3.16.13 \(\int \frac {(-1+x^6)^{2/3} (1+x^6) (-1-x^3+x^6)}{x^6 (-1+x^3+x^6)} \, dx\)

Optimal. Leaf size=105 \[ -\frac {2}{3} \log \left (\sqrt [3]{x^6-1}+x\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-1}-x}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-\sqrt [3]{x^6-1} x+\left (x^6-1\right )^{2/3}+x^2\right )+\frac {\left (x^6-1\right )^{2/3} \left (x^6-5 x^3-1\right )}{5 x^5} \]

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Rubi [C]  time = 1.19, antiderivative size = 380, normalized size of antiderivative = 3.62, number of steps used = 25, number of rules used = 12, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6728, 246, 245, 365, 364, 275, 1438, 430, 429, 465, 511, 510} \begin {gather*} -\frac {2 \left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 \left (1-x^6\right )^{2/3} x^5}-\frac {\left (x^6-1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{\left (1-x^6\right )^{2/3} x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(1 - Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, -2/3, 1, 7/6, x^6, (2*x^6)/(3 - Sqrt[5])])/((3 - Sqrt[5])*(
1 - x^6)^(2/3)) - (2*(1 + Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, 1, -2/3, 7/6, (2*x^6)/(3 + Sqrt[5]), x^6])
/((3 + Sqrt[5])*(1 - x^6)^(2/3)) + (x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 - Sqrt[5]
)])/((3 - Sqrt[5])*(1 - x^6)^(2/3)) + (x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 + Sqrt
[5])])/((3 + Sqrt[5])*(1 - x^6)^(2/3)) - ((-1 + x^6)^(2/3)*Hypergeometric2F1[-5/6, -2/3, 1/6, x^6])/(5*x^5*(1
- x^6)^(2/3)) - ((-1 + x^6)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, x^6])/(x^2*(1 - x^6)^(2/3)) + (x*(-1 + x^
6)^(2/3)*Hypergeometric2F1[-2/3, 1/6, 7/6, x^6])/(1 - x^6)^(2/3)

Rule 245

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 275

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

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 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(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] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !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 {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx &=\int \left (\left (-1+x^6\right )^{2/3}+\frac {\left (-1+x^6\right )^{2/3}}{x^6}+\frac {2 \left (-1+x^6\right )^{2/3}}{x^3}-\frac {2 \left (1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1+x^3+x^6}\right ) \, dx\\ &=2 \int \frac {\left (-1+x^6\right )^{2/3}}{x^3} \, dx-2 \int \frac {\left (1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1+x^3+x^6} \, dx+\int \left (-1+x^6\right )^{2/3} \, dx+\int \frac {\left (-1+x^6\right )^{2/3}}{x^6} \, dx\\ &=-\left (2 \int \left (\frac {2 \left (-1+x^6\right )^{2/3}}{1-\sqrt {5}+2 x^3}+\frac {2 \left (-1+x^6\right )^{2/3}}{1+\sqrt {5}+2 x^3}\right ) \, dx\right )+\frac {\left (-1+x^6\right )^{2/3} \int \left (1-x^6\right )^{2/3} \, dx}{\left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \int \frac {\left (1-x^6\right )^{2/3}}{x^6} \, dx}{\left (1-x^6\right )^{2/3}}+\operatorname {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \frac {\left (-1+x^6\right )^{2/3}}{1-\sqrt {5}+2 x^3} \, dx-4 \int \frac {\left (-1+x^6\right )^{2/3}}{1+\sqrt {5}+2 x^3} \, dx+\frac {\left (-1+x^6\right )^{2/3} \operatorname {Subst}\left (\int \frac {\left (1-x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}\\ &=-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \left (\frac {\left (1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6}\right ) \, dx-4 \int \left (\frac {\left (-1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6}\right ) \, dx\\ &=-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6} \, dx-4 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx+\left (2 \left (1-\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx-\left (2 \left (1+\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx\\ &=-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )-2 \operatorname {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )+\frac {\left (2 \left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}\\ &=-\frac {2 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}\\ &=-\frac {2 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {x^4 \left (-1+x^6\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {x^4 \left (-1+x^6\right )^{2/3} F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^6\right )}{\left (1-x^6\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.26, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{2/3} \left (-1-5 x^3+x^6\right )}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{3} \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)^(2/3)*(1 + x^6)*(-1 - x^3 + x^6))/(x^6*(-1 + x^3 + x^6)),x]

