\(\int \frac {(1+x^3)^{2/3} (1-2 x^3+x^6)}{x^6 (-2+x^6)} \, dx\) [1595]

Optimal result

Integrand size = 30, antiderivative size = 109 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\frac {\left (1-4 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{24} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(480\) vs. \(2(109)=218\).

Time = 0.54 (sec) , antiderivative size = 480, normalized size of antiderivative = 4.40, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {28, 6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=-\frac {\left (3+2 \sqrt {2}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{4 \sqrt {6}}+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{4 \sqrt {6}}+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{24+17 \sqrt {2}} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{2-\sqrt {2}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {\sqrt [3]{17 \sqrt {2}-24} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{2+\sqrt {2}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {1}{48} \sqrt [3]{17 \sqrt {2}-24} \log \left (\sqrt {2}-x^3\right )+\frac {1}{48} \sqrt [3]{24+17 \sqrt {2}} \log \left (x^3+\sqrt {2}\right )-\frac {1}{16} \sqrt [3]{24+17 \sqrt {2}} \log \left (\sqrt [3]{\frac {1}{2} \left (2-\sqrt {2}\right )} x-\sqrt [3]{x^3+1}\right )+\frac {1}{16} \sqrt [3]{17 \sqrt {2}-24} \log \left (\sqrt [3]{\frac {1}{2} \left (2+\sqrt {2}\right )} x-\sqrt [3]{x^3+1}\right )+\frac {1}{16} \left (4+3 \sqrt {2}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{16} \left (4-3 \sqrt {2}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]

[In]

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

[Out]

-1/2*(1 + x^3)^(2/3)/x^2 + (1 + x^3)^(5/3)/(10*x^5) + ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ((
3 - 2*Sqrt[2])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(4*Sqrt[6]) - ((3 + 2*Sqrt[2])*ArcTan[(1 + (2*x)/(
1 + x^3)^(1/3))/Sqrt[3]])/(4*Sqrt[6]) + ((24 + 17*Sqrt[2])^(1/3)*ArcTan[(1 + (2^(2/3)*(2 - Sqrt[2])^(1/3)*x)/(
1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[3]) - ((-24 + 17*Sqrt[2])^(1/3)*ArcTan[(1 + (2^(2/3)*(2 + Sqrt[2])^(1/3)*x)/
(1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[3]) - ((-24 + 17*Sqrt[2])^(1/3)*Log[Sqrt[2] - x^3])/48 + ((24 + 17*Sqrt[2])
^(1/3)*Log[Sqrt[2] + x^3])/48 - ((24 + 17*Sqrt[2])^(1/3)*Log[((2 - Sqrt[2])/2)^(1/3)*x - (1 + x^3)^(1/3)])/16
+ ((-24 + 17*Sqrt[2])^(1/3)*Log[((2 + Sqrt[2])/2)^(1/3)*x - (1 + x^3)^(1/3)])/16 - Log[-x + (1 + x^3)^(1/3)]/2
+ ((4 - 3*Sqrt[2])*Log[-x + (1 + x^3)^(1/3)])/16 + ((4 + 3*Sqrt[2])*Log[-x + (1 + x^3)^(1/3)])/16

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 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 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 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*} \text {integral}& = \int \frac {\left (-1+x^3\right )^2 \left (1+x^3\right )^{2/3}}{x^6 \left (-2+x^6\right )} \, dx \\ & = \int \left (-\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (-2+x^6\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\right )+\frac {1}{2} \int \frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{-2+x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{2} \int \left (-\frac {\left (3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}-x^3\right )}+\frac {\left (-3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}+x^3\right )}\right ) \, dx+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\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}{8} \left (4-3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}-x^3} \, dx-\frac {1}{8} \left (4+3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}+x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\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}{8} \left (-2+\sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {1}{8} \left (2+\sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (\sqrt {2}+x^3\right )} \, dx+\frac {1}{8} \left (-4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {1}{8} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {6}}-\frac {\left (3+2 \sqrt {2}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {6}}+\frac {\sqrt [3]{24+17 \sqrt {2}} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2-\sqrt {2}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {\sqrt [3]{-24+17 \sqrt {2}} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2+\sqrt {2}} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {1}{48} \sqrt [3]{-24+17 \sqrt {2}} \log \left (\sqrt {2}-x^3\right )+\frac {1}{48} \sqrt [3]{24+17 \sqrt {2}} \log \left (\sqrt {2}+x^3\right )-\frac {1}{16} \sqrt [3]{24+17 \sqrt {2}} \log \left (\sqrt [3]{\frac {1}{2} \left (2-\sqrt {2}\right )} x-\sqrt [3]{1+x^3}\right )+\frac {1}{16} \sqrt [3]{-24+17 \sqrt {2}} \log \left (\sqrt [3]{\frac {1}{2} \left (2+\sqrt {2}\right )} x-\sqrt [3]{1+x^3}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{16} \left (4-3 \sqrt {2}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{16} \left (4+3 \sqrt {2}\right ) \log \left (-x+\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\frac {\left (1-4 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{24} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

((1 - 4*x^3)*(1 + x^3)^(2/3))/(10*x^5) + RootSum[1 - 4*#1^3 + 2*#1^6 & , (-2*Log[x] + 2*Log[(1 + x^3)^(1/3) -
x*#1] + Log[x]*#1^3 - Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/24

Maple [N/A] (verified)

Time = 146.94 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}-48 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}+12 \left (x^{3}+1\right )^{\frac {2}{3}}}{120 x^{5}}\)	\(81\)
risch	\(\text {Expression too large to display}\)	\(9183\)
trager	\(\text {Expression too large to display}\)	\(10412\)

[In]

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

[Out]

1/120*(-5*sum((_R^3-2)*ln((-_R*x+(x^3+1)^(1/3))/x)/(_R^4-_R),_R=RootOf(2*_Z^6-4*_Z^3+1))*x^5-48*x^3*(x^3+1)^(2
/3)+12*(x^3+1)^(2/3))/x^5

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 23.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}{x^{6} \left (x^{6} - 2\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-2\,x^3+1\right )}{x^6\,\left (x^6-2\right )} \,d x \]

[In]

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

[Out]

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