{3}} + x^{\frac {1}{6}} + 1\right ) + \frac {6}{5} \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \]

[In]

integrate(x^(1/3)/(-1+x^(5/6)),x, algorithm="giac")

[Out]

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)*a
rctan((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))

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [3]{x}}{-1+x^{5/6}} \, dx=\frac {6\,\ln \left (1296\,x^{1/6}-1296\right )}{5}-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}-\frac {3\,\sqrt {5}}{10}+\frac {3}{10}\right )+\ln \left (750\,x^{1/6}\,{\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{10}+\frac {3\,\sqrt {5}}{10}-\frac {3}{10}\right )-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {5}}{10}-\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )-\ln \left (-750\,x^{1/6}\,{\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )}^3-1296\right )\,\left (\frac {3\,\sqrt {5}}{10}+\frac {3\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{10}+\frac {3}{10}\right )+2\,\sqrt {x} \]

[In]

int(x^(1/3)/(x^(5/6) - 1),x)

[Out]

(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)