\(\int \frac {(-2+x^6) \sqrt [3]{2+x^6}}{x^2 (2+2 x^3+x^6)} \, dx\) [1893]

Optimal result

Integrand size = 30, antiderivative size = 131 \[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx=\frac {\sqrt [3]{2+x^6}}{x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{2+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{2+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{2+x^6}-\sqrt [3]{2} \left (2+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.74 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.49, number of steps used = 29, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 371, 1576, 476, 440, 524} \[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx=\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x} \]

[In]

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

[Out]

((1/2 - I/2)*x^2*AppellF1[1/3, 1, -1/3, 4/3, (-1/2*I)*x^6, -1/2*x^6])/2^(2/3) + ((1/2 + I/2)*x^2*AppellF1[1/3,
1, -1/3, 4/3, (I/2)*x^6, -1/2*x^6])/2^(2/3) + ((I/5)*x^5*AppellF1[5/6, 1, -1/3, 11/6, (-1/2*I)*x^6, -1/2*x^6]
)/2^(2/3) - ((I/5)*x^5*AppellF1[5/6, 1, -1/3, 11/6, (I/2)*x^6, -1/2*x^6])/2^(2/3) + (2^(1/3)*Hypergeometric2F1
[-1/3, -1/6, 5/6, -1/2*x^6])/x

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 1576

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 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 (-\frac {\sqrt [3]{2+x^6}}{x^2}+\frac {2 x \left (1+x^3\right ) \sqrt [3]{2+x^6}}{2+2 x^3+x^6}\right ) \, dx \\ & = 2 \int \frac {x \left (1+x^3\right ) \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx-\int \frac {\sqrt [3]{2+x^6}}{x^2} \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+2 \int \left (\frac {x \sqrt [3]{2+x^6}}{2+2 x^3+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2+2 x^3+x^6}\right ) \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+2 \int \frac {x \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx+2 \int \frac {x^4 \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+2 \int \left (\frac {i x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3}+\frac {i x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3}\right ) \, dx+2 \int \left (-\frac {(1+i) x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3}+\frac {(1-i) x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3}\right ) \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+(-2-2 i) \int \frac {x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3} \, dx+2 i \int \frac {x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3} \, dx+2 i \int \frac {x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3} \, dx+(2-2 i) \int \frac {x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3} \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+(-2-2 i) \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{2 i+x^6}-\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (2 i+x^6\right )}\right ) \, dx+2 i \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{-2 i+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (-2 i+x^6\right )}\right ) \, dx+2 i \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{2 i+x^6}-\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (2 i+x^6\right )}\right ) \, dx+(2-2 i) \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{-2 i+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (-2 i+x^6\right )}\right ) \, dx \\ & = \frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+i \int \frac {x^4 \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx-i \int \frac {x^4 \sqrt [3]{2+x^6}}{2 i+x^6} \, dx+(1-i) \int \frac {x \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx+(1-i) \int \frac {x^4 \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx+(1+i) \int \frac {x \sqrt [3]{2+x^6}}{2 i+x^6} \, dx+(1+i) \int \frac {x^4 \sqrt [3]{2+x^6}}{2 i+x^6} \, dx-2 \int \frac {x \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx-2 \int \frac {x \sqrt [3]{2+x^6}}{2 i+x^6} \, dx \\ & = \frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x}+\left (\frac {1}{2}-\frac {i}{2}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{-2 i+x^3} \, dx,x,x^2\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{2 i+x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{-2 i+x^3} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{2 i+x^3} \, dx,x,x^2\right ) \\ & = \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x^2 \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{3},\frac {4}{3},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 \operatorname {AppellF1}\left (\frac {5}{6},1,-\frac {1}{3},\frac {11}{6},\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\sqrt [3]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-\frac {x^6}{2}\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx=\frac {\sqrt [3]{2+x^6}}{x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{2+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{2+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{2+x^6}-\sqrt [3]{2} \left (2+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 101.58 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (x^{6}+2\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{6}+2\right )^{\frac {1}{3}}}{x}\right ) x +2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{6}+2\right )^{\frac {1}{3}} x +\left (x^{6}+2\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{6}+2\right )^{\frac {1}{3}}}{6 x}\)	\(110\)
risch	\(\text {Expression too large to display}\)	\(1432\)
trager	\(\text {Expression too large to display}\)	\(1518\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (100) = 200\).

Time = 116.63 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.55 \[ \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx=\frac {2 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} \left (-2\right )^{\frac {2}{3}} {\left (x^{14} - 14 \, x^{11} + 8 \, x^{8} - 28 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 2\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} {\left (x^{13} - 2 \, x^{10} - 4 \, x^{7} - 4 \, x^{4} + 4 \, x\right )} {\left (x^{6} + 2\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 54 \, x^{12} - 112 \, x^{9} + 108 \, x^{6} - 120 \, x^{3} + 8\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 90 \, x^{12} + 32 \, x^{9} - 180 \, x^{6} + 24 \, x^{3} + 8\right )}}\right ) + 2 \, \left (-2\right )^{\frac {1}{3}} x \log \left (\frac {6 \, \left (-2\right )^{\frac {1}{3}} {\left (x^{6} + 2\right )}^{\frac {1}{3}} x^{2} - \left (-2\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 2\right )} - 6 \, {\left (x^{6} + 2\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x^{3} + 2}\right ) - \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {3 \, \left (-2\right )^{\frac {2}{3}} {\left (x^{7} - 4 \, x^{4} + 2 \, x\right )} {\left (x^{6} + 2\right )}^{\frac {2}{3}} + \left (-2\right )^{\frac {1}{3}} {\left (x^{12} - 14 \, x^{9} + 8 \, x^{6} - 28 \, x^{3} + 4\right )} - 12 \, {\left (x^{8} - x^{5} + 2 \, x^{2}\right )} {\left (x^{6} + 2\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 8 \, x^{6} + 8 \, x^{3} + 4}\right ) + 18 \, {\left (x^{6} + 2\right )}^{\frac {1}{3}}}{18 \, x} \]

[In]

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

[Out]

1/18*(2*sqrt(3)*(-2)^(1/3)*x*arctan(1/3*(6*sqrt(3)*(-2)^(2/3)*(x^14 - 14*x^11 + 8*x^8 - 28*x^5 + 4*x^2)*(x^6 +
2)^(1/3) + 6*sqrt(3)*(-2)^(1/3)*(x^13 - 2*x^10 - 4*x^7 - 4*x^4 + 4*x)*(x^6 + 2)^(2/3) + sqrt(3)*(x^18 - 30*x^
15 + 54*x^12 - 112*x^9 + 108*x^6 - 120*x^3 + 8))/(x^18 + 6*x^15 - 90*x^12 + 32*x^9 - 180*x^6 + 24*x^3 + 8)) +
2*(-2)^(1/3)*x*log((6*(-2)^(1/3)*(x^6 + 2)^(1/3)*x^2 - (-2)^(2/3)*(x^6 + 2*x^3 + 2) - 6*(x^6 + 2)^(2/3)*x)/(x^
6 + 2*x^3 + 2)) - (-2)^(1/3)*x*log(-(3*(-2)^(2/3)*(x^7 - 4*x^4 + 2*x)*(x^6 + 2)^(2/3) + (-2)^(1/3)*(x^12 - 14*
x^9 + 8*x^6 - 28*x^3 + 4) - 12*(x^8 - x^5 + 2*x^2)*(x^6 + 2)^(1/3))/(x^12 + 4*x^9 + 8*x^6 + 8*x^3 + 4)) + 18*(
x^6 + 2)^(1/3))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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