3.13.55 \(\int \frac {\sqrt [3]{-1+x^6} (1+x^6)}{x^2 (-1+x^3+x^6)} \, dx\)

Optimal. Leaf size=91 \[ \frac {\sqrt [3]{x^6-1}}{x}-\frac {1}{3} \log \left (\sqrt [3]{x^6-1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-1}-x}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (-\sqrt [3]{x^6-1} x+\left (x^6-1\right )^{2/3}+x^2\right ) \]

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Rubi [C]  time = 1.70, antiderivative size = 593, normalized size of antiderivative = 6.52, number of steps used = 38, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6728, 365, 364, 1562, 465, 430, 429, 511, 510} \begin {gather*} -\frac {2 \left (5-\sqrt {5}\right ) \sqrt [3]{x^6-1} x^5 F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \sqrt [3]{x^6-1} x^5 F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5+\sqrt {5}\right ) \sqrt [3]{x^6-1} x^5 F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {2 \sqrt [3]{x^6-1} x^5 F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {\left (5-\sqrt {5}\right ) \sqrt [3]{x^6-1} x^2 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{10 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {\sqrt [3]{x^6-1} x^2 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\sqrt {5} \sqrt [3]{1-x^6}}-\frac {\left (5+\sqrt {5}\right ) \sqrt [3]{x^6-1} x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{10 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{x^6-1} x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\sqrt {5} \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{x^6-1} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{\sqrt [3]{1-x^6} x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)),x]

[Out]

-((x^2*(-1 + x^6)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, x^6, (2*x^6)/(3 - Sqrt[5])])/(Sqrt[5]*(1 - x^6)^(1/3))) -
((5 - Sqrt[5])*x^2*(-1 + x^6)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, x^6, (2*x^6)/(3 - Sqrt[5])])/(10*(3 - Sqrt[5])
*(1 - x^6)^(1/3)) + (x^2*(-1 + x^6)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (2*x^6)/(3 + Sqrt[5]), x^6])/(Sqrt[5]*(1
- x^6)^(1/3)) - ((5 + Sqrt[5])*x^2*(-1 + x^6)^(1/3)*AppellF1[1/3, 1, -1/3, 4/3, (2*x^6)/(3 + Sqrt[5]), x^6])/
(10*(3 + Sqrt[5])*(1 - x^6)^(1/3)) - (2*x^5*(-1 + x^6)^(1/3)*AppellF1[5/6, -1/3, 1, 11/6, x^6, (2*x^6)/(3 - Sq
rt[5])])/(5*Sqrt[5]*(3 - Sqrt[5])*(1 - x^6)^(1/3)) - (2*(5 - Sqrt[5])*x^5*(-1 + x^6)^(1/3)*AppellF1[5/6, -1/3,
1, 11/6, x^6, (2*x^6)/(3 - Sqrt[5])])/(25*(3 - Sqrt[5])*(1 - x^6)^(1/3)) + (2*x^5*(-1 + x^6)^(1/3)*AppellF1[5
/6, -1/3, 1, 11/6, x^6, (2*x^6)/(3 + Sqrt[5])])/(5*Sqrt[5]*(3 + Sqrt[5])*(1 - x^6)^(1/3)) - (2*(5 + Sqrt[5])*x
^5*(-1 + x^6)^(1/3)*AppellF1[5/6, -1/3, 1, 11/6, x^6, (2*x^6)/(3 + Sqrt[5])])/(25*(3 + Sqrt[5])*(1 - x^6)^(1/3
)) + ((-1 + x^6)^(1/3)*Hypergeometric2F1[-1/3, -1/6, 5/6, x^6])/(x*(1 - x^6)^(1/3))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1562

