3.28.52 \(\int \frac {x^3}{\sqrt [3]{-x^2+x^3} (-1+x^6)} \, dx\) [2752]

Optimal. Leaf size=257 \[ -\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 2.12, antiderivative size = 1933, normalized size of antiderivative = 7.52, number of steps used = 42, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6857, 21, 49, 52, 61, 129, 490, 596, 544, 245, 384} \begin {gather*} -\frac {x^3}{2 \sqrt [3]{x^3-x^2}}-\frac {(1-x) x^2}{2 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (2+3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (2-3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{-1} \left (2-3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}-\frac {5 (1-x) x}{6 \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \text {ArcTan}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) x^{2/3}}{9 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\left (15 i+19 \sqrt {3}\right ) \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}+\frac {\left (3 i+13 \sqrt {3}\right ) \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}-\frac {\left (3 i-13 \sqrt {3}\right ) \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}+\frac {\left (15 i-19 \sqrt {3}\right ) \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}-\frac {4 \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{9 \sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {(-1)^{8/9} \sqrt [3]{x-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right ) x^{2/3}}{18 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (1+6 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{-1} \left (1-6 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (4+3 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{-1} \left (4-3 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {2 \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{9 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{2} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{8/9} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{8/9} \sqrt [3]{x-1} \log \left (\sqrt [3]{-1}-x\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \log \left (-x-(-1)^{2/3}\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \log (x) x^{2/3}}{54 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \log (x+1) x^{2/3}}{12 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \log \left (x+\sqrt [3]{-1}\right ) x^{2/3}}{12 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \log \left (x-(-1)^{2/3}\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*(1 - x)*x)/(6*(-x^2 + x^3)^(1/3)) + ((-1)^(1/3)*(2 - 3*(-1)^(1/3))*(1 - x)*x)/(18*(-x^2 + x^3)^(1/3)) - ((
-1)^(1/3)*(2 + 3*(-1)^(1/3))*(1 - x)*x)/(18*(-x^2 + x^3)^(1/3)) + ((-1)^(2/3)*(2 - 3*(-1)^(2/3))*(1 - x)*x)/(1
8*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(2 + 3*(-1)^(2/3))*(1 - x)*x)/(18*(-x^2 + x^3)^(1/3)) - ((1 - x)*x^2)/(2*(
-x^2 + x^3)^(1/3)) - x^3/(2*(-x^2 + x^3)^(1/3)) - (7*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/
3))/(Sqrt[3]*x^(1/3))])/(9*Sqrt[3]*(-x^2 + x^3)^(1/3)) - (4*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*x^(1/3))/(-1
+ x)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*(-x^2 + x^3)^(1/3)) + ((15*I - 19*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(1
+ (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(108*(-x^2 + x^3)^(1/3)) - ((3*I - 13*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*
ArcTan[(1 + (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(108*(-x^2 + x^3)^(1/3)) + ((3*I + 13*Sqrt[3])*(-1 + x)^(1/3
)*x^(2/3)*ArcTan[(1 + (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(108*(-x^2 + x^3)^(1/3)) - ((15*I + 19*Sqrt[3])*(-
1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(108*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1
/3)*x^(2/3)*ArcTan[(1 + (2*2^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(2*2^(1/3)*Sqrt[3]*(-x^2 + x^3)^(1/3)) -
((-1)^(2/3)*(-1 + (-1)^(1/3))^(2/3)*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*(1 - (-1)^(1/3))^(1/3)*x^(1/3))/(-1
