3.11.35 \(\int \frac {(-1+x^3)^{2/3} (1-2 x^3+x^6)}{x^6 (1-x^3+x^6)} \, dx\) [1035]

Optimal. Leaf size=78 \[ \frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.21, antiderivative size = 651, normalized size of antiderivative = 8.35, number of steps used = 14, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {28, 1534, 283, 245, 1442, 427, 544, 384} \begin {gather*} \frac {1}{18} \left (3 \sqrt {3}+7 i\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {1}{18} \left (-3 \sqrt {3}+7 i\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} i \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\frac {i \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}-\frac {i \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \log \left (2 x^3-i \sqrt {3}-1\right )}{6 \sqrt {3}}+\frac {i \log \left (2 x^3+i \sqrt {3}-1\right )}{6 \sqrt {3} \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}+\frac {i \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}\right )}{2 \sqrt {3}}-\frac {i \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x\right )}{2 \sqrt {3} \sqrt [3]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}}}-\frac {1}{36} \left (9+7 i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{36} \left (9-7 i \sqrt {3}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^3)^(2/3)/(2*x^2) + (-1 + x^3)^(5/3)/(5*x^5) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (
(7*I - 3*Sqrt[3])*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/18 + ((7*I + 3*Sqrt[3])*ArcTan[(1 + (2*x)/(-1
+ x^3)^(1/3))/Sqrt[3]])/18 - (I/3)*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*ArcTan[(1 + (2*x)/((-((I - Sqrt[3])/
(I + Sqrt[3])))^(1/3)*(-1 + x^3)^(1/3)))/Sqrt[3]] + ((I/3)*ArcTan[(1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/
3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3) - ((I/6)*(-((I - Sqrt[3])/(I + Sqrt[3
])))^(1/3)*Log[-1 - I*Sqrt[3] + 2*x^3])/Sqrt[3] + ((I/6)*Log[-1 + I*Sqrt[3] + 2*x^3])/(Sqrt[3]*(-((I - Sqrt[3]
)/(I + Sqrt[3])))^(1/3)) + ((I/2)*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*Log[x/(-((I - Sqrt[3])/(I + Sqrt[3]))
)^(1/3) - (-1 + x^3)^(1/3)])/Sqrt[3] - ((I/2)*Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x - (-1 + x^3)^(1/3)]
)/(Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) + Log[-x + (-1 + x^3)^(1/3)]/2 - ((9 - (7*I)*Sqrt[3])*Log[-
x + (-1 + x^3)^(1/3)])/36 - ((9 + (7*I)*Sqrt[3])*Log[-x + (-1 + x^3)^(1/3)])/36

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 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 1442

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[2*(c/r), Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[2*(c/r), Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rule 1534

Int[(((f_.)*(x_))^(m_)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Dist[d/a, Int[(f*x)^m*(d + e*x^n)^(q - 1), x], x] - Dist[1/(a*f^n), Int[(f*x)^(m + n)*(d + e*x^n)^(q - 1)*
(Simp[b*d - a*e + c*d*x^n, x]/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n]
&& NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx &=\int \frac {\left (-1+x^3\right )^{8/3}}{x^6 \left (1-x^3+x^6\right )} \, dx\\ &=-\int \frac {\left (-1+x^3\right )^{5/3}}{x^6} \, dx+\int \frac {\left (-1+x^3\right )^{5/3}}{1-x^3+x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {(2 i) \int \frac {\left (-1+x^3\right )^{5/3}}{-1-i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{5/3}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (2 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{5/3}}{-1-i \sqrt {3}+2 x^3} \, dx}{\sqrt {3} \left (1-x^3\right )^{2/3}}-\frac {\left (2 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{5/3}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3} \left (1-x^3\right )^{2/3}}-\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}+\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 78, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 22.01, size = 1839, normalized size = 23.58