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx &=\int \left (-\frac {\sqrt [3]{-1+x^6}}{x^2}+\frac {x \left (-1-2 x^3\right ) \sqrt [3]{-1+x^6}}{1-x^3-x^6}\right ) \, dx\\ &=-\int \frac {\sqrt [3]{-1+x^6}}{x^2} \, dx+\int \frac {x \left (-1-2 x^3\right ) \sqrt [3]{-1+x^6}}{1-x^3-x^6} \, dx\\ &=-\frac {\sqrt [3]{-1+x^6} \int \frac {\sqrt [3]{1-x^6}}{x^2} \, dx}{\sqrt [3]{1-x^6}}+\int \left (\frac {x \sqrt [3]{-1+x^6}}{-1+x^3+x^6}+\frac {2 x^4 \sqrt [3]{-1+x^6}}{-1+x^3+x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}+2 \int \frac {x^4 \sqrt [3]{-1+x^6}}{-1+x^3+x^6} \, dx+\int \frac {x \sqrt [3]{-1+x^6}}{-1+x^3+x^6} \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}+2 \int \left (-\frac {\left (-1+\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right )}+\frac {\left (1+\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right )}\right ) \, dx+\int \left (-\frac {2 x \sqrt [3]{-1+x^6}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^3\right )}-\frac {2 x \sqrt [3]{-1+x^6}}{\sqrt {5} \left (1+\sqrt {5}+2 x^3\right )}\right ) \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}-\frac {2 \int \frac {x \sqrt [3]{-1+x^6}}{-1+\sqrt {5}-2 x^3} \, dx}{\sqrt {5}}-\frac {2 \int \frac {x \sqrt [3]{-1+x^6}}{1+\sqrt {5}+2 x^3} \, dx}{\sqrt {5}}-\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {x \sqrt [3]{-1+x^6}}{-1+\sqrt {5}-2 x^3} \, dx+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {x \sqrt [3]{-1+x^6}}{1+\sqrt {5}+2 x^3} \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}-\frac {2 \int \left (\frac {\left (1+\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^4 \sqrt [3]{-1+x^6}}{-3-\sqrt {5}+2 x^6}\right ) \, dx}{\sqrt {5}}-\frac {2 \int \left (\frac {\left (1-\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{2 \left (-3+\sqrt {5}+2 x^6\right )}-\frac {x^4 \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6}\right ) \, dx}{\sqrt {5}}-\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \left (\frac {\left (1-\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{2 \left (-3+\sqrt {5}+2 x^6\right )}-\frac {x^4 \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6}\right ) \, dx+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \left (\frac {\left (1+\sqrt {5}\right ) x \sqrt [3]{-1+x^6}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^4 \sqrt [3]{-1+x^6}}{-3-\sqrt {5}+2 x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}-\frac {2 \int \frac {x^4 \sqrt [3]{-1+x^6}}{-3-\sqrt {5}+2 x^6} \, dx}{\sqrt {5}}+\frac {2 \int \frac {x^4 \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6} \, dx}{\sqrt {5}}-\frac {1}{5} \left (2 \left (5-3 \sqrt {5}\right )\right ) \int \frac {x \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6} \, dx+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {x^4 \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6} \, dx-\frac {1}{5} \left (-5+\sqrt {5}\right ) \int \frac {x \sqrt [3]{-1+x^6}}{-3+\sqrt {5}+2 x^6} \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {x \sqrt [3]{-1+x^6}}{3+\sqrt {5}-2 x^6} \, dx+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {x^4 \sqrt [3]{-1+x^6}}{-3-\sqrt {5}+2 x^6} \, dx+\frac {1}{5} \left (2 \left (5+3 \sqrt {5}\right )\right ) \int \frac {x \sqrt [3]{-1+x^6}}{3+\sqrt {5}-2 x^6} \, dx\\ &=\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )-\frac {1}{10} \left (-5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )-\frac {1}{10} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{3+\sqrt {5}-2 x^3} \, dx,x,x^2\right )+\frac {1}{5} \left (5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{3+\sqrt {5}-2 x^3} \, dx,x,x^2\right )-\frac {\left (2 \sqrt [3]{-1+x^6}\right ) \int \frac {x^4 \sqrt [3]{1-x^6}}{-3-\sqrt {5}+2 x^6} \, dx}{\sqrt {5} \sqrt [3]{1-x^6}}+\frac {\left (2 \sqrt [3]{-1+x^6}\right ) \int \frac {x^4 \sqrt [3]{1-x^6}}{-3+\sqrt {5}+2 x^6} \, dx}{\sqrt {5} \sqrt [3]{1-x^6}}+\frac {\left (2 \left (5-\sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \int \frac {x^4 \sqrt [3]{1-x^6}}{-3+\sqrt {5}+2 x^6} \, dx}{5 \sqrt [3]{1-x^6}}+\frac {\left (2 \left (5+\sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \int \frac {x^4 \sqrt [3]{1-x^6}}{-3-\sqrt {5}+2 x^6} \, dx}{5 \sqrt [3]{1-x^6}}\\ &=-\frac {2 x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5-\sqrt {5}\right ) x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {2 x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5+\sqrt {5}\right ) x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}-\frac {\left (\left (5-3 \sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{5 \sqrt [3]{1-x^6}}-\frac {\left (\left (-5+\sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{10 \sqrt [3]{1-x^6}}-\frac {\left (\left (5+\sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{3+\sqrt {5}-2 x^3} \, dx,x,x^2\right )}{10 \sqrt [3]{1-x^6}}+\frac {\left (\left (5+3 \sqrt {5}\right ) \sqrt [3]{-1+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{3+\sqrt {5}-2 x^3} \, dx,x,x^2\right )}{5 \sqrt [3]{1-x^6}}\\ &=-\frac {x^2 \sqrt [3]{-1+x^6} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\sqrt {5} \sqrt [3]{1-x^6}}-\frac {\left (5-\sqrt {5}\right ) x^2 \sqrt [3]{-1+x^6} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{10 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {x^2 \sqrt [3]{-1+x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\sqrt {5} \sqrt [3]{1-x^6}}-\frac {\left (5+\sqrt {5}\right ) x^2 \sqrt [3]{-1+x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{10 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{5 \sqrt {5} \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5-\sqrt {5}\right ) x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{25 \left (3-\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {2 x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{5 \sqrt {5} \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}-\frac {2 \left (5+\sqrt {5}\right ) x^5 \sqrt [3]{-1+x^6} F_1\left (\frac {5}{6};-\frac {1}{3},1;\frac {11}{6};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{25 \left (3+\sqrt {5}\right ) \sqrt [3]{1-x^6}}+\frac {\sqrt [3]{-1+x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};x^6\right )}{x \sqrt [3]{1-x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^2 \left (-1+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)),x]

