3.31.48 \(\int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx\)

Optimal. Leaf size=459 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3+7\& ,\frac {\text {$\#$1}^3 \log \left (x^3+1\right )-\text {$\#$1}^3 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )-5 \log \left (-\text {$\#$1} x^3-\text {$\#$1}+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}\right )+5 \log \left (x^3+1\right )}{2 \text {$\#$1}^4+\text {$\#$1}}\& \right ]-\frac {\log \left (x^3+1\right )}{3\ 2^{2/3}}+\frac {\log \left (x^6+2 x^3+1\right )}{6\ 2^{2/3}}+\frac {\log \left (-2 x^3+\sqrt [3]{2} \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}-2\right )}{3\ 2^{2/3}}-\frac {\log \left (4 x^6+8 x^3+2^{2/3} \left (-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1\right )^{2/3}+\left (2 \sqrt [3]{2} x^3+2 \sqrt [3]{2}\right ) \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}+4\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {\left (\sqrt {3} x+\sqrt {3}\right ) \left (x^2-x+1\right )}{x^3+\sqrt [3]{2} \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}+1}\right )}{2^{2/3} \sqrt {3}} \]

________________________________________________________________________________________

Rubi [C]  time = 1.05, antiderivative size = 1245, normalized size of antiderivative = 2.71, number of steps used = 31, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {6688, 6719, 2074, 56, 617, 204, 31, 843, 711, 50, 55} \begin {gather*} -\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1-3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1-3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1+3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{3 x-1} \left (x^3+1\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{3 x-1}}{\sqrt [3]{1+3 i \sqrt {3}}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \log (x+1)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{3 x-1}+2^{2/3}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}}-\frac {5 i \sqrt [3]{3 x-1} \left (x^3+1\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{3 x-1}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (x^3+1\right )^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(-1/3),x]

[Out]

-(((-1 + 3*x)^(1/3)*(1 + x^3)*ArcTan[(1 - 2^(1/3)*(-1 + 3*x)^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*((1 - 3*x)*(1 +
x^3)^3)^(1/3))) + (((5*I)/9)*2^(1/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*ArcTan[(1 + (2*2^(1/3)*(-1 + 3*x)^(1/3))/(1 -
(3*I)*Sqrt[3])^(1/3))/Sqrt[3]])/((1 - (3*I)*Sqrt[3])^(1/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - ((I/9)*(1 - (3*I)
*Sqrt[3])^(2/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*ArcTan[(1 + (2*2^(1/3)*(-1 + 3*x)^(1/3))/(1 - (3*I)*Sqrt[3])^(1/3))
/Sqrt[3]])/(2^(2/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - (((5*I)/9)*2^(1/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*ArcTan[(1 +
(2*2^(1/3)*(-1 + 3*x)^(1/3))/(1 + (3*I)*Sqrt[3])^(1/3))/Sqrt[3]])/((1 + (3*I)*Sqrt[3])^(1/3)*((1 - 3*x)*(1 +
x^3)^3)^(1/3)) + ((I/9)*(1 + (3*I)*Sqrt[3])^(2/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*ArcTan[(1 + (2*2^(1/3)*(-1 + 3*x)
^(1/3))/(1 + (3*I)*Sqrt[3])^(1/3))/Sqrt[3]])/(2^(2/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - (((5*I)/9)*(-1 + 3*x)^(
1/3)*(1 + x^3)*Log[1 - I*Sqrt[3] - 2*x])/(2^(2/3)*Sqrt[3]*(1 - (3*I)*Sqrt[3])^(1/3)*((1 - 3*x)*(1 + x^3)^3)^(1
/3)) + ((I/18)*(1 - (3*I)*Sqrt[3])^(2/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*Log[1 - I*Sqrt[3] - 2*x])/(2^(2/3)*Sqrt[3]
*((1 - 3*x)*(1 + x^3)^3)^(1/3)) + (((5*I)/9)*(-1 + 3*x)^(1/3)*(1 + x^3)*Log[1 + I*Sqrt[3] - 2*x])/(2^(2/3)*Sqr
t[3]*(1 + (3*I)*Sqrt[3])^(1/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - ((I/18)*(1 + (3*I)*Sqrt[3])^(2/3)*(-1 + 3*x)^(
1/3)*(1 + x^3)*Log[1 + I*Sqrt[3] - 2*x])/(2^(2/3)*Sqrt[3]*((1 - 3*x)*(1 + x^3)^3)^(1/3)) + ((-1 + 3*x)^(1/3)*(
1 + x^3)*Log[1 + x])/(6*2^(2/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - ((-1 + 3*x)^(1/3)*(1 + x^3)*Log[2^(2/3) + (-1
+ 3*x)^(1/3)])/(2*2^(2/3)*((1 - 3*x)*(1 + x^3)^3)^(1/3)) + (((5*I)/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*Log[(1 - (3*
I)*Sqrt[3])^(1/3) - 2^(1/3)*(-1 + 3*x)^(1/3)])/(2^(2/3)*Sqrt[3]*(1 - (3*I)*Sqrt[3])^(1/3)*((1 - 3*x)*(1 + x^3)
^3)^(1/3)) - ((I/6)*(1 - (3*I)*Sqrt[3])^(2/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*Log[(1 - (3*I)*Sqrt[3])^(1/3) - 2^(1/
3)*(-1 + 3*x)^(1/3)])/(2^(2/3)*Sqrt[3]*((1 - 3*x)*(1 + x^3)^3)^(1/3)) - (((5*I)/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*
Log[(1 + (3*I)*Sqrt[3])^(1/3) - 2^(1/3)*(-1 + 3*x)^(1/3)])/(2^(2/3)*Sqrt[3]*(1 + (3*I)*Sqrt[3])^(1/3)*((1 - 3*
x)*(1 + x^3)^3)^(1/3)) + ((I/6)*(1 + (3*I)*Sqrt[3])^(2/3)*(-1 + 3*x)^(1/3)*(1 + x^3)*Log[(1 + (3*I)*Sqrt[3])^(
1/3) - 2^(1/3)*(-1 + 3*x)^(1/3)])/(2^(2/3)*Sqrt[3]*((1 - 3*x)*(1 + x^3)^3)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 56

