\(\int \frac {(-1+x^3)^{2/3} (8+2 x^3+x^6)}{x^6 (-2+x^3)} \, dx\) [2547]

Optimal result

Integrand size = 30, antiderivative size = 214 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (8+7 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {2}{3} \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/10*(x^3-1)^(2/3)*(7*x^3+8)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)-2/3*3^(1/2)*arctan(3^(1/2)*
x/(x+2*2^(1/3)*(x^3-1)^(1/3)))*2^(1/3)-1/3*ln(-x+(x^3-1)^(1/3))+2/3*ln(-x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)+1/6*l
n(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))-1/3*ln(x^2+2^(1/3)*x*(x^3-1)^(1/3)+2^(2/3)*(x^3-1)^(2/3))*2^(1/3)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\frac {4 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (x^3-2\right )+\sqrt [3]{2} \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {4 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{2 x^2} \]

[In]

Int[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]

[Out]

(3*(-1 + x^3)^(2/3))/(2*x^2) - (4*(-1 + x^3)^(5/3))/(5*x^5) + (4*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])
/Sqrt[3] - Sqrt[3]*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - (2*2^(1/3)*ArcTan[(1 + (2^(2/3)*x)/(-1 + x^3
)^(1/3))/Sqrt[3]])/Sqrt[3] - (2^(1/3)*Log[-2 + x^3])/3 + 2^(1/3)*Log[x/2^(1/3) - (-1 + x^3)^(1/3)] - Log[-x +
(-1 + x^3)^(1/3)]/2

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {1}{30} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (8+7 x^3\right )}{x^5}+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-20 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-10 \log \left (-x+\sqrt [3]{-1+x^3}\right )+20 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )+5 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-10 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]

[Out]

((3*(-1 + x^3)^(2/3)*(8 + 7*x^3))/x^5 + 10*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] - 20*2^(1/3)*S
qrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))] - 10*Log[-x + (-1 + x^3)^(1/3)] + 20*2^(1/3)*Log[-
x + 2^(1/3)*(-1 + x^3)^(1/3)] + 5*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)] - 10*2^(1/3)*Log[x^2 + 2^(1
/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/30

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {20 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+20 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )-10 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-10 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )-10 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+21 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+24 \left (x^{3}-1\right )^{\frac {2}{3}}}{30 x^{5}}\)	\(222\)

[In]

int((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x,method=_RETURNVERBOSE)

[Out]

1/30*(20*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3-1)^(1/3)))*x^5-10*3^(1/2)*arctan(1/3*3^(1/2)/x
*(x+2*(x^3-1)^(1/3)))*x^5+20*2^(1/3)*x^5*ln((-2^(2/3)*x+2*(x^3-1)^(1/3))/x)-10*2^(1/3)*x^5*ln((2^(2/3)*x*(x^3-
1)^(1/3)+2^(1/3)*x^2+2*(x^3-1)^(2/3))/x^2)-10*2^(1/3)*x^5*ln(2)-10*ln((-x+(x^3-1)^(1/3))/x)*x^5+5*ln((x^2+x*(x
^3-1)^(1/3)+(x^3-1)^(2/3))/x^2)*x^5+21*x^3*(x^3-1)^(2/3)+24*(x^3-1)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (165) = 330\).

Time = 14.28 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.68 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {12 \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )}}{3 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 30 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) + 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{3} - 2}\right ) - 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 15 \, x^{5} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 9 \, {\left (7 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="fricas")

[Out]

1/90*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(12*sqrt(3)*2^(2/3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 6*sqrt(3)*
2^(1/3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3) + sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8))/(53*x^9 - 48*x^6
- 12*x^3 + 8)) + 30*sqrt(3)*x^5*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 - 1)^(2/3)*x +
sqrt(3)*(5831*x^3 - 7200))/(58653*x^3 - 8000)) + 20*2^(1/3)*x^5*log(-(3*2^(2/3)*(x^3 - 1)^(1/3)*x^2 - 6*(x^3
- 1)^(2/3)*x + 2^(1/3)*(x^3 - 2))/(x^3 - 2)) - 10*2^(1/3)*x^5*log((12*2^(1/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 2^
(2/3)*(19*x^6 - 22*x^3 + 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 15*x^5*log(-3*(x^3 - 1)^
(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) + 9*(7*x^3 + 8)*(x^3 - 1)^(2/3))/x^5

Sympy [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} + 8\right )}{x^{6} \left (x^{3} - 2\right )}\, dx \]

[In]

integrate((x**3-1)**(2/3)*(x**6+2*x**3+8)/x**6/(x**3-2),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + 2*x**3 + 8)/(x**6*(x**3 - 2)), x)

Maxima [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="maxima")

[Out]

integrate((x^6 + 2*x^3 + 8)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^6), x)

Giac [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="giac")

[Out]

integrate((x^6 + 2*x^3 + 8)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3+8\right )}{x^6\,\left (x^3-2\right )} \,d x \]

[In]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 + 8))/(x^6*(x^3 - 2)),x)

[Out]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 + 8))/(x^6*(x^3 - 2)), x)