+ x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-x^2 + x^3)^(1/3)) + ((-1)^(2/3)*(-(1 + (-1)^(1/3))^(-1))^(1/3)*(-1 + x)^(1
/3)*x^(2/3)*ArcTan[(1 + (2*(1 + (-1)^(1/3))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-x^2 + x^3)^(
1/3)) - ((-1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*(1 - (-1)^(2/3))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(2*Sq
rt[3]*(1 - (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1)^(8/9)*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(1 + (2*(1 + (-1)
^(2/3))^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-x^2 + x^3)^(1/3)) - (7*(-1 + x)^(1/3)*x^(2/3)*Lo
g[-1 + (-1 + x)^(1/3)/x^(1/3)])/(18*(-x^2 + x^3)^(1/3)) + (2*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - x^(1/
3)])/(9*(-x^2 + x^3)^(1/3)) - ((-1)^(1/3)*(4 - (3*I)*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - x^(1
/3)])/(36*(-x^2 + x^3)^(1/3)) + ((-1)^(2/3)*(4 + (3*I)*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - x^
(1/3)])/(36*(-x^2 + x^3)^(1/3)) + ((-1)^(1/3)*(1 - (6*I)*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) -
x^(1/3)])/(36*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(1 + (6*I)*Sqrt[3])*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)
- x^(1/3)])/(36*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - 2^(1/3)*x^(1/3)])/(4*2^(1/3
)*(-x^2 + x^3)^(1/3)) + ((-1)^(2/3)*(-1 + (-1)^(1/3))^(2/3)*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - (1 - (
-1)^(1/3))^(1/3)*x^(1/3)])/(4*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(-(1 + (-1)^(1/3))^(-1))^(1/3)*(-1 + x)^(1/3)*
x^(2/3)*Log[(-1 + x)^(1/3) - (1 + (-1)^(1/3))^(1/3)*x^(1/3)])/(4*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)
*Log[(-1 + x)^(1/3) - (1 - (-1)^(2/3))^(1/3)*x^(1/3)])/(4*(1 - (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1)^(
8/9)*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - (1 + (-1)^(2/3))^(1/3)*x^(1/3)])/(4*(-x^2 + x^3)^(1/3)) - ((-
1)^(8/9)*(-1 + x)^(1/3)*x^(2/3)*Log[(-1)^(1/3) - x])/(12*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(-1 + (-1)^(1/3))^(
2/3)*(-1 + x)^(1/3)*x^(2/3)*Log[-(-1)^(2/3) - x])/(12*(-x^2 + x^3)^(1/3)) - (7*(-1 + x)^(1/3)*x^(2/3)*Log[x])/
(54*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[1 + x])/(12*2^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3
)*x^(2/3)*Log[(-1)^(1/3) + x])/(12*(1 - (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1)^(2/3)*(-(1 + (-1)^(1/3))
^(-1))^(1/3)*(-1 + x)^(1/3)*x^(2/3)*Log[-(-1)^(2/3) + x])/(12*(-x^2 + x^3)^(1/3))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 61

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

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

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

Rule 490

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

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1+x^6\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {x^{7/3}}{2 \sqrt [3]{-1+x} \left (1-x^3\right )}-\frac {x^{7/3}}{2 \sqrt [3]{-1+x} \left (1+x^3\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1-x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1+x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {x^{7/3}}{3 (-1-x) \sqrt [3]{-1+x}}-\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )}-\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (\frac {x^{7/3}}{3 (1-x) \sqrt [3]{-1+x}}+\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )}+\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(-1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{(-1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(-1+x)^{4/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}\\ &=\frac {(1-x) x^2}{12 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{9 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(-1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-2 \frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+2 \frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {(1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {(1-x) x^2}{2 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{27 