3.20.60 \(\int \frac {(-1+x^3)^{2/3} (1+x^3+x^6)}{x^6 (-1+x^6)} \, dx\) [1960]

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

[Out]

1/10*(x^3-1)^(2/3)*(3*x^3+2)/x^5+1/6*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3-1)^(1/3)))*2^(2/3)*3^(1/2)-1/6*ln(-2*x+2
^(2/3)*(x^3-1)^(1/3))*2^(2/3)+1/12*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 [B] Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(138)=276\).
time = 0.51, antiderivative size = 407, normalized size of antiderivative = 2.95, number of steps used = 22, number of rules used = 16, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1600, 6857, 277, 270, 2174, 2183, 384, 502, 206, 31, 648, 631, 210, 642, 455, 58} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {2^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{18 \sqrt [3]{2}}+\frac {\log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )}{9 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )}{18 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{6 \sqrt [3]{2}}+\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 )}{18 \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]]/(3*2^(1/3)*Sqrt[3]) - ArcTan[(1 + (2*2^(1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sqrt[3]]/(3*2^(1/3)*Sqrt[3]) + (
2^(2/3)*ArcTan[(1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ArcTan[(1 - 2^(2/3)*(-1 + x^3)^(1/
3))/Sqrt[3]]/(3*2^(1/3)*Sqrt[3]) + Log[(1 - x)*(1 + x)^2]/(18*2^(1/3)) + Log[1 + x^3]/(18*2^(1/3)) + Log[1 - (
2^(1/3)*(1 - x))/(-1 + x^3)^(1/3)]/(9*2^(1/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(-1 + x^3)^(2/3) + (2^(1/3)*(1 -
x))/(-1 + x^3)^(1/3)]/(18*2^(1/3)) - Log[2^(1/3)*x - (-1 + x^3)^(1/3)]/(3*2^(1/3)) + Log[2^(1/3) + (-1 + x^3)^
(1/3)]/(6*2^(1/3)) - Log[1 - x + 2^(2/3)*(-1 + x^3)^(1/3)]/(6*2^(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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

Rule 206

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 270

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 277

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 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 455

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

Rule 502

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[-q^2/(3
*d), Int[1/((1 - q*x)*(a + b*x^3)^(1/3)), x], x] + Dist[q/d, Subst[Int[1/(1 + 2*a*x^3), x], x, (1 + q*x)/(a +
b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

Rule 631

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 642

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 648

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 1600

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 2174

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 2183

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/c^q, Int[E
xpandIntegrand[(c^3 - d^3*x^3)^q*(a + b*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&
PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6857

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+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 [A]
time = 0.31, 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 {\text {ArcTan}\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]

Integrate[((-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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 9.77, size = 914, normalized size = 6.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(914\)
trager	\(\text {Expression too large to display}\)	\(1137\)

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*(3*x^6-x^3-2)/x^5/(x^3-1)^(1/3)+1/6*RootOf(_Z^3+4)*ln(-(54*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36
*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*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+6*(x^3-1)^(1/3)*RootOf(Ro
otOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)*x^2+5*(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^2+18*RootOf(
RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3+RootOf(_Z^3+4)*x^3-2*x*(x^3-1)^(2/3)-18*RootOf(RootOf(_Z^3+4
)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)-RootOf(_Z^3+4))/(1+x)/(x^2-x+1))-1/6*ln((36*RootOf(RootOf(_Z^3+4)^2+6*_Z*Root
Of(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3-3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+
4)^3*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-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-(x^3-1)^(1/3)*RootOf(_Z^3+4)^2*x^
2-36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3+3*RootOf(_Z^3+4)*x^3+10*x*(x^3-1)^(2/3)+12*RootO
f(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)-RootOf(_Z^3+4))/(1+x)/(x^2-x+1))*RootOf(_Z^3+4)-ln((36*RootOf(
RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3-3*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^
3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3-12*(x^3-1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootO
f(_Z^3+4)^2*x-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-(x^3-1)
^(1/3)*RootOf(_Z^3+4)^2*x^2-36*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3+3*RootOf(_Z^3+4)*x^3+1
0*x*(x^3-1)^(2/3)+12*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)-RootOf(_Z^3+4))/(1+x)/(x^2-x+1))*Roo
tOf(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+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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (105) = 210\).
time = 1.79, 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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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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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+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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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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