\(\int \frac {(-1+x^6)^{2/3} (1+x^6)}{x^3 (-1-x^3+x^6)} \, dx\) [1307]

Optimal result

Integrand size = 30, antiderivative size = 94 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[Out]

1/2*(x^6-1)^(2/3)/x^2-1/3*arctan(3^(1/2)*x/(x+2*(x^6-1)^(1/3)))*3^(1/2)+1/3*ln(-x+(x^6-1)^(1/3))-1/6*ln(x^2+x*
(x^6-1)^(1/3)+(x^6-1)^(2/3))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.53 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.33, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6860, 281, 372, 371, 1452, 441, 440, 476, 525, 524} \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 \left (1-x^6\right )^{2/3} x^2} \]

[In]

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

[Out]

-(((1 - Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, -2/3, 1, 7/6, x^6, (2*x^6)/(3 - Sqrt[5])])/((3 - Sqrt[5])*(1
- x^6)^(2/3))) - ((1 + Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, 1, -2/3, 7/6, (2*x^6)/(3 + Sqrt[5]), x^6])/(
(3 + Sqrt[5])*(1 - x^6)^(2/3)) - (x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 - Sqrt[5])]
)/(2*(3 - Sqrt[5])*(1 - x^6)^(2/3)) - (x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 + Sqrt
[5])])/(2*(3 + Sqrt[5])*(1 - x^6)^(2/3)) + ((-1 + x^6)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, x^6])/(2*x^2*(
1 - x^6)^(2/3))

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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

Rule 440

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 441

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 476

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

Rule 524

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

Rule 525

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

Rule 1452

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 6860

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*} \text {integral}& = \int \left (-\frac {\left (-1+x^6\right )^{2/3}}{x^3}+\frac {\left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx \\ & = -\int \frac {\left (-1+x^6\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\int \left (\frac {2 \left (-1+x^6\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {2 \left (-1+x^6\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx \\ & = 2 \int \frac {\left (-1+x^6\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx+2 \int \frac {\left (-1+x^6\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx-\frac {\left (-1+x^6\right )^{2/3} \text {Subst}\left (\int \frac {\left (1-x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )}{2 \left (1-x^6\right )^{2/3}} \\ & = \frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+2 \int \left (\frac {\left (-1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6}\right ) \, dx+2 \int \left (\frac {\left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6}\right ) \, dx \\ & = \frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+2 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6} \, dx+2 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx+\left (-1-\sqrt {5}\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx+\left (1-\sqrt {5}\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx \\ & = \frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {\left (\left (-1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}+\frac {\left (\left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}+\text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right ) \\ & = -\frac {\left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {\left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3}}{2 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 27.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{2}}\)	\(95\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (\frac {27960018709626208768001196672 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}-9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}-5614428583490269128672312324 x^{6}-220185147338306394048009423792 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-45456256378905085721353124616 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +127282845030980663272778333724 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-88051906839210946445401966260 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+6818882387672964795952100759 x \left (x^{6}-1\right )^{\frac {2}{3}}+3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6327371895679509652948161508 x^{3}-27960018709626208768001196672 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+5614428583490269128672312324}{x^{6}-x^{3}-1}\right )}{3}+4 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {102663836955250635495722282496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{6}+78258464307289633232711371152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{6}+1723230319777309741777851531 x^{6}-808477716022598754528812974656 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-45456256378905085721353124616 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -81826588652075577551425209108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-9793442531921313212690595108 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-10606903752581721939398194477 x \left (x^{6}-1\right )^{\frac {2}{3}}+3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+194166796594626449777786088 x^{3}-102663836955250635495722282496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-78258464307289633232711371152 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-1723230319777309741777851531}{x^{6}-x^{3}-1}\right )\)	\(406\)
risch	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}-1}\right )}{3}-\frac {\ln \left (-\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{x^{6}-x^{3}-1}\right )}{3}-\ln \left (-\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-2 x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{x^{6}-x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(406\)

[In]

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

[Out]

1/6*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^6-1)^(1/3)))*x^2+2*ln((-x+(x^6-1)^(1/3))/x)*x^2-ln((x^2+x*(x^6-1)^
(1/3)+(x^6-1)^(2/3))/x^2)*x^2+3*(x^6-1)^(2/3))/x^2

Fricas [A] (verification not implemented)

none

Time = 9.82 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\right )}}\right ) - x^{2} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - x^{3} - 1}\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]

[In]

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

[Out]

-1/6*(2*sqrt(3)*x^2*arctan(1/3*(473996388635948633452428917614298985996886224511260115036680453514888144148250
*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 19325031480489228255674265966448835967818926087643600184123099965366515892788*s
qrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(77122577980774102085597780297263121642836874020275522160397193158871803614
4*x^6 + 245889484278411189833195613987401279765924206559249102388797804808538611984375*x^3 - 77122577980774102
0855977802972631216428368740202755221603971931588718036144))/(154075137855386652020330175695521646369068967401
49986002803824712402669144*x^6 - 22735108609151524126357935884149462717917055610854840741228148059947321679687
5*x^3 - 15407513785538665202033017569552164636906896740149986002803824712402669144)) - x^2*log((x^6 - x^3 + 3*
(x^6 - 1)^(1/3)*x^2 - 3*(x^6 - 1)^(2/3)*x - 1)/(x^6 - x^3 - 1)) - 3*(x^6 - 1)^(2/3))/x^2

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right )}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )}{x^3\,\left (-x^6+x^3+1\right )} \,d x \]

[In]

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

[Out]

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