\(\int \frac {(-1+x^3) (1+x^3)^{2/3}}{x^3 (-1-x^3+x^6)} \, dx\) [1398]

Optimal result

Integrand size = 30, antiderivative size = 100 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(100)=200\).

Time = 0.34 (sec) , antiderivative size = 505, normalized size of antiderivative = 5.05, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (3+\sqrt {5}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\left (3-\sqrt {5}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{\sqrt {5}-1} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (2 x^3-\sqrt {5}-1\right )}{6 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (2 x^3+\sqrt {5}-1\right )}{6 \sqrt {5}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (-\sqrt [3]{x^3+1}-\sqrt [3]{\frac {1}{2} \left (\sqrt {5}-1\right )} x\right )}{2 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt {5}}+\frac {1}{20} \left (5+3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{20} \left (5-3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ((3 - Sqrt[5])*ArcTan[(1 + (2
*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2*Sqrt[15]) - ((3 + Sqrt[5])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2*S
qrt[15]) + (((7 + 3*Sqrt[5])/2)^(1/3)*ArcTan[(1 - (2^(2/3)*(-1 + Sqrt[5])^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/
Sqrt[15] - (((7 - 3*Sqrt[5])/2)^(1/3)*ArcTan[(1 + (2^(2/3)*(1 + Sqrt[5])^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/S
qrt[15] - (((7 - 3*Sqrt[5])/2)^(1/3)*Log[-1 - Sqrt[5] + 2*x^3])/(6*Sqrt[5]) + (((7 + 3*Sqrt[5])/2)^(1/3)*Log[-
1 + Sqrt[5] + 2*x^3])/(6*Sqrt[5]) - (((7 + 3*Sqrt[5])/2)^(1/3)*Log[-(((-1 + Sqrt[5])/2)^(1/3)*x) - (1 + x^3)^(
1/3)])/(2*Sqrt[5]) + (((7 - 3*Sqrt[5])/2)^(1/3)*Log[((1 + Sqrt[5])/2)^(1/3)*x - (1 + x^3)^(1/3)])/(2*Sqrt[5])
- Log[-x + (1 + x^3)^(1/3)]/2 + ((5 - 3*Sqrt[5])*Log[-x + (1 + x^3)^(1/3)])/20 + ((5 + 3*Sqrt[5])*Log[-x + (1
+ x^3)^(1/3)])/20

Rule 245

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 283

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 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 399

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

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^3\right )^{2/3}}{x^3}+\frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx \\ & = \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (-1+\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {\left (-1-\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \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 )+\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \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 )+\frac {2 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1-\sqrt {5}+2 x^3\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+\sqrt {5}+2 x^3\right )} \, dx}{\sqrt {5}}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (3-\sqrt {5}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {15}}-\frac {\left (3+\sqrt {5}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {2 \sqrt [3]{2} \arctan \left (\frac {1-\frac {2^{2/3} \sqrt [3]{-1+\sqrt {5}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {15} \left (-1+\sqrt {5}\right )^{4/3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {15} \left (1+\sqrt {5}\right )^{4/3}}-\frac {\sqrt [3]{2} \log \left (-1-\sqrt {5}+2 x^3\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right )^{4/3}}+\frac {\sqrt [3]{2} \log \left (-1+\sqrt {5}+2 x^3\right )}{3 \sqrt {5} \left (-1+\sqrt {5}\right )^{4/3}}-\frac {\sqrt [3]{2} \log \left (-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x-\sqrt [3]{1+x^3}\right )}{\sqrt {5} \left (-1+\sqrt {5}\right )^{4/3}}+\frac {\sqrt [3]{2} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt [3]{1+x^3}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )^{4/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{20} \left (5-3 \sqrt {5}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{20} \left (5+3 \sqrt {5}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 - RootSum[-1 - #1^3 + #1^6 & , (-Log[x] + Log[(1 + x^3)^(1/3) - x*#1] + Log[x]*#1^3 -
Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + 2*#1^4) & ]/3

Maple [N/A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{4}-\textit {\_R}}\right ) x^{2}-3 \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{2}}\)	\(69\)
risch	\(\text {Expression too large to display}\)	\(9436\)
trager	\(\text {Expression too large to display}\)	\(14259\)

[In]

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

[Out]

1/6*(2*sum((_R^3-1)*ln((-_R*x+(x^3+1)^(1/3))/x)/(2*_R^4-_R),_R=RootOf(_Z^6-_Z^3-1))*x^2-3*(x^3+1)^(2/3))/x^2

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.30 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.30 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx=-\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^{2/3}}{x^3\,\left (-x^6+x^3+1\right )} \,d x \]

[In]

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

[Out]

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