method	result	size
risch	\(\text {Expression too large to display}\)	\(1839\)
trager	\(\text {Expression too large to display}\)	\(4954\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(7*x^6-9*x^3+2)/x^5/(x^3-1)^(1/3)-729*RootOf(19683*_Z^6+243*_Z^3+1)^5*ln((-4374*RootOf(19683*_Z^6+243*_Z^
3+1)^5*x^3+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x+2187*RootOf(19683*_Z^6+243*_Z^3+1)^5-18*RootOf(1
9683*_Z^6+243*_Z^3+1)^2*x^3+3*RootOf(19683*_Z^6+243*_Z^3+1)*(x^3-1)^(1/3)*x^2+x*(x^3-1)^(2/3)+9*RootOf(19683*_
Z^6+243*_Z^3+1)^2)/(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+27*RootOf(
19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+54*RootOf(19683*_Z^6+243*_Z^3+1)^2*x+1))
+1458*RootOf(19683*_Z^6+243*_Z^3+1)^5*ln((729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x^3+243*(x^3-1)^(1/3)*RootOf(196
83*_Z^6+243*_Z^3+1)^3*x^2+27*RootOf(19683*_Z^6+243*_Z^3+1)^2*(x^3-1)^(2/3)*x+6*RootOf(19683*_Z^6+243*_Z^3+1)*x
^3+x^2*(x^3-1)^(1/3)-3*RootOf(19683*_Z^6+243*_Z^3+1))/(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19
683*_Z^6+243*_Z^3+1)^5*x+27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+54*Ro
otOf(19683*_Z^6+243*_Z^3+1)^2*x+1))+243*ln(-(-59049*RootOf(19683*_Z^6+243*_Z^3+1)^7*x^3-4374*RootOf(19683*_Z^6
+243*_Z^3+1)^5*(x^3-1)^(1/3)*x^2-972*RootOf(19683*_Z^6+243*_Z^3+1)^4*x^3+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+24
3*_Z^3+1)^3*x-27*RootOf(19683*_Z^6+243*_Z^3+1)^2*(x^3-1)^(1/3)*x^2-243*RootOf(19683*_Z^6+243*_Z^3+1)^4-3*RootO
f(19683*_Z^6+243*_Z^3+1)*x^3+x*(x^3-1)^(2/3)-3*RootOf(19683*_Z^6+243*_Z^3+1))/(729*RootOf(19683*_Z^6+243*_Z^3+
1)^4*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*x+1)/(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+1)/(1458*RootOf(19683*_Z^6+
243*_Z^3+1)^4*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*x-1))*RootOf(19683*_Z^6+243*_Z^3+1)^4-9*RootOf(19683*_Z^6+243*
_Z^3+1)^2*ln((-4374*RootOf(19683*_Z^6+243*_Z^3+1)^5*x^3+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x+218
7*RootOf(19683*_Z^6+243*_Z^3+1)^5-18*RootOf(19683*_Z^6+243*_Z^3+1)^2*x^3+3*RootOf(19683*_Z^6+243*_Z^3+1)*(x^3-
1)^(1/3)*x^2+x*(x^3-1)^(2/3)+9*RootOf(19683*_Z^6+243*_Z^3+1)^2)/(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561
*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19683*_Z^6+243*_Z^3+1)
^5*x+54*RootOf(19683*_Z^6+243*_Z^3+1)^2*x+1))+9*RootOf(19683*_Z^6+243*_Z^3+1)^2*ln((729*RootOf(19683*_Z^6+243*
_Z^3+1)^4*x^3+243*(x^3-1)^(1/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x^2+27*RootOf(19683*_Z^6+243*_Z^3+1)^2*(x^3-1)
^(2/3)*x+6*RootOf(19683*_Z^6+243*_Z^3+1)*x^3+x^2*(x^3-1)^(1/3)-3*RootOf(19683*_Z^6+243*_Z^3+1))/(27*RootOf(196
83*_Z^6+243*_Z^3+1)^2*x-1)/(6561*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(65
61*RootOf(19683*_Z^6+243*_Z^3+1)^5*x+54*RootOf(19683*_Z^6+243*_Z^3+1)^2*x+1))+RootOf(19683*_Z^6+243*_Z^3+1)*ln
(-(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*(x^3-1)^(1/3)*x^2+324*RootOf(19683*_Z^6+243*_Z^3+1)^3*x^3+27*RootOf(196
83*_Z^6+243*_Z^3+1)^2*(x^3-1)^(2/3)*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*(x^3-1)^(1/3)*x^2-162*RootOf(19683*_Z^6+
243*_Z^3+1)^3+2*x^3-1)/(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*x+1)/(729*RootOf
(19683*_Z^6+243*_Z^3+1)^4*x+1)/(1458*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*x-1))+l
n(-(-59049*RootOf(19683*_Z^6+243*_Z^3+1)^7*x^3-4374*RootOf(19683*_Z^6+243*_Z^3+1)^5*(x^3-1)^(1/3)*x^2-972*Root
Of(19683*_Z^6+243*_Z^3+1)^4*x^3+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x-27*RootOf(19683*_Z^6+243*_Z
^3+1)^2*(x^3-1)^(1/3)*x^2-243*RootOf(19683*_Z^6+243*_Z^3+1)^4-3*RootOf(19683*_Z^6+243*_Z^3+1)*x^3+x*(x^3-1)^(2
/3)-3*RootOf(19683*_Z^6+243*_Z^3+1))/(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+9*RootOf(19683*_Z^6+243*_Z^3+1)*x+
1)/(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+1)/(1458*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+9*RootOf(19683*_Z^6+243*_
Z^3+1)*x-1))*RootOf(19683*_Z^6+243*_Z^3+1)

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

integrate((x^6 - 2*x^3 + 1)*(x^3 - 1)^(2/3)/((x^6 - x^3 + 1)*x^6), 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^6-2\,x^3+1\right )}{x^6\,\left (x^6-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)*(x^6 - 2*x^3 + 1))/(x^6*(x^6 - x^3 + 1)),x)

[Out]

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

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