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{9 \sqrt [3]{-x^2+x^3}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}\right )\\ &=-\frac {5 (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(1-x) x^2}{2 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}+\frac {4 \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{9 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {2 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{9 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{12 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {2 \sqrt [3]{-1+x} x^{2/3} \log (x)}{27 \sqrt [3]{-x^2+x^3}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{27 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {5 (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(1-x) x^2}{2 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{3 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{12 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{18 \sqrt [3]{-x^2+x^3}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 281, normalized size = 1.09 \begin {gather*} \frac {x^{2/3} \left (-12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt [3]{-1+x} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+2^{2/3} \sqrt [3]{-1+x} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+4 \sqrt [3]{-1+x} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-4 \sqrt [3]{-1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{(-1+x) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(2/3)*(-12*x^(1/3) + 2*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^
(1/3))] - 2*2^(2/3)*(-1 + x)^(1/3)*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + 2^(2/3)*(-1 + x)^(1/3)*Log[2^(1/3
)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3)] + 4*(-1 + x)^(1/3)*RootSum[3 - 3*#1^3 + #1^6 &
, (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ] - 4*(-1 + x)^(1/3)*RootSum[1 - #1^3 + #1^6 & , (-L
og[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(24*((-1 + x)*x^2)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 13.62, size = 4486, normalized size = 17.46

method	result	size
trager	\(\text {Expression too large to display}\)	\(4486\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^3-x^2)^(2/3)/(-1+x)/x+81/64*RootOf(729*_Z^6+216*_Z^3+64)^5*ln((81*RootOf(729*_Z^6+216*_Z^3+64)^5*x^2-1
62*RootOf(729*_Z^6+216*_Z^3+64)^5*x+48*x^2*RootOf(729*_Z^6+216*_Z^3+64)^2-24*RootOf(729*_Z^6+216*_Z^3+64)^2*x-
48*RootOf(729*_Z^6+216*_Z^3+64)*(x^3-x^2)^(1/3)*x+32*(x^3-x^2)^(2/3))/(27*RootOf(729*_Z^6+216*_Z^3+64)^3*x-54*
RootOf(729*_Z^6+216*_Z^3+64)^3-8*x-8)/x)+3/8*RootOf(729*_Z^6+216*_Z^3+64)^2*ln((162*RootOf(729*_Z^6+216*_Z^3+6
4)^5*x^2-81*RootOf(729*_Z^6+216*_Z^3+64)^5*x-81*RootOf(729*_Z^6+216*_Z^3+64)^4*(x^3-x^2)^(1/3)*x+54*RootOf(729
*_Z^6+216*_Z^3+64)^3*(x^3-x^2)^(2/3)+24*x^2*RootOf(729*_Z^6+216*_Z^3+64)^2-12*RootOf(729*_Z^6+216*_Z^3+64)^2*x
-24*RootOf(729*_Z^6+216*_Z^3+64)*(x^3-x^2)^(1/3)*x+16*(x^3-x^2)^(2/3))/(27*RootOf(729*_Z^6+216*_Z^3+64)^3*x-54
*RootOf(729*_Z^6+216*_Z^3+64)^3-8*x-8)/x)+1/4*RootOf(729*_Z^6+216*_Z^3+64)*ln(-(243*RootOf(729*_Z^6+216*_Z^3+6
4)^4*(x^3-x^2)^(1/3)*x+216*x^2*RootOf(729*_Z^6+216*_Z^3+64)^3-108*RootOf(729*_Z^6+216*_Z^3+64)^3*x+108*(x^3-x^
2)^(2/3)*RootOf(729*_Z^6+216*_Z^3+64)^2+72*RootOf(729*_Z^6+216*_Z^3+64)*(x^3-x^2)^(1/3)*x+32*x^2-16*x)/(27*Roo
tOf(729*_Z^6+216*_Z^3+64)^3*x-54*RootOf(729*_Z^6+216*_Z^3+64)^3+16*x-8)/x)-27/32*ln(-(81*RootOf(729*_Z^6+216*_
Z^3+64)^5*x^2-162*RootOf(729*_Z^6+216*_Z^3+64)^5*x-108*(x^3-x^2)^(1/3)*RootOf(729*_Z^6+216*_Z^3+64)^3*x-24*x^2
*RootOf(729*_Z^6+216*_Z^3+64)^2-24*RootOf(729*_Z^6+216*_Z^3+64)^2*x-48*RootOf(729*_Z^6+216*_Z^3+64)*(x^3-x^2)^
(2/3)-32*x*(x^3-x^2)^(1/3))/(27*RootOf(729*_Z^6+216*_Z^3+64)^3*x-54*RootOf(729*_Z^6+216*_Z^3+64)^3+16*x-8)/x)*
RootOf(729*_Z^6+216*_Z^3+64)^4-1/4*ln(-(81*RootOf(729*_Z^6+216*_Z^3+64)^5*x^2-162*RootOf(729*_Z^6+216*_Z^3+64)
^5*x-108*(x^3-x^2)^(1/3)*RootOf(729*_Z^6+216*_Z^3+64)^3*x-24*x^2*RootOf(729*_Z^6+216*_Z^3+64)^2-24*RootOf(729*
