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

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

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Rubi [B]  time = 2.32, antiderivative size = 1568, normalized size of antiderivative = 6.10, number of steps used = 62, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 6725, 21, 47, 50, 59, 105, 91}

result too large to display

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)*(1 - x)*x)/(3*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(1 - x)*x)
/(3*(-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)) - ((-1 + x)^(1/3)*
x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(3*Sqrt[3]*(-x^2 + x^3)^(1/3)) + ((-1)^(1/3)
*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(3*Sqrt[3]*(-x^2 + x^3)^(1/3
)) - ((-1)^(2/3)*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(3*Sqrt[3]*(
-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(
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/Sq
rt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)*x^(1/3))])/(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/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 +
(-1)^(1/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(
-1 + x)^(1/3))/(Sqrt[3]*(1 - (-1)^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[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/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*x
^(1/3))])/(2*Sqrt[3]*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1 + x)^(1/3)/x^(1/3)])/(6*(-x^2
+ x^3)^(1/3)) + ((-1)^(1/3)*(-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1 + x)^(1/3)/x^(1/3)])/(6*(-x^2 + x^3)^(1/3)) -
((-1)^(2/3)*(-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1 + x)^(1/3)/x^(1/3)])/(6*(-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 + 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[x])/(18*(-x^2 + x^3)^(1/3)) + ((-1)^(1/3)*(-1 + x)^(1/3)*x^(2/3)*Log
[x])/(18*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3)*(-1 + x)^(1/3)*x^(2/3)*Log[x])/(18*(-x^2 + x^3)^(1/3)) - ((-1)^(2/3
)*(-1 + (-1)^(1/3))^(2/3)*(-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1)^(1/3)*x])/(12*(-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 + (-1)^(1/3)*x])/(12*(-x^2 + x^3)^(1/3)) - ((-1)^
(8/9)*(-1 + x)^(1/3)*x^(2/3)*Log[-1 - (-1)^(2/3)*x])/(12*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[1 -
(-1)^(2/3)*x])/(12*(1 - (-1)^(2/3))^(1/3)*(-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 47

