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

Optimal. Leaf size=131 \[ -\frac {2 \log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{\sqrt [3]{3}}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-17 x^3-2\right )}{10 x^5}+\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{\sqrt [3]{3}} \]

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Rubi [A]  time = 0.53, antiderivative size = 134, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1586, 6725, 271, 264, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )}{\sqrt [3]{3}}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}-\frac {17 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )}{\sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(1 + x^3)^(2/3)/x^5 - (17*(1 + x^3)^(2/3))/(10*x^2) + 2*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(1 + x^
3)^(1/3))] - (2*Log[1 - (3^(1/3)*x)/(1 + x^3)^(1/3)])/3^(1/3) + Log[1 + (3^(2/3)*x^2)/(1 + x^3)^(2/3) + (3^(1/
3)*x)/(1 + x^3)^(1/3)]/3^(1/3)

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 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 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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx &=\int \frac {-1-2 x^3+2 x^6}{x^6 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=\int \left (\frac {1}{x^6 \sqrt [3]{1+x^3}}+\frac {4}{x^3 \sqrt [3]{1+x^3}}-\frac {6}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )}\right ) \, dx\\ &=4 \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx-6 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx-6 \operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-2 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-2 \operatorname {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+3 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}-\left (2\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.32, size = 131, normalized size = 1.00 \begin {gather*} \frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 - 17*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 2*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - (2*
Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)])/3^(1/3) + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)
]/3^(1/3)

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fricas [B]  time = 2.76, size = 292, normalized size = 2.23 \begin {gather*} \frac {20 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 60 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 9 \, {\left (17 \, x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90*(20*3^(2/3)*(-1)^(1/3)*x^5*log((9*3^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 + 3^(2/3)*(-1)^(1/3)*(2*x^3 - 1)
- 9*(x^3 + 1)^(2/3)*x)/(2*x^3 - 1)) - 10*3^(2/3)*(-1)^(1/3)*x^5*log(-(3*3^(2/3)*(-1)^(1/3)*(7*x^4 + x)*(x^3 +
1)^(2/3) - 3^(1/3)*(-1)^(2/3)*(31*x^6 + 23*x^3 + 1) - 9*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1))
- 60*3^(1/6)*(-1)^(1/3)*x^5*arctan(1/3*3^(1/6)*(6*3^(2/3)*(-1)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 1
8*(-1)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - 3^(1/3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 + 23
1*x^6 + 6*x^3 - 1)) - 9*(17*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 (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, 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)*(2*x^6-2*x^3-1)/x^6/(2*x^6+x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 17.01, size = 893, normalized size = 6.82

method	result	size
risch	\(-\frac {17 x^{6}+19 x^{3}+2}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {2 \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}-81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )}{3}-6 \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}-81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )+\frac {2 \RootOf \left (\textit {\_Z}^{3}+9\right ) \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )^{3} x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} x^{3}-15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x +7 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-6 x \left (x^{3}+1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+9\right )+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+9\right )+81 \textit {\_Z}^{2}\right )}{2 x^{3}-1}\right )}{3}\)	\(893\)
trager	\(\text {Expression too large to display}\)	\(1104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(17*x^6+19*x^3+2)/x^5/(x^3+1)^(1/3)-2/3*ln((3*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*RootO
f(_Z^3+9)^3*x^3-81*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x^3+15*(x^3+1)^(2/3
)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x+2*RootOf(_Z^3+9)^2*(x^3+1)^(1/3)*x^2
+63*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2-4*RootOf(_Z^3+9)*x^3
+108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3-21*x*(x^3+1)^(2/3)-RootOf(_Z^3+9)+27*RootOf(Root
Of(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))*RootOf(_Z^3+9)-6*ln((3*RootOf(RootOf(_Z^3+9)^2+9*_Z*Root
Of(_Z^3+9)+81*_Z^2)*RootOf(_Z^3+9)^3*x^3-81*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3
+9)^2*x^3+15*(x^3+1)^(2/3)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x+2*RootOf(_Z
^3+9)^2*(x^3+1)^(1/3)*x^2+63*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)
*x^2-4*RootOf(_Z^3+9)*x^3+108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3-21*x*(x^3+1)^(2/3)-Root
Of(_Z^3+9)+27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))*RootOf(RootOf(_Z^3+9)^2+9*_Z*Ro
otOf(_Z^3+9)+81*_Z^2)+2/3*RootOf(_Z^3+9)*ln((3*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*RootOf(_Z^
3+9)^3*x^3+108*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x^3-15*(x^3+1)^(2/3)*Ro
otOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x+7*RootOf(_Z^3+9)^2*(x^3+1)^(1/3)*x^2+18*
(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2+RootOf(_Z^3+9)*x^3+36*Ro
otOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3-6*x*(x^3+1)^(2/3)+RootOf(_Z^3+9)+36*RootOf(RootOf(_Z^3+
9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))

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

[Out]

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

[Out]

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

[Out]

Timed out

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