\(\int \frac {\sqrt [3]{-x+x^3} (8-10 x^2+x^4)}{x^4 (4-2 x^2+x^4)} \, dx\) [1779]

Optimal result

Integrand size = 37, antiderivative size = 120 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 725, normalized size of antiderivative = 6.04, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2081, 6860, 270, 477, 476, 486, 597, 12, 503} \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [6]{3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [6]{3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3 \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x}+\frac {3 \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x}-\frac {\left (\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x^3}-\frac {\left (-\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x^3}-\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3}-\frac {\left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2-i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2+i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {3^{2/3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3^{2/3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}} \]

[In]

Int[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]

[Out]

-1/4*((6*I - Sqrt[3])*(-x + x^3)^(1/3))/((I + Sqrt[3])*x^3) - ((6*I + Sqrt[3])*(-x + x^3)^(1/3))/(4*(I - Sqrt[
3])*x^3) + (3*(I - 2*Sqrt[3])*(-x + x^3)^(1/3))/(4*(I - Sqrt[3])*x) + (3*(I + 2*Sqrt[3])*(-x + x^3)^(1/3))/(4*
(I + Sqrt[3])*x) - (3*(1 - x^2)*(-x + x^3)^(1/3))/(8*x^3) - (3^(1/6)*(I - 2*Sqrt[3])*(-x + x^3)^(1/3)*ArcTan[(
1 + (2*3^(1/6)*x^(2/3))/((-I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)))/Sqrt[3]])/((-I + Sqrt[3])^(7/3)*x^(1/3)*(-1 +
x^2)^(1/3)) + (3^(1/6)*(I + 2*Sqrt[3])*(-x + x^3)^(1/3)*ArcTan[(1 + (2*3^(1/6)*x^(2/3))/((I + Sqrt[3])^(1/3)*
(-1 + x^2)^(1/3)))/Sqrt[3]])/((I + Sqrt[3])^(7/3)*x^(1/3)*(-1 + x^2)^(1/3)) - ((I + 2*Sqrt[3])*(-x + x^3)^(1/3
)*Log[1 - I*Sqrt[3] - x^2])/(2*3^(1/3)*(I + Sqrt[3])^(7/3)*x^(1/3)*(-1 + x^2)^(1/3)) + ((I - 2*Sqrt[3])*(-x +
x^3)^(1/3)*Log[1 + I*Sqrt[3] - x^2])/(2*3^(1/3)*(-I + Sqrt[3])^(7/3)*x^(1/3)*(-1 + x^2)^(1/3)) - (3^(2/3)*(I -
2*Sqrt[3])*(-x + x^3)^(1/3)*Log[3^(1/6)*x^(2/3) - (-I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)])/(2*(-I + Sqrt[3])^(
7/3)*x^(1/3)*(-1 + x^2)^(1/3)) + (3^(2/3)*(I + 2*Sqrt[3])*(-x + x^3)^(1/3)*Log[3^(1/6)*x^(2/3) - (I + Sqrt[3])
^(1/3)*(-1 + x^2)^(1/3)])/(2*(I + Sqrt[3])^(7/3)*x^(1/3)*(-1 + x^2)^(1/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

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

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 477

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

Rule 486

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

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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}& = \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2} \left (8-10 x^2+x^4\right )}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{-1+x^2}}{x^{11/3}}+\frac {4 \left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \frac {\left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \left (\frac {\left (-2+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )}+\frac {\left (-2-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2-2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2+2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\left (\left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {2 \left (3-i \sqrt {3}\right )-6 x^3}{x^2 \left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (\left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {2 \left (3+i \sqrt {3}\right )-6 x^3}{x^2 \left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (\left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {16 i \sqrt {3} x}{\left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{8 \left (1+i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (\left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int -\frac {16 i \sqrt {3} x}{\left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{8 \left (1-i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 i \sqrt {3} \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\left (1+i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (2 i \sqrt {3} \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\left (1-i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt [6]{3} \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{\left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [6]{3} \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{\left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (1-i \sqrt {3}-x^2\right )}{2 \sqrt [3]{3} \left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (1+i \sqrt {3}-x^2\right )}{2 \sqrt [3]{3} \left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {3^{2/3} \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{-1+x^2}\right )}{2 \left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {3^{2/3} \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{i+\sqrt {3}} \sqrt [3]{-1+x^2}\right )}{2 \left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (18 \left (1-4 x^2\right ) \sqrt [3]{-1+x^2}+x^{8/3} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-18 \log (x)+27 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{-1+x^2}} \]

[In]

Integrate[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]

[Out]

-1/24*((x*(-1 + x^2))^(1/3)*(18*(1 - 4*x^2)*(-1 + x^2)^(1/3) + x^(8/3)*RootSum[3 - 6*#1^3 + 4*#1^6 & , (-18*Lo
g[x] + 27*Log[(-1 + x^2)^(1/3) - x^(2/3)*#1] + 8*Log[x]*#1^3 - 12*Log[(-1 + x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-3
*#1^2 + 4*#1^5) & ]))/(x^3*(-1 + x^2)^(1/3))

Maple [N/A] (verified)

Time = 42.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {\left (24 x^{2}-6\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-6 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\left (4 \textit {\_R}^{3}-9\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-3\right )}\right ) x^{3}}{8 x^{3}}\)	\(83\)
trager	\(\text {Expression too large to display}\)	\(2117\)
risch	\(\text {Expression too large to display}\)	\(3327\)

[In]

int((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x,method=_RETURNVERBOSE)

[Out]

1/8*((24*x^2-6)*(x^3-x)^(1/3)+sum((4*_R^3-9)*ln((-_R*x+(x^3-x)^(1/3))/x)/_R^2/(4*_R^3-3),_R=RootOf(4*_Z^6-6*_Z
^3+3))*x^3)/x^3

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [N/A]

Not integrable

Time = 2.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \]

[In]

integrate((x**3-x)**(1/3)*(x**4-10*x**2+8)/x**4/(x**4-2*x**2+4),x)

[Out]

Integral((x*(x - 1)*(x + 1))**(1/3)*(x**4 - 10*x**2 + 8)/(x**4*(x**4 - 2*x**2 + 4)), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}} \,d x } \]

[In]

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="maxima")

[Out]

integrate((x^4 - 10*x^2 + 8)*(x^3 - x)^(1/3)/((x^4 - 2*x^2 + 4)*x^4), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Invalid _EXT in r
eplace_ext Error: Bad Argument Value(3072*(-(1/sageVARx)^2+1)^(1/3)*(-(1/sageVARx)^2+1)+9216*(-(1/sageVARx)^2+
1)^(1/3))/4096

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \]

[In]

int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)),x)

[Out]

int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)), x)