3.21.18 \(\int \frac {(-1+x^3)^{2/3} (4+4 x^3+x^6)}{x^9 (1+x^3)} \, dx\) [2018]

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

[Out]

1/10*(x^3-1)^(2/3)*(-2*x^6+2*x^3-5)/x^8+1/3*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3-1)^(1/3)))*2^(2/3)*3^(1/2)-1/3*ln
(-2*x+2^(2/3)*(x^3-1)^(1/3))*2^(2/3)+1/6*ln(2*x^2+2^(2/3)*x*(x^3-1)^(1/3)+2^(1/3)*(x^3-1)^(2/3))*2^(2/3)

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Rubi [A]
time = 0.19, antiderivative size = 213, normalized size of antiderivative = 1.49, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {28, 600, 594, 597, 12, 384} \begin {gather*} \frac {2\ 2^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {1}{3} 2^{2/3} \log \left (x^3+1\right )-\frac {\log \left (x^3+1\right )}{24 \sqrt [3]{2}}-2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{8 \sqrt [3]{2}}-\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} \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 + x^3]/(24*2^(1/3)) + (2^(2/3)*Log[1 + x^3])/3 + Log[2^(1/3)*x - (-1 + x^3)^(1/3
)]/(8*2^(1/3)) - 2^(2/3)*Log[2^(1/3)*x - (-1 + 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 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 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 594

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 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 600

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]

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} \text {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} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{12} \text {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 \text {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} \text {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} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{3} \text {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 {\text {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 \text {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 {\text {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} \text {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 ) \text {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 = 0.30, 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} \text {ArcTan}\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]

Integrate[((-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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 8.99, size = 913, normalized size = 6.38

method	result	size
risch	\(\text {Expression too large to display}\)	\(913\)
trager	\(\text {Expression too large to display}\)	\(1125\)

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^9-4*x^6+7*x^3-5)/x^8/(x^3-1)^(1/3)+1/3*RootOf(_Z^3+4)*ln((-3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z
^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-54*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2
*x^3+12*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x-5*(x^3-1)^(1/3)*
RootOf(_Z^3+4)^2*x^2-6*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)*x^2-R
ootOf(_Z^3+4)*x^3-18*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3+2*x*(x^3-1)^(2/3)+RootOf(_Z^3+4)
+18*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(1+x)/(x^2-x+1))-1/3*ln(-(3*RootOf(RootOf(_Z^3+4)^2+
6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*Ro
otOf(_Z^3+4)^2*x^3+12*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^2*x+(x
^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2+30*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(
_Z^3+4)*x^2-3*RootOf(_Z^3+4)*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3-10*x*(x^3-1)^(2/3
)+RootOf(_Z^3+4)-12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(1+x)/(x^2-x+1))*RootOf(_Z^3+4)-2*ln
(-(3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-36*RootOf(RootOf(_Z^3+4)^2+6*_Z
*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+12*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+3
6*_Z^2)*RootOf(_Z^3+4)^2*x+(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2+30*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*Ro
otOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)*x^2-3*RootOf(_Z^3+4)*x^3+36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+3
6*_Z^2)*x^3-10*x*(x^3-1)^(2/3)+RootOf(_Z^3+4)-12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(1+x)/(
x^2-x+1))*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (110) = 220\).
time = 1.91, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(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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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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