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

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

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Rubi [C]  time = 0.14, antiderivative size = 223, normalized size of antiderivative = 2.86, number of steps used = 10, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {28, 1520, 277, 239, 1428, 430, 429} \begin {gather*} -\frac {2 x \left (x^3-1\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 (\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {2 x \left (x^3-1\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 (-\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[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) - (2*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -5/3, 1, 4/3, x^3, (
2*x^3)/(1 - I*Sqrt[3])])/(Sqrt[3]*(I + Sqrt[3])*(1 - x^3)^(2/3)) + (2*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -5/3, 1
, 4/3, x^3, (2*x^3)/(1 + I*Sqrt[3])])/(Sqrt[3]*(I - Sqrt[3])*(1 - x^3)^(2/3)) - ArcTan[(1 + (2*x)/(-1 + x^3)^(
1/3))/Sqrt[3]]/Sqrt[3] + Log[-x + (-1 + x^3)^(1/3)]/2

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 239

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 277

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 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 1428

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 1520

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 [C]  time = 1.94, size = 415, normalized size = 5.32 \begin {gather*} \left (x^3-1\right )^{2/3} \left (\frac {7}{10 x^2}-\frac {1}{5 x^5}\right )+\frac {i \left (-\frac {2 \log \left (\sqrt [3]{\sqrt {3}+i}-\frac {\sqrt [3]{\sqrt {3}-i} x}{\sqrt [3]{x^3-1}}\right )}{\sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}+i}}}+2 \sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}+i}} \log \left (\sqrt [3]{\sqrt {3}-i}-\frac {\sqrt [3]{\sqrt {3}+i} x}{\sqrt [3]{x^3-1}}\right )+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt {3}-i\right )^{2/3} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}-i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}+i\right )^{2/3}\right )}{\left (\frac {1}{2} \left (\sqrt {3}-i\right )\right )^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt {3}+i\right )^{2/3} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}+i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}-i\right )^{2/3}\right )}{\left (\frac {1}{2} \left (\sqrt {3}+i\right )\right )^{2/3}}\right )}{6 \sqrt {3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/5*1/x^5 + 7/(10*x^2))*(-1 + x^3)^(2/3) + ((I/6)*((2*Sqrt[3]*ArcTan[(1 + (2^(1/3)*(-I + Sqrt[3])^(2/3)*x)/(
-1 + x^3)^(1/3))/Sqrt[3]] + Log[(I + Sqrt[3])^(2/3) + ((-I + Sqrt[3])^(2/3)*x^2)/(-1 + x^3)^(2/3) + (2^(2/3)*x
)/(-1 + x^3)^(1/3)])/((-I + Sqrt[3])/2)^(2/3) - (2*Sqrt[3]*ArcTan[(1 + (2^(1/3)*(I + Sqrt[3])^(2/3)*x)/(-1 + x
^3)^(1/3))/Sqrt[3]] + Log[(-I + Sqrt[3])^(2/3) + ((I + Sqrt[3])^(2/3)*x^2)/(-1 + x^3)^(2/3) + (2^(2/3)*x)/(-1
+ x^3)^(1/3)])/((I + Sqrt[3])/2)^(2/3) - (2*Log[(I + Sqrt[3])^(1/3) - ((-I + Sqrt[3])^(1/3)*x)/(-1 + x^3)^(1/3
)])/((-I + Sqrt[3])/(I + Sqrt[3]))^(1/3) + 2*((-I + Sqrt[3])/(I + Sqrt[3]))^(1/3)*Log[(-I + Sqrt[3])^(1/3) - (
(I + Sqrt[3])^(1/3)*x)/(-1 + x^3)^(1/3)]))/Sqrt[3]

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IntegrateAlgebraic [A]  time = 0.21, 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]

IntegrateAlgebraic[((-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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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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + 1\right )} x^{6}}\,{d x} \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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maple [B]  time = 26.34, size = 1461, normalized size = 18.73

method	result	size
risch	\(\frac {7 x^{6}-9 x^{3}+2}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \ln \left (\frac {-177147 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{7} x^{3}-2916 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x^{3}-243 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x^{2}+27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {2}{3}} x +729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4}-12 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}-2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )+1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \ln \left (\frac {729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+81 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x -18 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x^{3}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+x \left (x^{3}-1\right )^{\frac {2}{3}}+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2}}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )-243 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \ln \left (-\frac {27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \ln \left (\frac {729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+81 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x -18 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x^{3}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+x \left (x^{3}-1\right )^{\frac {2}{3}}+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2}}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )+\RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \ln \left (-\frac {177147 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{7} x^{3}+6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x^{3}+243 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )-2 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \ln \left (-\frac {27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )\)	\(1461\)
trager	\(\text {Expression too large to display}\)	\(4455\)

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((-177147*RootOf(19683*_Z^6+243*_
Z^3+1)^7*x^3-2916*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+729*RootOf(19683*_Z^6+243*_Z^3+1)^4-12*RootOf(19683*_Z^6+243*
_Z^3+1)*x^3-2*x^2*(x^3-1)^(1/3)+6*RootOf(19683*_Z^6+243*_Z^3+1))/(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*x-1)/(656
1*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^3-1)^(1/3)*x^2+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x-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))-243*RootOf(1968
3*_Z^6+243*_Z^3+1)^4*ln(-(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*(x^3-1)^(1/3)*x^2-6*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))/(1458*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+9*RootOf(19683*_Z^6+243*_Z^3+1)*x+1)/(729*RootOf(1
9683*_Z^6+243*_Z^3+1)^4*x+1))+9*RootOf(19683*_Z^6+243*_Z^3+1)^2*ln((729*RootOf(19683*_Z^6+243*_Z^3+1)^4*(x^3-1
)^(1/3)*x^2+81*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x-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))+RootOf(19683*_Z^6+243*_Z^3+1)*ln(-(
177147*RootOf(19683*_Z^6+243*_Z^3+1)^7*x^3+6561*RootOf(19683*_Z^6+243*_Z^3+1)^5*(x^3-1)^(1/3)*x^2+1458*RootOf(
19683*_Z^6+243*_Z^3+1)^4*x^3+243*(x^3-1)^(2/3)*RootOf(19683*_Z^6+243*_Z^3+1)^3*x+54*RootOf(19683*_Z^6+243*_Z^3
+1)^2*(x^3-1)^(1/3)*x^2+729*RootOf(19683*_Z^6+243*_Z^3+1)^4+3*RootOf(19683*_Z^6+243*_Z^3+1)*x^3+2*x*(x^3-1)^(2
/3)+3*RootOf(19683*_Z^6+243*_Z^3+1))/(1458*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+9*RootOf(19683*_Z^6+243*_Z^3+1)*x+1)/(729*RootOf(19683*_Z^6+243*_Z^
3+1)^4*x+1))-2*RootOf(19683*_Z^6+243*_Z^3+1)*ln(-(27*RootOf(19683*_Z^6+243*_Z^3+1)^2*(x^3-1)^(1/3)*x^2-6*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))/(1458*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+9*RootOf(19683*_Z^6+243*_Z
^3+1)*x+1)/(729*RootOf(19683*_Z^6+243*_Z^3+1)^4*x+1))

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