3.21.18 \(\int \frac {(-1+x^3)^{2/3} (4+4 x^3+x^6)}{x^9 (1+x^3)} \, dx\)

Optimal. Leaf size=143 \[ -\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (-2 x^6+2 x^3-5\right )}{10 x^8} \]

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Rubi [A]  time = 0.42, antiderivative size = 277, normalized size of antiderivative = 1.94, number of steps used = 23, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {28, 586, 580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )}{12 \sqrt [3]{2}}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {\left (x^3-1\right )^{2/3}}{2 x^8}+\frac {11 \left (x^3-1\right )^{2/3}}{20 x^5}-\frac {79 \left (x^3-1\right )^{2/3}}{80 x^2}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )-\frac {\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{24 \sqrt [3]{2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^3)^(2/3)*(4 + 4*x^3 + x^6))/(x^9*(1 + x^3)),x]

[Out]

-1/2*(-1 + x^3)^(2/3)/x^8 + (11*(-1 + x^3)^(2/3))/(20*x^5) - (79*(-1 + x^3)^(2/3))/(80*x^2) - ArcTan[(1 + (2*2
^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(4*2^(1/3)*Sqrt[3]) + (2*2^(2/3)*ArcTan[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/
3))/Sqrt[3]])/Sqrt[3] + Log[1 - (2^(1/3)*x)/(-1 + x^3)^(1/3)]/(12*2^(1/3)) - (2*2^(2/3)*Log[1 - (2^(1/3)*x)/(-
1 + x^3)^(1/3)])/3 - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + (2^(1/3)*x)/(-1 + x^3)^(1/3)]/(24*2^(1/3)) + (2^
(2/3)*Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + (2^(1/3)*x)/(-1 + x^3)^(1/3)])/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 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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 377

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

Rule 580

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)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

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 586

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Dist[e, Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] + Dist[f/e^n,
Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p,
q}, x] && IGtQ[n, 0] && IGtQ[r, 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx &=\int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )^2}{x^9 \left (1+x^3\right )} \, dx\\ &=\frac {1}{8} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+x^3\right )} \, dx+2 \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9 \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}-\frac {\left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{40} \int \frac {9-x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{4} \int \frac {12-4 x^3}{x^6 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {9 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{80} \int -\frac {20}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{20} \int \frac {-44+36 x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{40} \int \frac {160}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+4 \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{12} \operatorname {Subst}\left (\int \frac {2+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+4 \operatorname {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {2+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\left (2\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 139, normalized size = 0.97 \begin {gather*} \frac {1}{30} \left (5\ 2^{2/3} \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )-\frac {3 \left (x^3-1\right )^{2/3} \left (2 x^6-2 x^3+5\right )}{x^8}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^3)^(2/3)*(4 + 4*x^3 + x^6))/(x^9*(1 + x^3)),x]

[Out]

((-3*(-1 + x^3)^(2/3)*(5 - 2*x^3 + 2*x^6))/x^8 + 5*2^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(1 - x^3)^(1/3
))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*x)/(1 - x^3)^(1/3)] + Log[1 + (2^(2/3)*x^2)/(1 - x^3)^(2/3) + (2^(1/3)*x)/(1
- x^3)^(1/3)]))/30

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IntegrateAlgebraic [A]  time = 0.34, size = 143, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-5+2 x^3-2 x^6\right )}{10 x^8}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + 4*x^3 + x^6))/(x^9*(1 + x^3)),x]

[Out]

((-1 + x^3)^(2/3)*(-5 + 2*x^3 - 2*x^6))/(10*x^8) + (2^(2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))]
)/Sqrt[3] - (2^(2/3)*Log[-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + Log[2*x^2 + 2^(2/3)*x*(-1 + x^3)^(1/3) + 2^(1/3
)*(-1 + x^3)^(2/3)]/(3*2^(1/3))

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fricas [B]  time = 2.64, size = 276, normalized size = 1.93 \begin {gather*} -\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{8} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 5 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (2 \, x^{6} - 2 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4*x^3+4)/x^9/(x^3+1),x, algorithm="fricas")

[Out]