_Z^6+216*_Z^3+64)^2*x-48*RootOf(729*_Z^6+216*_Z^3+64)*(x^3-x^2)^(2/3)-32*x*(x^3-x^2)^(1/3))/(27*RootOf(729*_Z^
6+216*_Z^3+64)^3*x-54*RootOf(729*_Z^6+216*_Z^3+64)^3+16*x-8)/x)*RootOf(729*_Z^6+216*_Z^3+64)-3/32*ln((2916*Roo
tOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x^2-5832*RootOf(_Z^3+243*RootOf
(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x+2943*(x^3-x^2)^(1/3)*RootOf(_Z^3+243*RootOf(729*
_Z^6+216*_Z^3+64)^3-72)^2*RootOf(729*_Z^6+216*_Z^3+64)^3*x+12015*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^
3-72)*RootOf(729*_Z^6+216*_Z^3+64)^3*x^2-7614*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z
^6+216*_Z^3+64)^3*x+31590*RootOf(729*_Z^6+216*_Z^3+64)^3*(x^3-x^2)^(2/3)+964*(x^3-x^2)^(1/3)*RootOf(_Z^3+243*R
ootOf(729*_Z^6+216*_Z^3+64)^3-72)^2*x+5856*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*x^2-2440*RootOf(
_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*x+22032*(x^3-x^2)^(2/3))/(27*RootOf(729*_Z^6+216*_Z^3+64)^3*x-54*R
ootOf(729*_Z^6+216*_Z^3+64)^3+24*x+8)/x)*RootOf(729*_Z^6+216*_Z^3+64)^3*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z
^3+64)^3-72)-1/36*ln((2916*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x
^2-5832*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x+2943*(x^3-x^2)^(1/
3)*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)^2*RootOf(729*_Z^6+216*_Z^3+64)^3*x+12015*RootOf(_Z^3+243
*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^3*x^2-7614*RootOf(_Z^3+243*RootOf(729*_Z^6+21
6*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^3*x+31590*RootOf(729*_Z^6+216*_Z^3+64)^3*(x^3-x^2)^(2/3)+964*(x^
3-x^2)^(1/3)*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)^2*x+5856*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_
Z^3+64)^3-72)*x^2-2440*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*x+22032*(x^3-x^2)^(2/3))/(27*RootOf(
729*_Z^6+216*_Z^3+64)^3*x-54*RootOf(729*_Z^6+216*_Z^3+64)^3+24*x+8)/x)*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^
3+64)^3-72)+3/32*RootOf(729*_Z^6+216*_Z^3+64)^3*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*ln(-(-29889
*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x^2+59778*RootOf(_Z^3+243*R
ootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^6*x+6372*(x^3-x^2)^(1/3)*RootOf(_Z^3+243*RootOf
(729*_Z^6+216*_Z^3+64)^3-72)^2*RootOf(729*_Z^6+216*_Z^3+64)^3*x+16632*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3
+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^3*x^2+24300*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(
729*_Z^6+216*_Z^3+64)^3*x+36936*RootOf(729*_Z^6+216*_Z^3+64)^3*(x^3-x^2)^(2/3)+2864*(x^3-x^2)^(1/3)*RootOf(_Z^
3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)^2*x+8832*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*x^2+1472*
RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*x+57024*(x^3-x^2)^(2/3))/(27*RootOf(729*_Z^6+216*_Z^3+64)^3
*x-54*RootOf(729*_Z^6+216*_Z^3+64)^3+24*x+8)/x)-27/128*RootOf(729*_Z^6+216*_Z^3+64)^5*RootOf(_Z^3+243*RootOf(7
29*_Z^6+216*_Z^3+64)^3-72)*ln((-10935*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_
Z^3+64)^8*x^2+21870*RootOf(_Z^3+243*RootOf(729*_Z^6+216*_Z^3+64)^3-72)*RootOf(729*_Z^6+216*_Z^3+64)^8*x+6588*(
x^3-x^2)^(1/3)*RootOf(_Z^3+243*RootOf(729*_Z^6+...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.61, size = 2380, normalized size = 9.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(2*12^(1/6)*6^(2/3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2))*log(12*(4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)
^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*c
os(2/3*arctan(sqrt(3) - 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)
^(2/3))/x^2) - 8*12^(1/6)*6^(2/3)*(x^2 - x)*arctan(1/108*(12*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3)*cos(2/
3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3) - sqrt(3)*(2*12^(2/3)*6^(2/3)*sqrt(3)*
x*cos(2/3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) -
2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3)
- 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 +
