\(\int \frac {1}{x^3 (1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [2522]

Optimal result

Integrand size = 24, antiderivative size = 211 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.64, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}+\frac {1-x}{20 x \sqrt [3]{x^3-x^2}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{x^3-x^2}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {17 (1-x)}{40 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{2} \sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{-1}-x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (-x-(-1)^{2/3}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{6 \sqrt [3]{2} \sqrt [3]{x^3-x^2}} \]

[In]

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

[Out]

(-17*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - ((5 - (16*I)*Sqrt[3])*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - ((5 + (16*I)*
Sqrt[3])*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - (3*(1 - x))/(8*x^2*(-x^2 + x^3)^(1/3)) + (1 - x)/(20*x*(-x^2 + x^3
)^(1/3)) - ((5 - (2*I)*Sqrt[3])*(1 - x))/(20*x*(-x^2 + x^3)^(1/3)) - ((5 + (2*I)*Sqrt[3])*(1 - x))/(20*x*(-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^(1/3)*Sq
rt[3]*(-x^2 + x^3)^(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]])/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)*(-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]])/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) +
((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - 2^(1/3)*x^(1/3)])/(2*2^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3
)*x^(2/3)*Log[(-1 + x)^(1/3) - (1 - (-1)^(1/3))^(1/3)*x^(1/3)])/(2*(1 - (-1)^(1/3))^(1/3)*(-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)])/(2*(1 + (-1)^(2/3))^(1/3)*(-x^
2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[(-1)^(1/3) - x])/(6*(1 + (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) -
((-1 + x)^(1/3)*x^(2/3)*Log[-(-1)^(2/3) - x])/(6*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)
*x^(2/3)*Log[1 + x])/(6*2^(1/3)*(-x^2 + x^3)^(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 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 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 491

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 + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

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 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*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (1+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt [3]{-1+x} x^{11/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\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^{11/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2+6 x^3}{x^6 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {2 \left (5+2 i \sqrt {3}\right )-6 \sqrt [3]{-1} x^3}{x^6 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {2 \left (5-2 i \sqrt {3}\right )+6 (-1)^{2/3} x^3}{x^6 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-34+6 x^3}{x^3 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (5+16 i \sqrt {3}\right )+6 \sqrt [3]{-1} \left (5+2 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (5-16 i \sqrt {3}\right )-3 \left (1+7 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{2} \sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1}-x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-(-1)^{2/3}-x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (1+x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {-45-9 x-27 x^2+81 x^3-20\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{8/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+20\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )-10\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+40 \sqrt [3]{-1+x} x^{8/3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 x^2 \sqrt [3]{(-1+x) x^2}} \]

[In]

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

[Out]

(-45 - 9*x - 27*x^2 + 81*x^3 - 20*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*x^(8/3)*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1
+ x)^(1/3) + x^(1/3))] + 20*2^(2/3)*(-1 + x)^(1/3)*x^(8/3)*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] - 10*2^(2/
3)*(-1 + x)^(1/3)*x^(8/3)*Log[2^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3)] + 40*(-1 +
x)^(1/3)*x^(8/3)*RootSum[1 - #1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*x
^2*((-1 + x)*x^2)^(1/3))

Maple [N/A] (verified)

Time = 14.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-10 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}+40 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+20 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{4}+20 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) x^{4}+81 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}} \left (x^{2}+\frac {2}{3} x +\frac {5}{9}\right )}{120 x^{4}}\)	\(176\)
trager	\(\text {Expression too large to display}\)	\(1758\)
risch	\(\text {Expression too large to display}\)	\(4518\)

[In]

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

[Out]

1/120*(-10*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+((-1+x)*x^2)^(2/3))/x^2)*x^4+40*sum(ln((-_R*x+
((-1+x)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))*x^4+20*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x
^2)^(1/3)+x)/x)*x^4+20*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)*x^4+81*((-1+x)*x^2)^(2/3)*(x^2+2/3*x+5/9)
)/x^4

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.28 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {20 \cdot 2^{\frac {2}{3}} x^{4} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 20 \cdot 2^{\frac {2}{3}} x^{4} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 20 \, \sqrt {6} 2^{\frac {1}{6}} x^{4} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 20 \cdot 2^{\frac {2}{3}} x^{4} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 10 \cdot 2^{\frac {2}{3}} x^{4} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 9 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (9 \, x^{2} + 6 \, x + 5\right )}}{120 \, x^{4}} \]

[In]

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

[Out]

1/120*(20*2^(2/3)*x^4*(I*sqrt(3) + 1)^(1/3)*log(((I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(I*sqrt(3) + 1)^(2/3) + 4*(
x^3 - x^2)^(1/3))/x) + 20*2^(2/3)*x^4*(-I*sqrt(3) + 1)^(1/3)*log(((-I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(-I*sqrt(
3) + 1)^(2/3) + 4*(x^3 - x^2)^(1/3))/x) + 20*sqrt(6)*2^(1/6)*x^4*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqr
t(6)*(x^3 - x^2)^(1/3))/x) + 20*2^(2/3)*x^4*log(-(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - 10*2^(2/3)*x^4*log((2^(2
/3)*x^2 + 2^(1/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 10*2^(2/3)*(sqrt(-3)*x^4 + x^4)*(I*sqrt(3) +
1)^(1/3)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x - I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(I*sqrt(3) + 1)^(2/3) + 8*(x^
3 - x^2)^(1/3))/x) + 10*2^(2/3)*(sqrt(-3)*x^4 - x^4)*(I*sqrt(3) + 1)^(1/3)*log(((sqrt(3)*2^(1/3)*(-I*sqrt(-3)*
x - I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(I*sqrt(3) + 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) + 10*2^(2/3)*(sqrt(-3)*x^
4 - x^4)*(-I*sqrt(3) + 1)^(1/3)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x + I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(-I*sqr
t(3) + 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) - 10*2^(2/3)*(sqrt(-3)*x^4 + x^4)*(-I*sqrt(3) + 1)^(1/3)*log(((sqrt(
3)*2^(1/3)*(-I*sqrt(-3)*x + I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(-I*sqrt(3) + 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x)
+ 9*(x^3 - x^2)^(2/3)*(9*x^2 + 6*x + 5))/x^4

Sympy [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 38.66 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.77 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/8*(1/x - 1)^2*(-1/x + 1)^(2/3) + 1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3
))) - 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^
4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqr
t(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x + 1)^(1/3))
/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 +
5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*p
i)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)
*cos(2/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt
(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1
/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9
*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1
/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos
(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)
+ cos(4/9*pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3
) + 1) - 1/6*(5*sqrt(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*p
i)^5 + cos(2/9*pi)^5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*si
n(2/9*pi) + cos(2/9*pi)^2 - sin(2/9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x + 1)^(1/3) + (-1/x
+ 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)
*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(
1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x + 1)^(1/3
) + (-1/x + 1)^(2/3) + 1) - 6/5*(-1/x + 1)^(5/3) - 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x + 1)^(1/3) + (-1/x
+ 1)^(2/3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (-1/x + 1)^(1/3))) + 3/2*(-1/x + 1)^(2/3)

Mupad [N/A]

Not integrable

Time = 7.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^3\,\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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