-1/90*(10*sqrt(3)*(-4)^(1/3)*x^8*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(5*x^7 + 4*x^4 - x)*(x^3 - 1)^(2/3) + 6*sqrt
(3)*(-4)^(1/3)*(19*x^8 - 16*x^5 + x^2)*(x^3 - 1)^(1/3) - sqrt(3)*(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 1
05*x^6 + 3*x^3 + 1)) - 10*(-4)^(1/3)*x^8*log(-(3*(-4)^(2/3)*(x^3 - 1)^(1/3)*x^2 - 6*(x^3 - 1)^(2/3)*x + (-4)^(
1/3)*(x^3 + 1))/(x^3 + 1)) + 5*(-4)^(1/3)*x^8*log(-(6*(-4)^(1/3)*(5*x^4 - x)*(x^3 - 1)^(2/3) - (-4)^(2/3)*(19*
x^6 - 16*x^3 + 1) - 24*(2*x^5 - x^2)*(x^3 - 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 9*(2*x^6 - 2*x^3 + 5)*(x^3 - 1)^(2/
3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 4 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4*x^3+4)/x^9/(x^3+1),x, algorithm="giac")

[Out]

integrate((x^6 + 4*x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 + 1)*x^9), x)

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maple [C]  time = 16.73, size = 765, normalized size = 5.35

method	result	size
trager	\(-\frac {\left (2 x^{6}-2 x^{3}+5\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{10 x^{8}}+512 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \ln \left (\frac {10384382361600 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-394193756160 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x -1588743277056 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-513273120 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-83075058892800 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-175429257216 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )-743673216000 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) x^{3}-1570411645 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-1042129902 x \left (x^{3}-1\right )^{\frac {2}{3}}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {16841208692736 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-197096878080 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x [phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-256636560 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-134729669541888 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-27042662400 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right ) x^{3}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}[phone] x \left (x^{3}-1\right )^{\frac {2}{3}}-153500600064 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1536 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+2359296 \textit {\_Z}^{2}\right )-30810325 \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}\)	\(765\)
risch	\(-\frac {2 x^{9}-4 x^{6}+7 x^{3}-5}{10 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+30 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-10 x \left (x^{3}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+4\right )-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )}{3}-2 \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +\left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+30 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-10 x \left (x^{3}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+4\right )-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+54 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}-12 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +5 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+6 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 x \left (x^{3}-1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+4\right )-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}\)	\(915\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+4*x^3+4)/x^9/(x^3+1),x,method=_RETURNVERBOSE)

[Out]

-1/10*(2*x^6-2*x^3+5)/x^8*(x^3-1)^(2/3)+512*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*ln((1
0384382361600*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+21928657152*
RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)^3*x^3-394193756160*(x^3-1)^(2/3)*R
ootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)^2*x-1588743277056*RootOf(RootOf(_Z^
3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2-513273120*(x^3-1)^(1/3)*RootOf(_Z
^3+4)^2*x^2-83075058892800*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)^2*RootOf(_Z^3+4)^2-175
429257216*RootOf(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)-743673216000*RootOf(Ro
otOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*x^3-1570411645*RootOf(_Z^3+4)*x^3-1042129902*x*(x^3-1)^(2/
3)[phone]*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)[phone]*RootOf(_Z^3+4))/(1+x)/(
x^2-x+1))+1/3*RootOf(_Z^3+4)*ln(-(16841208692736*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)^
2*RootOf(_Z^3+4)^2*x^3+3380332800*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)^
3*x^3-197096878080*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)^2
*x+400177882368*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*RootOf(_Z^3+4)*(x^3-1)^(1/3)*x^2-
256636560*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-134729669541888*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+23
59296*_Z^2)^2*RootOf(_Z^3+4)^2-27042662400*RootOf(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+235
9296*_Z^2)[phone]*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)*x^3+233278175*RootOf(_Z^
3+4)*x^3+1034338071*x*(x^3-1)^(2/3)-153500600064*RootOf(RootOf(_Z^3+4)^2+1536*_Z*RootOf(_Z^3+4)+2359296*_Z^2)-
30810325*RootOf(_Z^3+4))/(1+x)/(x^2-x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 4 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4*x^3+4)/x^9/(x^3+1),x, algorithm="maxima")

[Out]

integrate((x^6 + 4*x^3 + 4)*(x^3 - 1)^(2/3)/((x^3 + 1)*x^9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\,x^3+4\right )}{x^9\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(4*x^3 + x^6 + 4))/(x^9*(x^3 + 1)),x)

[Out]

int(((x^3 - 1)^(2/3)*(4*x^3 + x^6 + 4))/(x^9*(x^3 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )^{2}}{x^{9} \left (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**3-1)**(2/3)*(x**6+4*x**3+4)/x**9/(x**3+1),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**3 + 2)**2/(x**9*(x + 1)*(x**2 - x + 1)), x)

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