\(\int \frac {(-1+x^3)^{2/3} (2+x^3)}{x^6 (4+x^6)} \, dx\) [1614]

Optimal result

Integrand size = 25, antiderivative size = 110 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.55, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (2-i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2-i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (2+i)^{2/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{2+i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\left (\frac {1}{48}-\frac {i}{48}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-x^3+2 i\right )+\left (\frac {1}{48}+\frac {i}{48}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (x^3+2 i\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1-\frac {i}{2}} x\right )-\left (\frac {1}{16}-\frac {i}{16}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1+\frac {i}{2}} x\right )+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \]

[In]

Int[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \]

[In]

Integrate[((-1 + x^3)^(2/3)*(2 + x^3))/(x^6*(4 + x^6)),x]

[Out]

((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[5 - 8*#1^3 + 4*#1^6 & , (-5*Log[x] + 5*Log[(-1 + x^3)^(1/3) -
x*#1] + 6*Log[x]*#1^3 - 6*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/96

Maple [N/A] (verified)

Time = 183.59 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(\frac {-5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-8 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\left (6 \textit {\_R}^{3}-5\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{4}-\textit {\_R}}\right ) x^{5}-12 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}-48 \left (x^{3}-1\right )^{\frac {2}{3}}}{480 x^{5}}\)	\(83\)
risch	\(\text {Expression too large to display}\)	\(7713\)
trager	\(\text {Expression too large to display}\)	\(10429\)

[In]

int((x^3-1)^(2/3)*(x^3+2)/x^6/(x^6+4),x,method=_RETURNVERBOSE)

[Out]

1/480*(-5*sum((6*_R^3-5)*ln((-_R*x+(x^3-1)^(1/3))/x)/(_R^4-_R),_R=RootOf(4*_Z^6-8*_Z^3+5))*x^5-12*x^3*(x^3-1)^
(2/3)-48*(x^3-1)^(2/3))/x^5

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((x^3-1)^(2/3)*(x^3+2)/x^6/(x^6+4),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )}{x^{6} \left (x^{6} + 4\right )}\, dx \]

[In]

integrate((x**3-1)**(2/3)*(x**3+2)/x**6/(x**6+4),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**3 + 2)/(x**6*(x**6 + 4)), x)

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^3+2)/x^6/(x^6+4),x, algorithm="maxima")

[Out]

integrate((x^3 + 2)*(x^3 - 1)^(2/3)/((x^6 + 4)*x^6), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^3+2)/x^6/(x^6+4),x, algorithm="giac")

[Out]

integrate((x^3 + 2)*(x^3 - 1)^(2/3)/((x^6 + 4)*x^6), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^6\,\left (x^6+4\right )} \,d x \]

[In]

int(((x^3 - 1)^(2/3)*(x^3 + 2))/(x^6*(x^6 + 4)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^3 + 2))/(x^6*(x^6 + 4)), x)