[Out]

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

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fricas [A]  time = 17.40, size = 142, normalized size = 1.35 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \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 ) + 5 \, x^{5} \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 ) - 3 \, {\left (x^{6} - 5 \, x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(10*sqrt(3)*x^5*arctan(1/3*(1770797931534669154710348707860106628265705908272667327884196338930088849705
9669011634*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 187790748244649020235189729458750340135644526059641258771848641124055
50428883609929964*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(8791266734992875261237504664599259772605087326251698970
792557525513888268399719816592*x^6 + 9326814489551980499445247598236243638058784087870425269964007887066219234
311690275757*x^3 - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592))/(99
23243904393545413458713816471868889492119646716071835561526356358143878603884871272*x^6 - 83202831655122513718
52516195766181258618636197629327742451887394495075584367754599527*x^3 - 99232439043935454134587138164718688894
92119646716071835561526356358143878603884871272)) + 5*x^5*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)) - 3*(x^6 - 5*x^3 - 1)*(x^6 - 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 28.88, size = 298, normalized size = 2.84

method	result	size
risch	\(\frac {x^{12}-5 x^{9}-2 x^{6}+5 x^{3}+1}{5 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}-\frac {2 \ln \left (-\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+x^{6}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}-1}\right )}{3}+2 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+x^{6}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-3 x \left (x^{6}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}-1}\right )\)	\(298\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (x^{6}-5 x^{3}-1\right )}{5 x^{5}}+\frac {2 \ln \left (\frac {241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{6}+14360235166545558502709989692576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{6}-209561394779200926893843501540 x^{6}-1900883571636169558491668255628288 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+25558529270134496713387197451392 \RootOf \left (9216 \textit {\_Z}^{2}-96 \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}-241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-14360235166545558502709989692576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+209561394779200926893843501540}{x^{6}+x^{3}-1}\right )}{3}-64 \ln \left (\frac {241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{6}+14360235166545558502709989692576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{6}-209561394779200926893843501540 x^{6}-1900883571636169558491668255628288 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+25558529270134496713387197451392 \RootOf \left (9216 \textit {\_Z}^{2}-96 \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}-241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-14360235166545558502709989692576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+209561394779200926893843501540}{x^{6}+x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+64 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (\frac {241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{6}-19389027684101562625703821056672 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{6}-33783984098747171016688237221 x^{6}-1900883571636169558491668255628288 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -11116972693065830085107918919360 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+14043211805619035755189224540864 \RootOf \left (9216 \textit {\_Z}^{2}-96 \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}-241382040842688197903703905476608 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+19389027684101562625703821056672 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+33783984098747171016688237221}{x^{6}+x^{3}-1}\right )\)	\(610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(x^12-5*x^9-2*x^6+5*x^3+1)/x^5/(x^6-1)^(1/3)-2/3*ln(-(3*RootOf(9*_Z^2-3*_Z+1)*x^6+x^6+9*RootOf(9*_Z^2-3*_Z
+1)^2*x^3+9*RootOf(9*_Z^2-3*_Z+1)*(x^6-1)^(2/3)*x+9*RootOf(9*_Z^2-3*_Z+1)*(x^6-1)^(1/3)*x^2-3*x^2*(x^6-1)^(1/3
)-x^3-3*RootOf(9*_Z^2-3*_Z+1)-1)/(x^6+x^3-1))+2*RootOf(9*_Z^2-3*_Z+1)*ln((3*RootOf(9*_Z^2-3*_Z+1)*x^6+x^6-9*Ro
otOf(9*_Z^2-3*_Z+1)^2*x^3+9*RootOf(9*_Z^2-3*_Z+1)*(x^6-1)^(2/3)*x-3*RootOf(9*_Z^2-3*_Z+1)*x^3-3*x*(x^6-1)^(2/3
)-3*x^2*(x^6-1)^(1/3)-3*RootOf(9*_Z^2-3*_Z+1)-1)/(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} - x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^6 - x^3 - 1)*(x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 + x^3 - 1)*x^6), 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 )}^{2/3}\,\left (x^6+1\right )\,\left (-x^6+x^3+1\right )}{x^6\,\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)^(2/3)*(x^6 + 1)*(x^3 - x^6 + 1))/(x^6*(x^3 + x^6 - 1)),x)

[Out]

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

[Out]

Timed out

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