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

Optimal. Leaf size=102 \[ \log \left (\sqrt [3]{x^6-1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^6-1} x+\left (x^6-1\right )^{2/3}+x^2\right )+\frac {\left (x^6-1\right )^{2/3} \left (4 x^6+15 x^3-4\right )}{10 x^5} \]

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Rubi [C]  time = 1.29, antiderivative size = 389, normalized size of antiderivative = 3.81, number of steps used = 25, number of rules used = 12, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6728, 246, 245, 365, 364, 275, 1438, 430, 429, 465, 511, 510} \begin {gather*} -\frac {3 \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 {3 \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 {3 \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 )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \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 )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {2 \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 {2 \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 {3 \left (x^6-1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 \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)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]

[Out]

(-3*(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)) - (3*(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)) - (3*x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 - Sqrt[
5])])/(2*(3 - Sqrt[5])*(1 - x^6)^(2/3)) - (3*x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3
+ Sqrt[5])])/(2*(3 + Sqrt[5])*(1 - x^6)^(2/3)) - (2*(-1 + x^6)^(2/3)*Hypergeometric2F1[-5/6, -2/3, 1/6, x^6])/
(5*x^5*(1 - x^6)^(2/3)) + (3*(-1 + x^6)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, x^6])/(2*x^2*(1 - x^6)^(2/3))
+ (2*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 (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx &=\int \left (2 \left (-1+x^6\right )^{2/3}+\frac {2 \left (-1+x^6\right )^{2/3}}{x^6}-\frac {3 \left (-1+x^6\right )^{2/3}}{x^3}+\frac {3 \left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx\\ &=2 \int \left (-1+x^6\right )^{2/3} \, dx+2 \int \frac {\left (-1+x^6\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (-1+x^6\right )^{2/3}}{x^3} \, dx+3 \int \frac {\left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6} \, dx\\ &=-\left (\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+3 \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+\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \int \left (1-x^6\right )^{2/3} \, dx}{\left (1-x^6\right )^{2/3}}+\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{x^6} \, dx}{\left (1-x^6\right )^{2/3}}\\ &=-\frac {2 \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 {2 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}}+6 \int \frac {\left (-1+x^6\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx+6 \int \frac {\left (-1+x^6\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx-\frac {\left (3 \left (-1+x^6\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )}{2 \left (1-x^6\right )^{2/3}}\\ &=-\frac {2 \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 {3 \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 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}}+6 \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+6 \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 {2 \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 {3 \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 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}}+6 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6} \, dx+6 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx+\left (3 \left (1-\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx-\left (3 \left (1+\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx\\ &=-\frac {2 \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 {3 \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 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}}+3 \operatorname {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )+3 \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 (3 \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 (3 \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 {3 \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 {3 \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 {2 \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 {3 \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 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 (3 \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 (3 \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 {3 \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 {3 \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 {3 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 )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 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 )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \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 {3 \left (-1+x^6\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 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 = 0.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 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)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.27, size = 102, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \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)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]

[Out]

((-1 + x^6)^(2/3)*(-4 + 15*x^3 + 4*x^6))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^6)^(1/3))] + Log
[-x + (-1 + x^6)^(1/3)] - Log[x^2 + x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/2

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fricas [A]  time = 13.27, size = 148, normalized size = 1.45 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\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 ) - {\left (4 \, x^{6} + 15 \, x^{3} - 4\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(10*sqrt(3)*x^5*arctan(1/3*(4739963886359486334524289176142989859968862245112601150366804535148881441482
50*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 19325031480489228255674265966448835967818926087643600184123099965366515892788
*sqrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(771225779807741020855977802972631216428368740202755221603971931588718036
144*x^6 + 245889484278411189833195613987401279765924206559249102388797804808538611984375*x^3 - 771225779807741
020855977802972631216428368740202755221603971931588718036144))/(1540751378553866520203301756955216463690689674
0149986002803824712402669144*x^6 - 227351086091515241263579358841494627179170556108548407412281480599473216796
875*x^3 - 15407513785538665202033017569552164636906896740149986002803824712402669144)) - 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)) - (4*x^6 + 15*x^3 - 4)*(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 (2 \, x^{6} + x^{3} - 2\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)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 21.09, size = 490, normalized size = 4.80

method	result	size
risch	\(\frac {4 x^{12}+15 x^{9}-8 x^{6}-15 x^{3}+4}{10 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}-3 \ln \left (\frac {6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-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 ) \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 \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}}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \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^{6}+x^{6}+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 ) \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}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right )-\ln \left (\frac {6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-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 ) \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 \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}}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right )\)	\(490\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (4 x^{6}+15 x^{3}-4\right )}{10 x^{5}}-96 \ln \left (-\frac {6570485565136040671726226079744 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{6}-489182598517982885200727925888 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{6}-4085365060307585836672246881 x^{6}-51742573825446320289844030377984 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+363650051031240685770824996928 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +654612709216604620411401672864 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-999622747774761167003559205344 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-6818882387672964795952100759 x \left (x^{6}-1\right )^{\frac {2}{3}}+10606903752581721939398194477 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-4604141575902199911170309977 x^{3}-6570485565136040671726226079744 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+489182598517982885200727925888 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+4085365060307585836672246881}{x^{6}-x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+\ln \left (\frac {1789441197416077361152076587008 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{6}+115627565201538784058859689760 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{6}-4604141575902199911170309977 x^{6}-14091849429651609219072603122688 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+363650051031240685770824996928 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -1018262760247845306182226669792 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+410835058262612379499203165024 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+10606903752581721939398194477 x \left (x^{6}-1\right )^{\frac {2}{3}}-6818882387672964795952100759 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-518776515594614074498063096 x^{3}-1789441197416077361152076587008 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-115627565201538784058859689760 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+4604141575902199911170309977}{x^{6}-x^{3}-1}\right )-\ln \left (-\frac {6570485565136040671726226079744 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{6}-489182598517982885200727925888 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{6}-4085365060307585836672246881 x^{6}-51742573825446320289844030377984 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+363650051031240685770824996928 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +654612709216604620411401672864 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-999622747774761167003559205344 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-6818882387672964795952100759 x \left (x^{6}-1\right )^{\frac {2}{3}}+10606903752581721939398194477 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-4604141575902199911170309977 x^{3}-6570485565136040671726226079744 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+489182598517982885200727925888 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+4085365060307585836672246881}{x^{6}-x^{3}-1}\right )\)	\(607\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(4*x^12+15*x^9-8*x^6-15*x^3+4)/x^5/(x^6-1)^(1/3)-3*ln((6*RootOf(9*_Z^2+3*_Z+1)*x^6+x^6-18*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*RootOf(9*_Z^2
+3*_Z+1)*x^3+3*x*(x^6-1)^(2/3)+3*x^2*(x^6-1)^(1/3)-6*RootOf(9*_Z^2+3*_Z+1)-1)/(x^6-x^3-1))*RootOf(9*_Z^2+3*_Z+
1)+3*RootOf(9*_Z^2+3*_Z+1)*ln(-(6*RootOf(9*_Z^2+3*_Z+1)*x^6+x^6+18*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+9*RootOf(9*_Z^2+3*_Z+1)*x^3+x^3-6*RootOf(9*
_Z^2+3*_Z+1)-1)/(x^6-x^3-1))-ln((6*RootOf(9*_Z^2+3*_Z+1)*x^6+x^6-18*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*RootOf(9*_Z^2+3*_Z+1)*x^3+3*x*(x^6-1)^(2
/3)+3*x^2*(x^6-1)^(1/3)-6*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 (2 \, x^{6} + x^{3} - 2\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)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Timed out

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