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 50

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 59

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

Rule 91

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 2056

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 6725

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 [C]  time = 4.74, size = 928, normalized size = 3.61 \begin {gather*} \frac {x \left (\frac {2^{2/3} \left (6 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1}{\sqrt {3}}\right )+6 \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{1-i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \sqrt [6]{3} \sqrt [3]{3-i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{3-i \sqrt {3}}}+1}{\sqrt {3}}\right )+6 \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{1+i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \sqrt [6]{3} \sqrt [3]{3+i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{3+i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \log \left (1-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )-6 \sqrt [3]{1-i \sqrt {3}} \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )-6 \sqrt [3]{1+i \sqrt {3}} \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )+2\ 3^{2/3} \sqrt [3]{3-i \sqrt {3}} \log \left (\sqrt [3]{3-i \sqrt {3}}-\sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}\right )+2\ 3^{2/3} \sqrt [3]{3+i \sqrt {3}} \log \left (\sqrt [3]{3+i \sqrt {3}}-\sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}\right )+3 \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1\right )+3 \sqrt [3]{1-i \sqrt {3}} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2-2 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (1-i \sqrt {3}\right )^{2/3}\right )+3 \sqrt [3]{1+i \sqrt {3}} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2+2 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (1+i \sqrt {3}\right )^{2/3}\right )-3^{2/3} \sqrt [3]{3-i \sqrt {3}} \log \left (6^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{18-6 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (3-i \sqrt {3}\right )^{2/3}\right )-3^{2/3} \sqrt [3]{3+i \sqrt {3}} \log \left (6^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{18+6 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (3+i \sqrt {3}\right )^{2/3}\right )\right )}{\sqrt [3]{\frac {x}{x-1}}}-36\right )}{72 \sqrt [3]{(x-1) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-36 + (2^(2/3)*(6*Sqrt[3]*ArcTan[(1 + 2*2^(1/3)*(x/(-1 + x))^(1/3))/Sqrt[3]] + 6*Sqrt[3]*(1 - I*Sqrt[3])^(
1/3)*ArcTan[(1 + (2*2^(1/3)*(x/(-1 + x))^(1/3))/(1 - I*Sqrt[3])^(1/3))/Sqrt[3]] - 6*3^(1/6)*(3 - I*Sqrt[3])^(1
/3)*ArcTan[(1 + (2*6^(1/3)*(x/(-1 + x))^(1/3))/(3 - I*Sqrt[3])^(1/3))/Sqrt[3]] + 6*Sqrt[3]*(1 + I*Sqrt[3])^(1/
3)*ArcTan[(1 + (2*2^(1/3)*(x/(-1 + x))^(1/3))/(1 + I*Sqrt[3])^(1/3))/Sqrt[3]] - 6*3^(1/6)*(3 + I*Sqrt[3])^(1/3
)*ArcTan[(1 + (2*6^(1/3)*(x/(-1 + x))^(1/3))/(3 + I*Sqrt[3])^(1/3))/Sqrt[3]] - 6*Log[1 - 2^(1/3)*(x/(-1 + x))^
(1/3)] - 6*(1 - I*Sqrt[3])^(1/3)*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*(x/(-1 + x))^(1/3)] - 6*(1 + I*Sqrt[3])^(
1/3)*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*(x/(-1 + x))^(1/3)] + 2*3^(2/3)*(3 - I*Sqrt[3])^(1/3)*Log[(3 - I*Sqrt
[3])^(1/3) - 6^(1/3)*(x/(-1 + x))^(1/3)] + 2*3^(2/3)*(3 + I*Sqrt[3])^(1/3)*Log[(3 + I*Sqrt[3])^(1/3) - 6^(1/3)
*(x/(-1 + x))^(1/3)] + 3*Log[1 + 2^(1/3)*(x/(-1 + x))^(1/3) + 2^(2/3)*(x/(-1 + x))^(2/3)] + 3*(1 - I*Sqrt[3])^
(1/3)*Log[(1 - I*Sqrt[3])^(2/3) + (2 - (2*I)*Sqrt[3])^(1/3)*(x/(-1 + x))^(1/3) + 2^(2/3)*(x/(-1 + x))^(2/3)] +
3*(1 + I*Sqrt[3])^(1/3)*Log[(1 + I*Sqrt[3])^(2/3) + (2 + (2*I)*Sqrt[3])^(1/3)*(x/(-1 + x))^(1/3) + 2^(2/3)*(x
/(-1 + x))^(2/3)] - 3^(2/3)*(3 - I*Sqrt[3])^(1/3)*Log[(3 - I*Sqrt[3])^(2/3) + (18 - (6*I)*Sqrt[3])^(1/3)*(x/(-
1 + x))^(1/3) + 6^(2/3)*(x/(-1 + x))^(2/3)] - 3^(2/3)*(3 + I*Sqrt[3])^(1/3)*Log[(3 + I*Sqrt[3])^(2/3) + (18 +
(6*I)*Sqrt[3])^(1/3)*(x/(-1 + x))^(1/3) + 6^(2/3)*(x/(-1 + x))^(2/3)]))/(x/(-1 + x))^(1/3)))/(72*((-1 + x)*x^2
)^(1/3))

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IntegrateAlgebraic [A]  time = 0.61, size = 257, normalized size = 1.00 \begin {gather*} -\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.87, size = 2380, normalized size = 9.26

result too large to display

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*(sqrt(3)*(x^2 - x)*sin(2/9*pi) + (x^2 - x)*cos(2/9*pi))*log(64
*(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) + 6*(sqrt(3)*(x^2 - x)*sin(2/9*pi) - (x^2 - x)*cos(2/9*pi))*log(64*(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) - 36*(x^3 - x^2)^(2/3))/(x^2 - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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maple [B]  time = 141.39, size = 9712, normalized size = 37.79

method	result	size
trager	\(\text {Expression too large to display}\)	\(9712\)
risch	\(\text {Expression too large to display}\)	\(15043\)

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]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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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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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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