[Out]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)), x]

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IntegrateAlgebraic [A]  time = 1.00, size = 91, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^6}}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6))/(x^2*(-1 + x^3 + x^6)),x]

[Out]

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

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fricas [A]  time = 13.84, size = 128, normalized size = 1.41 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {17707979315346691547103487078601066282657059082726673278841963389300888497059669011634 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 18779074824464902023518972945875034013564452605964125877184864112405550428883609929964 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (8791266734992875261237504664599259772605087326251698970792557525513888268399719816592 \, x^{6} + 9326814489551980499445247598236243638058784087870425269964007887066219234311690275757 \, x^{3} - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592\right )}}{3 \, {\left (9923243904393545413458713816471868889492119646716071835561526356358143878603884871272 \, x^{6} - 8320283165512251371852516195766181258618636197629327742451887394495075584367754599527 \, x^{3} - 9923243904393545413458713816471868889492119646716071835561526356358143878603884871272\right )}}\right ) - x \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) + 6 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^2/(x^6+x^3-1),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(3)*x*arctan(1/3*(177079793153466915471034870786010662826570590827266732788419633893008884970596690
11634*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 18779074824464902023518972945875034013564452605964125877184864112405550428
883609929964*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(879126673499287526123750466459925977260508732625169897079255
7525513888268399719816592*x^6 + 932681448955198049944524759823624363805878408787042526996400788706621923431169
0275757*x^3 - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592))/(9923243
904393545413458713816471868889492119646716071835561526356358143878603884871272*x^6 - 8320283165512251371852516
195766181258618636197629327742451887394495075584367754599527*x^3 - 9923243904393545413458713816471868889492119
646716071835561526356358143878603884871272)) - x*log((x^6 + x^3 + 3*(x^6 - 1)^(1/3)*x^2 + 3*(x^6 - 1)^(2/3)*x
- 1)/(x^6 + x^3 - 1)) + 6*(x^6 - 1)^(1/3))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^2/(x^6+x^3-1),x, algorithm="giac")

[Out]

integrate((x^6 + 1)*(x^6 - 1)^(1/3)/((x^6 + x^3 - 1)*x^2), x)

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maple [C]  time = 28.27, size = 599, normalized size = 6.58

method	result	size
trager	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}}}{x}+\frac {\ln \left (\frac {942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-1211814230256347664106488816042 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-33783984098747171016688237221 x^{6}-7425326451703787337858079123548 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+877700737851189734699326533804 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}+295728634269156474191717482137 x \left (x^{6}-1\right )^{\frac {2}{3}}-295728634269156474191717482137 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+29493954371922133427267508685 x^{3}-942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+1211814230256347664106488816042 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+33783984098747171016688237221}{x^{6}+x^{3}-1}\right )}{3}-2 \ln \left (\frac {942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}-1211814230256347664106488816042 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-33783984098747171016688237221 x^{6}-7425326451703787337858079123548 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}+694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+877700737851189734699326533804 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}+295728634269156474191717482137 x \left (x^{6}-1\right )^{\frac {2}{3}}-295728634269156474191717482137 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+29493954371922133427267508685 x^{3}-942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+1211814230256347664106488816042 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+33783984098747171016688237221}{x^{6}+x^{3}-1}\right ) \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+2 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{6}+897514697909097406419374355786 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{6}-209561394779200926893843501540 x^{6}-7425326451703787337858079123548 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +694810793316614380319244932460 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+1597408079383406044586699840712 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{3}+411530433155258870911591637547 x \left (x^{6}-1\right )^{\frac {2}{3}}-411530433155258870911591637547 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-30481657422429225730013600224 x^{3}-942898597041750773061343380768 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-897514697909097406419374355786 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+209561394779200926893843501540}{x^{6}+x^{3}-1}\right )\)	\(599\)
risch	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (\frac {\ln \left (-\frac {6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{12}-x^{12}+18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{9}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+x^{9}-12 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}-18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )}{3}-\ln \left (-\frac {6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{12}-x^{12}+18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{9}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+x^{9}-12 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}-18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{6}+x^{3}-1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{12}-x^{12}-18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{9}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{9}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}+3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x^{7}-12 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}+18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{12}-2 x^{6}+1\right )^{\frac {2}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x -3 \left (x^{12}-2 x^{6}+1\right )^{\frac {1}{3}} x +6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{6}+x^{3}-1\right )}\right )\right ) \left (\left (x^{6}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{6}-1\right )^{\frac {2}{3}}}\)	\(866\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/3)*(x^6+1)/x^2/(x^6+x^3-1),x,method=_RETURNVERBOSE)

