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

Optimal. Leaf size=138 \[ -\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )}{3 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (3 x^3+2\right )}{10 x^5}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{6 \sqrt [3]{2}} \]

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Rubi [C]  time = 0.78, antiderivative size = 402, normalized size of antiderivative = 2.91, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1586, 6725, 271, 264, 2148, 6728} \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2\ 2^{2/3} \sqrt [3]{x^3-1}-2 x-i \sqrt {3}-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2\ 2^{2/3} \sqrt [3]{x^3-1}-2 x+i \sqrt {3}-1\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (2 x-i \sqrt {3}+1\right )}{\sqrt [3]{x^3-1}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (2 x+i \sqrt {3}+1\right )}{\sqrt [3]{x^3-1}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{10 x^2}+\frac {\log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (-2 x-i \sqrt {3}+1\right )^2 \left (2 x-i \sqrt {3}+1\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (-2 x+i \sqrt {3}+1\right )^2 \left (2 x+i \sqrt {3}+1\right )\right )}{12 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^3)^(2/3)/(5*x^5) + (3*(-1 + x^3)^(2/3))/(10*x^2) + ArcTan[(1 - (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sqr
t[3]]/(2*2^(1/3)*Sqrt[3]) + ArcTan[(2 + (2^(1/3)*(1 - I*Sqrt[3] + 2*x))/(-1 + x^3)^(1/3))/(2*Sqrt[3])]/(2*2^(1
/3)*Sqrt[3]) + ArcTan[(2 + (2^(1/3)*(1 + I*Sqrt[3] + 2*x))/(-1 + x^3)^(1/3))/(2*Sqrt[3])]/(2*2^(1/3)*Sqrt[3])
+ Log[(1 - x)*(1 + x)^2]/(12*2^(1/3)) + Log[-((1 - I*Sqrt[3] - 2*x)^2*(1 - I*Sqrt[3] + 2*x))]/(12*2^(1/3)) + L
og[-((1 + I*Sqrt[3] - 2*x)^2*(1 + I*Sqrt[3] + 2*x))]/(12*2^(1/3)) - Log[1 - x + 2^(2/3)*(-1 + x^3)^(1/3)]/(4*2
^(1/3)) - Log[-1 - I*Sqrt[3] - 2*x + 2*2^(2/3)*(-1 + x^3)^(1/3)]/(4*2^(1/3)) - Log[-1 + I*Sqrt[3] - 2*x + 2*2^
(2/3)*(-1 + x^3)^(1/3)]/(4*2^(1/3))

Rule 264

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

Rule 271

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.37, size = 138, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\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 )}{6 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.44, size = 301, normalized size = 2.18 \begin {gather*} -\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (3 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{180 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/180*(10*sqrt(6)*2^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(-1)^(2/3)*(5*x^7 + 4*x^4 - x)
*(x^3 - 1)^(2/3) + 12*sqrt(6)*(-1)^(1/3)*(19*x^8 - 16*x^5 + x^2)*(x^3 - 1)^(1/3) - sqrt(6)*2^(1/3)*(71*x^9 - 1
11*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(-1)^(1/3)*x^5*log(-(6*2^(1/3)*(-1)^(2/3)*
(x^3 - 1)^(1/3)*x^2 + 2^(2/3)*(-1)^(1/3)*(x^3 + 1) - 6*(x^3 - 1)^(2/3)*x)/(x^3 + 1)) + 5*2^(2/3)*(-1)^(1/3)*x^
5*log(-(3*2^(2/3)*(-1)^(1/3)*(5*x^4 - x)*(x^3 - 1)^(2/3) - 2^(1/3)*(-1)^(2/3)*(19*x^6 - 16*x^3 + 1) - 12*(2*x^
5 - x^2)*(x^3 - 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 18*(3*x^3 + 2)*(x^3 - 1)^(2/3))/x^5

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

[Out]

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

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maple [C]  time = 18.06, size = 760, normalized size = 5.51

method	result	size
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (3 x^{3}+2\right )}{10 x^{5}}+224 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \ln \left (-\frac {180768 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+9934848 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +14784 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-7011648 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{2}-1446144 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )-4234051584 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-14795 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-43317120 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{3}-40002 x \left (x^{3}-1\right )^{\frac {2}{3}}+6187 \RootOf \left (\textit {\_Z}^{3}+4\right )[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (\frac {-2957791200 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-12894050405376 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}[phone] \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x [phone] \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-350155647072 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{2}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-233278175 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-1016941475424 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right ) x^{3}-1034338071 x \left (x^{3}-1\right )^{\frac {2}{3}}[phone] \RootOf \left (\textit {\_Z}^{3}+4\right )[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+1344 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+1806336 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{6}\)	\(760\)
risch	\(\frac {3 x^{6}-x^{3}-2}{10 x^{5} \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 ) \ln \left (-\frac {-9 \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}-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 )^{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}-24 \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}+9 \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}-10 x \left (x^{3}-1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{3}+4\right )-6 \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 )-\frac {\ln \left (\frac {-6 \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}+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 )^{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}-24 \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}-2 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+6 \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}}+2 \RootOf \left (\textit {\_Z}^{3}+4\right )-6 \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 )}{6}-\ln \left (\frac {-6 \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}+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 )^{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}-24 \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}-2 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+6 \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}}+2 \RootOf \left (\textit {\_Z}^{3}+4\right )-6 \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 )\)	\(930\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(x^3-1)^(2/3)*(3*x^3+2)/x^5+224*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*ln(-(180768*
RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^3*x^3+529256448*RootOf(RootOf(_Z^3
+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+9934848*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^
2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^2*x+14784*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-7011648*(x^
3-1)^(1/3)*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^2-1446144*RootOf(_Z^3
+4)^3*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)-4234051584*RootOf(RootOf(_Z^3+4)^2+1344*_Z*
RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+4)^2-14795*RootOf(_Z^3+4)*x^3-43317120*RootOf(RootOf(_Z^3+4)^2+1344
*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^3-40002*x*(x^3-1)^(2/3)+6187*RootOf(_Z^3+4)[phone]*RootOf(RootOf(_Z^3+4)^
2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2))/(1+x)/(x^2-x+1))+1/6*RootOf(_Z^3+4)*ln((-2957791200*RootOf(RootOf(_Z^3
+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^3*x^3-12894050405376*RootOf(RootOf(_Z^3+4)^2+1344*_Z
*RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+172459768320*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+1344*_
Z*RootOf(_Z^3+4)+1806336*_Z^2)*RootOf(_Z^3+4)^2*x+256636560*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2-350155647072*(x
^3-1)^(1/3)*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^2+23662329600*RootOf
(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)[phone]*RootOf(RootOf(_Z^3+4)^
2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)^2*RootOf(_Z^3+4)^2-233278175*RootOf(_Z^3+4)*x^3-1016941475424*RootOf(Ro
otOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2)*x^3-1034338071*x*(x^3-1)^(2/3)[phone]*RootOf(_Z^3+4)+134
313025056*RootOf(RootOf(_Z^3+4)^2+1344*_Z*RootOf(_Z^3+4)+1806336*_Z^2))/(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} + x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 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+x^3+1)/x^6/(x^6-1),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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