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 711

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^
m, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[m]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx &=\int \frac {1}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}} \, dx\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\sqrt [3]{-1+3 x} \left (1+x^3\right )} \, dx}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {1}{3 (1+x) \sqrt [3]{-1+3 x}}+\frac {2-x}{3 \sqrt [3]{-1+3 x} \left (1-x+x^2\right )}\right ) \, dx}{\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{(1+x) \sqrt [3]{-1+3 x}} \, dx}{3 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {2-x}{\sqrt [3]{-1+3 x} \left (1-x+x^2\right )} \, dx}{3 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{1-x+x^2} \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}-2^{2/3} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{2 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\sqrt [3]{-1+3 x} \left (1-x+x^2\right )} \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {2 i (-1+3 x)^{2/3}}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right )}+\frac {2 i (-1+3 x)^{2/3}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \left (\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}}+\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}}\right ) \, dx}{9 \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (\sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{-2+6 x}\right )}{2^{2/3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (2 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (2 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {(-1+3 x)^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (10 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (10 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1-3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1-3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 i \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [3]{-1+3 x}} \, dx}{9 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\left (5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1-3 i \sqrt {3}}}\right )}{3 \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1-3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1-3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{12 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1+3 i \sqrt {3}}}\right )}{3 \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )}-x} \, dx,x,\sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}+\frac {\left (i \left (1+3 i \sqrt {3}\right ) \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2} \left (1+3 i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+3 i \sqrt {3}\right )} x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )}{12 \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\left (i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1-3 i \sqrt {3}}}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}-\frac {\left (i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{-2+6 x}}{\sqrt [3]{1+3 i \sqrt {3}}}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}}\\ &=-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1-\sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1-3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{2} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9 \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{-1+3 x}}{\sqrt [3]{1+3 i \sqrt {3}}}}{\sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1-i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{9\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (1+i \sqrt {3}-2 x\right )}{18\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log (1+x)}{6\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {\sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (2^{2/3}+\sqrt [3]{-1+3 x}\right )}{2\ 2^{2/3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {i \left (1-3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1-3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}-\frac {5 i \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}+\frac {i \left (1+3 i \sqrt {3}\right )^{2/3} \sqrt [3]{-1+3 x} \left (1+x^3\right ) \log \left (\sqrt [3]{1+3 i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{-1+3 x}\right )}{6\ 2^{2/3} \sqrt {3} \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 10.13, size = 175, normalized size = 0.38 \begin {gather*} \frac {\sqrt [3]{3 x-1} \left (x^3+1\right ) \left (\sqrt [3]{2} \left (-2 \log \left (\sqrt [3]{6 x-2}+2\right )+\log \left ((6 x-2)^{2/3}-2 \sqrt [3]{6 x-2}+4\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{6 x-2}-1}{\sqrt {3}}\right )\right )-4 \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+7\&,\frac {\text {$\#$1}^3 \log \left (\sqrt [3]{3 x-1}-\text {$\#$1}\right )-5 \log \left (\sqrt [3]{3 x-1}-\text {$\#$1}\right )}{2 \text {$\#$1}^4-\text {$\#$1}}\&\right ]\right )}{12 \sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(-1/3),x]