[Out]

(x^6-1)^(1/3)/x+1/3*ln((942898597041750773061343380768*RootOf(36*_Z^2-6*_Z+1)^2*x^6-12118142302563476641064888
16042*RootOf(36*_Z^2-6*_Z+1)*x^6-33783984098747171016688237221*x^6-7425326451703787337858079123548*RootOf(36*_
Z^2-6*_Z+1)^2*x^3+694810793316614380319244932460*RootOf(36*_Z^2-6*_Z+1)*(x^6-1)^(2/3)*x-6948107933166143803192
44932460*RootOf(36*_Z^2-6*_Z+1)*(x^6-1)^(1/3)*x^2+877700737851189734699326533804*RootOf(36*_Z^2-6*_Z+1)*x^3+29
5728634269156474191717482137*x*(x^6-1)^(2/3)-295728634269156474191717482137*x^2*(x^6-1)^(1/3)+2949395437192213
3427267508685*x^3-942898597041750773061343380768*RootOf(36*_Z^2-6*_Z+1)^2+1211814230256347664106488816042*Root
Of(36*_Z^2-6*_Z+1)+33783984098747171016688237221)/(x^6+x^3-1))-2*ln((942898597041750773061343380768*RootOf(36*
_Z^2-6*_Z+1)^2*x^6-1211814230256347664106488816042*RootOf(36*_Z^2-6*_Z+1)*x^6-33783984098747171016688237221*x^
6-7425326451703787337858079123548*RootOf(36*_Z^2-6*_Z+1)^2*x^3+694810793316614380319244932460*RootOf(36*_Z^2-6
*_Z+1)*(x^6-1)^(2/3)*x-694810793316614380319244932460*RootOf(36*_Z^2-6*_Z+1)*(x^6-1)^(1/3)*x^2+877700737851189
734699326533804*RootOf(36*_Z^2-6*_Z+1)*x^3+295728634269156474191717482137*x*(x^6-1)^(2/3)-29572863426915647419
1717482137*x^2*(x^6-1)^(1/3)+29493954371922133427267508685*x^3-942898597041750773061343380768*RootOf(36*_Z^2-6
*_Z+1)^2+1211814230256347664106488816042*RootOf(36*_Z^2-6*_Z+1)+33783984098747171016688237221)/(x^6+x^3-1))*Ro
otOf(36*_Z^2-6*_Z+1)+2*RootOf(36*_Z^2-6*_Z+1)*ln((942898597041750773061343380768*RootOf(36*_Z^2-6*_Z+1)^2*x^6+
897514697909097406419374355786*RootOf(36*_Z^2-6*_Z+1)*x^6-209561394779200926893843501540*x^6-74253264517037873
37858079123548*RootOf(36*_Z^2-6*_Z+1)^2*x^3-694810793316614380319244932460*RootOf(36*_Z^2-6*_Z+1)*(x^6-1)^(2/3
)*x+694810793316614380319244932460*RootOf(36*_Z^2-6*_Z+1)*(x^6-1)^(1/3)*x^2+1597408079383406044586699840712*Ro
otOf(36*_Z^2-6*_Z+1)*x^3+411530433155258870911591637547*x*(x^6-1)^(2/3)-411530433155258870911591637547*x^2*(x^
6-1)^(1/3)-30481657422429225730013600224*x^3-942898597041750773061343380768*RootOf(36*_Z^2-6*_Z+1)^2-897514697
909097406419374355786*RootOf(36*_Z^2-6*_Z+1)+209561394779200926893843501540)/(x^6+x^3-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^2/(x^6+x^3-1),x, algorithm="maxima")

[Out]

integrate((x^6 + 1)*(x^6 - 1)^(1/3)/((x^6 + x^3 - 1)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^2\,\left (x^6+x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/(x^2*(x^3 + x^6 - 1)),x)

[Out]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/(x^2*(x^3 + x^6 - 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(1/3)*(x**6+1)/x**2/(x**6+x**3-1),x)

[Out]

Timed out

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