\(\int \frac {(-1+x^3) (1+x^3)^3 (1+x^6)^{2/3}}{x^6 (1-x^3+x^6)} \, dx\) [1449]

Optimal result

Integrand size = 37, antiderivative size = 102 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2+15 x^3+2 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )+\log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]

[Out]

1/10*(x^6+1)^(2/3)*(2*x^6+15*x^3+2)/x^5-arctan(3^(1/2)*x/(x+2*(x^6+1)^(1/3)))*3^(1/2)+ln(-x+(x^6+1)^(1/3))-1/2
*ln(x^2+x*(x^6+1)^(1/3)+(x^6+1)^(2/3))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.72 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.82, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {6860, 251, 371, 281, 1452, 440, 476, 524} \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {3 \left (-\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{\sqrt {3}+i}+\frac {3 \left (\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{-\sqrt {3}+i}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2} \]

[In]

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

[Out]

(3*(I - Sqrt[3])*x*AppellF1[1/6, 1, -2/3, 7/6, (-2*x^6)/(1 - I*Sqrt[3]), -x^6])/(I + Sqrt[3]) + (3*(I + Sqrt[3
])*x*AppellF1[1/6, 1, -2/3, 7/6, (-2*x^6)/(1 + I*Sqrt[3]), -x^6])/(I - Sqrt[3]) + (3*x^4*AppellF1[2/3, -2/3, 1
, 5/3, -x^6, (-2*x^6)/(1 - I*Sqrt[3])])/(2*(1 - I*Sqrt[3])) + (3*x^4*AppellF1[2/3, -2/3, 1, 5/3, -x^6, (-2*x^6
)/(1 + I*Sqrt[3])])/(2*(1 + I*Sqrt[3])) + Hypergeometric2F1[-5/6, -2/3, 1/6, -x^6]/(5*x^5) + (3*Hypergeometric
2F1[-2/3, -1/3, 2/3, -x^6])/(2*x^2) + x*Hypergeometric2F1[-2/3, 1/6, 7/6, -x^6]

Rule 251

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 281

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 440

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 476

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 1452

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 6860

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*} \text {integral}& = \int \left (\left (1+x^6\right )^{2/3}-\frac {\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 \\ & = -\left (3 \int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx\right )+3 \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 \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )-\frac {3}{2} \text {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )+3 \int \left (\frac {2 \left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^3}+\frac {2 \left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^3}\right ) \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+6 \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^3} \, dx+6 \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^3} \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+6 \int \left (\frac {\left (i-\sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{2 \left (i+\sqrt {3}+2 i x^6\right )}+\frac {x^3 \left (1+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x^6}\right ) \, dx+6 \int \left (\frac {\left (-i-\sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{2 \left (-i+\sqrt {3}-2 i x^6\right )}+\frac {x^3 \left (1+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x^6}\right ) \, dx \\ & = \frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+6 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x^6} \, dx+6 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x^6} \, dx+\left (3 \left (i-\sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{i+\sqrt {3}+2 i x^6} \, dx-\left (3 \left (i+\sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-i+\sqrt {3}-2 i x^6} \, dx \\ & = \frac {3 \left (i-\sqrt {3}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{i+\sqrt {3}}+\frac {3 \left (i+\sqrt {3}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{i-\sqrt {3}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+3 \text {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x^3} \, dx,x,x^2\right )+3 \text {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x^3} \, dx,x,x^2\right ) \\ & = \frac {3 \left (i-\sqrt {3}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{i+\sqrt {3}}+\frac {3 \left (i+\sqrt {3}\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{i-\sqrt {3}}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {3 x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2+15 x^3+2 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )+\log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]

[In]

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

[Out]

((1 + x^6)^(2/3)*(2 + 15*x^3 + 2*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

Maple [A] (verified)

Time = 4.96 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {2 x^{6} \left (x^{6}+1\right )^{\frac {2}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{6}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}-5 \ln \left (\frac {x^{2}+x \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{6}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+15 \left (x^{6}+1\right )^{\frac {2}{3}} x^{3}+2 \left (x^{6}+1\right )^{\frac {2}{3}}}{10 x^{5}}\)	\(119\)
trager	\(\frac {\left (x^{6}+1\right )^{\frac {2}{3}} \left (2 x^{6}+15 x^{3}+2\right )}{10 x^{5}}+\ln \left (-\frac {6624 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}-2763 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-1102 x^{6}-13248 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +6057 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9930 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3495 x \left (x^{6}+1\right )^{\frac {2}{3}}-1476 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-1653 x^{3}+6624 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-2763 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1102}{x^{6}-x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+7167 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+2208 x^{6}-9918 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x -10485 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-2763 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2019 x \left (x^{6}+1\right )^{\frac {2}{3}}-1476 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+736 x^{3}+4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7167 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2208}{x^{6}-x^{3}+1}\right )\)	\(416\)
risch	\(\frac {2 x^{12}+15 x^{9}+4 x^{6}+15 x^{3}+2}{10 x^{5} \left (x^{6}+1\right )^{\frac {1}{3}}}-3 \ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-3 \operatorname {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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right )-\ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +9 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-3 \operatorname {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 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right )\)	\(490\)

[In]

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

[Out]

1/10*(2*x^6*(x^6+1)^(2/3)+10*3^(1/2)*arctan(1/3*(x+2*(x^6+1)^(1/3))*3^(1/2)/x)*x^5-5*ln((x^2+x*(x^6+1)^(1/3)+(
x^6+1)^(2/3))/x^2)*x^5+10*ln((-x+(x^6+1)^(1/3))/x)*x^5+15*(x^6+1)^(2/3)*x^3+2*(x^6+1)^(2/3))/x^5

Fricas [A] (verification not implemented)

none

Time = 4.49 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1078 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (32 \, x^{6} + 605 \, x^{3} + 32\right )}}{8 \, x^{6} - 1331 \, x^{3} + 8}\right ) - 5 \, x^{5} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{3} + 1}\right ) - {\left (2 \, x^{6} + 15 \, x^{3} + 2\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]

[In]

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

[Out]

-1/10*(10*sqrt(3)*x^5*arctan((1078*sqrt(3)*(x^6 + 1)^(1/3)*x^2 + 196*sqrt(3)*(x^6 + 1)^(2/3)*x + sqrt(3)*(32*x
^6 + 605*x^3 + 32))/(8*x^6 - 1331*x^3 + 8)) - 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)) - (2*x^6 + 15*x^3 + 2)*(x^6 + 1)^(2/3))/x^5

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^3\,{\left (x^6+1\right )}^{2/3}}{x^6\,\left (x^6-x^3+1\right )} \,d x \]

[In]

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

[Out]

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