[Out]

((-1 + 3*x)^(1/3)*(1 + x^3)*(2^(1/3)*(2*Sqrt[3]*ArcTan[(-1 + (-2 + 6*x)^(1/3))/Sqrt[3]] - 2*Log[2 + (-2 + 6*x)
^(1/3)] + Log[4 - 2*(-2 + 6*x)^(1/3) + (-2 + 6*x)^(2/3)]) - 4*RootSum[7 - #1^3 + #1^6 & , (-5*Log[(-1 + 3*x)^(
1/3) - #1] + Log[(-1 + 3*x)^(1/3) - #1]*#1^3)/(-#1 + 2*#1^4) & ]))/(12*(-((-1 + 3*x)*(1 + x^3)^3))^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.65, size = 459, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\left (\sqrt {3}+\sqrt {3} x\right ) \left (1-x+x^2\right )}{1+x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (1+2 x^3+x^6\right )}{6\ 2^{2/3}}+\frac {\log \left (-2-2 x^3+\sqrt [3]{2} \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}\right )}{3\ 2^{2/3}}-\frac {\log \left (4+8 x^3+4 x^6+\left (2 \sqrt [3]{2}+2 \sqrt [3]{2} x^3\right ) \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}+2^{2/3} \left (1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right )^{2/3}\right )}{6\ 2^{2/3}}+\frac {1}{3} \text {RootSum}\left [7+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {5 \log \left (1+x^3\right )-5 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right )+\log \left (1+x^3\right ) \text {$\#$1}^3-\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(-1/3),x]

[Out]

-(ArcTan[((Sqrt[3] + Sqrt[3]*x)*(1 - x + x^2))/(1 + x^3 + 2^(1/3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x
^9 - 3*x^10)^(1/3))]/(2^(2/3)*Sqrt[3])) - Log[1 + x^3]/(3*2^(2/3)) + Log[1 + 2*x^3 + x^6]/(6*2^(2/3)) + Log[-2
- 2*x^3 + 2^(1/3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)]/(3*2^(2/3)) - Log[4 + 8*x^3
+ 4*x^6 + (2*2^(1/3) + 2*2^(1/3)*x^3)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) + 2^(2/3
)*(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(2/3)]/(6*2^(2/3)) + RootSum[7 + #1^3 + #1^6 & , (5
*Log[1 + x^3] - 5*Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1] + Log[1 +
x^3]*#1^3 - Log[(1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) - #1 - x^3*#1]*#1^3)/(#1 + 2*#1
^4) & ]/3

________________________________________________________________________________________

fricas [B]  time = 0.98, size = 3411, normalized size = 7.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="fricas")

[Out]

