\(\int \frac {(1+x^3)^{2/3} (-1+2 x^6)}{x^6 (-1+2 x^3)} \, dx\) [2509]

Optimal result

Integrand size = 29, antiderivative size = 209 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+2 x^6\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-1-6 x^3\right ) \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {\log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{3}} \]

[Out]

1/5*(-6*x^3-1)*(x^3+1)^(2/3)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))*3^(1/2)+3^(1/6)*arctan(3^(5/6)*x/(3
^(1/3)*x+2*(x^3+1)^(1/3)))-1/3*ln(-x+(x^3+1)^(1/3))-1/3*ln(-3*x+3^(2/3)*(x^3+1)^(1/3))*3^(2/3)+1/6*ln(x^2+x*(x
^3+1)^(1/3)+(x^3+1)^(2/3))+1/6*ln(3*x^2+3^(2/3)*x*(x^3+1)^(1/3)+3^(1/3)*(x^3+1)^(2/3))*3^(2/3)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6857, 270, 283, 245, 399, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+2 x^6\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {\log \left (2 x^3-1\right )}{2 \sqrt [3]{3}}-\frac {1}{2} 3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {\left (x^3+1\right )^{2/3}}{x^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+2 x^6\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-1-6 x^3\right ) \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {\log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{3}} \]

[In]

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

[Out]

((-1 - 6*x^3)*(1 + x^3)^(2/3))/(5*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + 3^(1/6)*ArcTan[
(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - Log[-x + (1 + x^3)^(1/3)]/3 - Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3
)]/3^(1/3) + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6 + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3
)*(1 + x^3)^(2/3)]/(2*3^(1/3))

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {-10 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \,3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-30 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-10 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-36 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\)	\(207\)

[In]

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

[Out]

1/30*(-10*3^(2/3)*ln((-3^(1/3)*x+(x^3+1)^(1/3))/x)*x^5+5*3^(2/3)*ln((3^(2/3)*x^2+3^(1/3)*(x^3+1)^(1/3)*x+(x^3+
1)^(2/3))/x^2)*x^5-10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3+1)^(1/3)))*x^5-30*3^(1/6)*arctan(1/9*3^(1/2)*(2*3
^(2/3)*(x^3+1)^(1/3)+3*x)/x)*x^5-10*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^5-36*x^3*(x^3+1)^(2/3)-6*(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. 382 vs. \(2 (164) = 328\).

Time = 18.43 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.83 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1+2 x^6\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 5 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 30 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) + 30 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 15 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) - 18 \, {\left (6 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]

[In]

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

[Out]

1/90*(10*3^(2/3)*(-1)^(1/3)*x^5*log((9*3^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 + 3^(2/3)*(-1)^(1/3)*(2*x^3 - 1)
- 9*(x^3 + 1)^(2/3)*x)/(2*x^3 - 1)) - 5*3^(2/3)*(-1)^(1/3)*x^5*log(-(3*3^(2/3)*(-1)^(1/3)*(7*x^4 + x)*(x^3 +
1)^(2/3) - 3^(1/3)*(-1)^(2/3)*(31*x^6 + 23*x^3 + 1) - 9*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1))
- 30*3^(1/6)*(-1)^(1/3)*x^5*arctan(1/3*3^(1/6)*(6*3^(2/3)*(-1)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 18
*(-1)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - 3^(1/3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 + 231
*x^6 + 6*x^3 - 1)) + 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)) - 15*x^5*log(3*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x +
1) - 18*(6*x^3 + 1)*(x^3 + 1)^(2/3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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