12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)^(2/3))/x^2) + 36*(48*x*cos(2/3*
arctan(sqrt(3) - 2))^3 - (12^(2/3)*6^(2/3)*(x^3 - x^2)^(1/3) + 24*x)*cos(2/3*arctan(sqrt(3) - 2)))*sin(2/3*arc
tan(sqrt(3) - 2)) - 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) - 2))^4 - 16*x*cos(2/3*arctan(sqrt(3) - 2))^2
+ x))*sin(2/3*arctan(sqrt(3) - 2)) - 6*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^2 - x)*arctan(-1/6*2^(1/6)*(sqrt(6)*2^(1/
3)*x - 2*sqrt(6)*(-1)^(1/3)*(x^3 - x^2)^(1/3))/x) + 6*2^(2/3)*(-1)^(1/3)*(x^2 - x)*log(-(2^(1/3)*(-1)^(2/3)*x
- (x^3 - x^2)^(1/3))/x) - 3*2^(2/3)*(-1)^(1/3)*(x^2 - x)*log(-(2^(2/3)*(-1)^(1/3)*x^2 - 2^(1/3)*(-1)^(2/3)*(x^
3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) + 12*(x^2 - x)*cos(2/9*pi)*log(16*(x^2 + (2*sqrt(3)*x*cos(2/9*pi)*s
in(2/9*pi) - 2*x*cos(2/9*pi)^2 + x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 48*(x^2 - x)*arctan((8*(2*x*
cos(2/9*pi)^3 - x*cos(2/9*pi))*sin(2/9*pi) - sqrt(3)*x - 2*(2*sqrt(3)*x*cos(2/9*pi)^2 - 2*x*cos(2/9*pi)*sin(2/
9*pi) - sqrt(3)*x)*sqrt((x^2 + (2*sqrt(3)*x*cos(2/9*pi)*sin(2/9*pi) - 2*x*cos(2/9*pi)^2 + x)*(x^3 - x^2)^(1/3)
+ (x^3 - x^2)^(2/3))/x^2) + 2*(x^3 - x^2)^(1/3)*(2*sqrt(3)*cos(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi) - sqrt(3
)))/(16*x*cos(2/9*pi)^4 - 16*x*cos(2/9*pi)^2 + 3*x))*sin(2/9*pi) - 4*(12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*cos(2
/3*arctan(sqrt(3) - 2)) + 12^(1/6)*6^(2/3)*(x^2 - x)*sin(2/3*arctan(sqrt(3) - 2)))*arctan(-1/108*(12*12^(2/3)*
6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3)*cos(2/3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3
) - sqrt(3)*(2*12^(2/3)*6^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan
(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^
3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(
1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^
3 - x^2)^(2/3))/x^2) + 36*(48*x*cos(2/3*arctan(sqrt(3) - 2))^3 + (12^(2/3)*6^(2/3)*(x^3 - x^2)^(1/3) - 24*x)*c
os(2/3*arctan(sqrt(3) - 2)))*sin(2/3*arctan(sqrt(3) - 2)) + 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) - 2))^
4 - 16*x*cos(2/3*arctan(sqrt(3) - 2))^2 + x)) - 4*(12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) -
2)) - 12^(1/6)*6^(2/3)*(x^2 - x)*sin(2/3*arctan(sqrt(3) - 2)))*arctan(1/72*(144*x*cos(2/3*arctan(sqrt(3) - 2)
)*sin(2/3*arctan(sqrt(3) - 2)) + 12^(2/3)*6^(2/3)*x*sqrt(-(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(
2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 - x^2)^(2/3))/x^2) - 2*
12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3))/(2*x*cos(2/3*arctan(sqrt(3) - 2))^2 - x)) - 24*(sqrt(3)*(x^2 - x)*
cos(2/9*pi) - (x^2 - x)*sin(2/9*pi))*arctan((8*(2*x*cos(2/9*pi)^3 - x*cos(2/9*pi))*sin(2/9*pi) + sqrt(3)*x + 2
*(2*sqrt(3)*x*cos(2/9*pi)^2 + 2*x*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)*x)*sqrt((x^2 - (2*sqrt(3)*x*cos(2/9*pi)*si
n(2/9*pi) + 2*x*cos(2/9*pi)^2 - x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 2*(x^3 - x^2)^(1/3)*(2*sqrt(3
)*cos(2/9*pi)^2 + 2*cos(2/9*pi)*sin(2/9*pi) - sqrt(3)))/(16*x*cos(2/9*pi)^4 - 16*x*cos(2/9*pi)^2 + 3*x)) + 24*
(sqrt(3)*(x^2 - x)*cos(2/9*pi) + (x^2 - x)*sin(2/9*pi))*arctan(-1/2*(2*x*cos(2/9*pi)^2 - x*sqrt((x^2 + 2*(x^3
- x^2)^(1/3)*(2*x*cos(2/9*pi)^2 - x) + (x^3 - x^2)^(2/3))/x^2) - x + (x^3 - x^2)^(1/3))/(x*cos(2/9*pi)*sin(2/9
*pi))) - (12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*sin(2/3*arctan(sqrt(3) - 2)) + 12^(1/6)*6^(2/3)*(x^2 - x)*cos(2/3
*arctan(sqrt(3) - 2)))*log(-48*(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*si
n(2/3*arctan(sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 - x^2)^(2/3))/x^2) + (12^(1/6)*6^(2/3)*sqrt(3)*(x^
2 - x)*sin(2/3*arctan(sqrt(3) - 2)) - 12^(1/6)*6^(2/3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2)))*log(48*(4*12^(1
/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 12*12^(1/3
)*6^(1/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3
- x^2)^(1/3)*x + 12*(x^3 - x^2)^(2/3))/x^2) - 6...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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