1/42*28^(1/6)*14^(2/3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*log(7*(8*28^(1/3)*14^(1/3)*sqrt(3)*(
-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) +
1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) + 16*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7
+ 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 28^
(2/3)*14^(2/3)*(x^6 + 2*x^3 + 1) - 8*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x +
1)^(1/3)*(x^3 + 1) + 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(2/3))/(x^6 + 2*x^3 + 1)) +
2/21*28^(1/6)*14^(2/3)*arctan(-1/392*(28*28^(2/3)*14^(2/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*
x^3 - 3*x + 1)^(1/3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 14*28^(2/3)*14^(2/3)*sqrt(3)*(-3*x
^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) - sqrt(7)*(2*28^(2/3)*14^(2/3)*sqrt(3)*(x^3 + 1)*co
s(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 4*28^(2/3)*14^(2/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*
sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 28^(2/3)*14^(2/3)*sqrt(3)*(x^3 +
1))*sqrt((8*28^(1/3)*14^(1/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1
)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) + 16*2
8^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*
sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 28^(2/3)*14^(2/3)*(x^6 + 2*x^3 + 1) - 8*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 -
9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1) + 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3
- 3*x + 1)^(2/3))/(x^6 + 2*x^3 + 1)) + 56*(196*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^3
- (98*x^3 - 28^(2/3)*14^(2/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) + 98)*cos(2/3*ar
ctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) + 784*sqrt(3)*(x^
3 + 1))/(28*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^4 + 3*x^3 - 28*(x^3 + 1)*cos(2/3*arct
an(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 3))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 1/21*(28^(
1/6)*14^(2/3)*sqrt(3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) + 28^(1/6)*14^(2/3)*sin(2/3*arctan(2/
9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))))*arctan(1/196*(84*28^(2/3)*14^(2/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 -
9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 42*28^(2/3)*14^(2/3)*sq
rt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) - sqrt(7)*(6*28^(2/3)*14^(2/3)*sqrt(3)*(
x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 2*28^(2/3)*14^(2/3)*(x^3 + 1)*cos(2/3*arctan(2
/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 3*28^(2/3)*14^(2/3)*sq
rt(3)*(x^3 + 1))*sqrt(-(12*28^(1/3)*14^(1/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)
^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*s
qrt(3))) - 4*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2
/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 28^(2/3)*14^(2/3)*(x^6 + 2*x^3 + 1) + 2*28^(1/3)*14^(1/3)*(-
3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1) - 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 -
9*x^4 + 3*x^3 - 3*x + 1)^(2/3))/(x^6 + 2*x^3 + 1)) - 28*(784*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/
9*sqrt(3)))^3 - (392*x^3 - 28^(2/3)*14^(2/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) +
392)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) +
588*sqrt(3)*(x^3 + 1))/(112*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^4 + 27*x^3 - 112*(x^
3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 27)) - 1/21*(28^(1/6)*14^(2/3)*sqrt(3)*cos(2/3*a
rctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 28^(1/6)*14^(2/3)*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)
)))*arctan(1/196*(28*28^(2/3)*14^(2/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)
*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 14*28^(2/3)*14^(2/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 +
3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) - sqrt(7)*(2*28^(2/3)*14^(2/3)*sqrt(3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqr
t(7)*sqrt(3) + 1/9*sqrt(3)))^2 - 10*28^(2/3)*14^(2/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(
3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 28^(2/3)*14^(2/3)*sqrt(3)*(x^3 + 1))*sqrt((4*28^(1/3
)*14^(1/3)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/
9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 20*28^(1/3)*14^(1/3)*(-
3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1
/9*sqrt(3)))^2 + 28^(2/3)*14^(2/3)*(x^6 + 2*x^3 + 1) + 10*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9
*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1) + 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(2/3))/
(x^6 + 2*x^3 + 1)) + 28*(784*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^3 - (392*x^3 + 5*28^
(2/3)*14^(2/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3) + 392)*cos(2/3*arctan(2/9*sqrt(
7)*sqrt(3) + 1/9*sqrt(3))))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 980*sqrt(3)*(x^3 + 1))/(112*(
x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^4 + 3*x^3 - 112*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(
7)*sqrt(3) + 1/9*sqrt(3)))^2 + 3)) + 1/84*(28^(1/6)*14^(2/3)*sqrt(3)*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*
sqrt(3))) - 28^(1/6)*14^(2/3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))))*log(-28*(12*28^(1/3)*14^(1/3
)*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)
*sqrt(3) + 1/9*sqrt(3)))*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 4*28^(1/3)*14^(1/3)*(-3*x^10 + x
^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)
))^2 - 28^(2/3)*14^(2/3)*(x^6 + 2*x^3 + 1) + 2*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^
3 - 3*x + 1)^(1/3)*(x^3 + 1) - 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(2/3))/(x^6 + 2*x^
3 + 1)) - 1/84*(28^(1/6)*14^(2/3)*sqrt(3)*sin(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) + 28^(1/6)*14^(2/
3)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))))*log(28*(4*28^(1/3)*14^(1/3)*sqrt(3)*(-3*x^10 + x^9 - 9*
x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))*sin(
2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3))) - 20*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 +
3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1)*cos(2/3*arctan(2/9*sqrt(7)*sqrt(3) + 1/9*sqrt(3)))^2 + 28^(2/3)*14^(2/3)*(x^
6 + 2*x^3 + 1) + 10*28^(1/3)*14^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1
) + 28*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(2/3))/(x^6 + 2*x^3 + 1)) + 1/6*4^(1/6)*sqrt(
3)*arctan(1/6*4^(1/6)*(4^(1/3)*sqrt(3)*(x^3 + 1) + 2*sqrt(3)*(-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 -
3*x + 1)^(1/3))/(x^3 + 1)) - 1/24*4^(2/3)*log((4^(2/3)*(x^6 + 2*x^3 + 1) + 4^(1/3)*(-3*x^10 + x^9 - 9*x^7 + 3*
x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3)*(x^3 + 1) + (-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(2/
3))/(x^6 + 2*x^3 + 1)) + 1/12*4^(2/3)*log(-(4^(1/3)*(x^3 + 1) - (-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3
- 3*x + 1)^(1/3))/(x^3 + 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="giac")

[Out]

integrate((-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(-1/3), x)

________________________________________________________________________________________

maple [B]  time = 12.28, size = 17220, normalized size = 37.52

method	result	size
trager	\(\text {Expression too large to display}\)	\(17220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(-1/3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3),x)

[Out]

int(1/(3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**10+x**9-9*x**7+3*x**6-9*x**4+3*x**3-3*x+1)**(1/3),x)

[Out]

Integral((-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(-1/3), x)

________________________________________________________________________________________