\(\int \frac {(-1+x^3)^{2/3} (1-5 x^3+4 x^6)}{x^6 (-1+2 x^3)^2} \, dx\) [1756]

Optimal result

Integrand size = 34, antiderivative size = 118 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (6-33 x^3+62 x^6\right )}{30 x^5 \left (-1+2 x^3\right )}+\frac {7 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {7}{9} \log \left (x+\sqrt [3]{-1+x^3}\right )-\frac {7}{18} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/30*(x^3-1)^(2/3)*(62*x^6-33*x^3+6)/x^5/(2*x^3-1)+7/9*arctan(3^(1/2)*x/(-x+2*(x^3-1)^(1/3)))*3^(1/2)+7/9*ln(x
+(x^3-1)^(1/3))-7/18*ln(x^2-x*(x^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6874, 270, 283, 245, 386, 384, 399} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=-\frac {7 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2 x \left (x^3-1\right )^{2/3}}{3 \left (1-2 x^3\right )}-\frac {7}{18} \log \left (2 x^3-1\right )+\frac {7}{6} \log \left (-\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \]

[In]

Int[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]

[Out]

(-1 + x^3)^(2/3)/(2*x^2) - (2*x*(-1 + x^3)^(2/3))/(3*(1 - 2*x^3)) + (-1 + x^3)^(5/3)/(5*x^5) - (7*ArcTan[(1 -
(2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - (7*Log[-1 + 2*x^3])/18 + (7*Log[-x - (-1 + x^3)^(1/3)])/6

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 386

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

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{x^3}-\frac {2 \left (-1+x^3\right )^{2/3}}{\left (-1+2 x^3\right )^2}+\frac {2 \left (-1+x^3\right )^{2/3}}{-1+2 x^3}\right ) \, dx \\ & = -\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+2 x^3\right )^2} \, dx\right )+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-\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx-\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = \frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {7 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {7}{18} \log \left (-1+2 x^3\right )+\frac {7}{6} \log \left (-x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\frac {1}{90} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (6-33 x^3+62 x^6\right )}{x^5 \left (-1+2 x^3\right )}-70 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+70 \log \left (x+\sqrt [3]{-1+x^3}\right )-35 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^3)^(2/3)*(1 - 5*x^3 + 4*x^6))/(x^6*(-1 + 2*x^3)^2),x]

[Out]

((3*(-1 + x^3)^(2/3)*(6 - 33*x^3 + 62*x^6))/(x^5*(-1 + 2*x^3)) - 70*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^
3)^(1/3))] + 70*Log[x + (-1 + x^3)^(1/3)] - 35*Log[x^2 - x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/90

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(\frac {\left (140 x^{8}-70 x^{5}\right ) \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (186 x^{6}-99 x^{3}+18\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+35 x^{5} \left (2 x^{3}-1\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{90 \left (x +\left (x^{3}-1\right )^{\frac {1}{3}}\right ) \left (\left (x^{3}-1\right )^{\frac {2}{3}}+x \left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right ) x^{5}}\)	\(153\)
risch	\(\frac {62 x^{9}-95 x^{6}+39 x^{3}-6}{30 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}} \left (2 x^{3}-1\right )}+\frac {7 \ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1}{2 x^{3}-1}\right )}{9}+\frac {7 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}-1}\right )}{3}\)	\(274\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (62 x^{6}-33 x^{3}+6\right )}{30 x^{5} \left (2 x^{3}-1\right )}+\frac {7 \ln \left (-\frac {65382681600 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-73232 x^{3}-523061452800 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-212378400 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )+64078}{2 x^{3}-1}\right )}{9}+1120 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \ln \left (\frac {-18981734400 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -606368160 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}+106800 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-220717 x^{3}[phone] \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )^{2}-350055360 \operatorname {RootOf}\left (2073600 \textit {\_Z}^{2}+1440 \textit {\_Z} +1\right )-31531}{2 x^{3}-1}\right )\)	\(343\)

[In]

int((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x,method=_RETURNVERBOSE)

[Out]

1/90*((140*x^8-70*x^5)*ln((x+(x^3-1)^(1/3))/x)+(186*x^6-99*x^3+18)*(x^3-1)^(2/3)+35*x^5*(2*x^3-1)*(2*3^(1/2)*a
rctan(1/3*(x-2*(x^3-1)^(1/3))*3^(1/2)/x)-ln((x^2-x*(x^3-1)^(1/3)+(x^3-1)^(2/3))/x^2)))/(x+(x^3-1)^(1/3))/((x^3
-1)^(2/3)+x*(x-(x^3-1)^(1/3)))/x^5

Fricas [A] (verification not implemented)

none

Time = 0.60 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=-\frac {70 \, \sqrt {3} {\left (2 \, x^{8} - x^{5}\right )} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 35 \, {\left (2 \, x^{8} - x^{5}\right )} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (62 \, x^{6} - 33 \, x^{3} + 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (2 \, x^{8} - x^{5}\right )}} \]

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="fricas")

[Out]

-1/90*(70*sqrt(3)*(2*x^8 - x^5)*arctan((4*sqrt(3)*(x^3 - 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*
(x^3 - 1))/(7*x^3 + 1)) - 35*(2*x^8 - x^5)*log((2*x^3 + 3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x - 1)/(2*x^
3 - 1)) - 3*(62*x^6 - 33*x^3 + 6)*(x^3 - 1)^(2/3))/(2*x^8 - x^5)

Sympy [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (4 x^{3} - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \left (2 x^{3} - 1\right )^{2}}\, dx \]

[In]

integrate((x**3-1)**(2/3)*(4*x**6-5*x**3+1)/x**6/(2*x**3-1)**2,x)

[Out]

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

Maxima [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int { \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="maxima")

[Out]

integrate((4*x^6 - 5*x^3 + 1)*(x^3 - 1)^(2/3)/((2*x^3 - 1)^2*x^6), x)

Giac [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int { \frac {{\left (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )}^{2} x^{6}} \,d x } \]

[In]

integrate((x^3-1)^(2/3)*(4*x^6-5*x^3+1)/x^6/(2*x^3-1)^2,x, algorithm="giac")

[Out]

integrate((4*x^6 - 5*x^3 + 1)*(x^3 - 1)^(2/3)/((2*x^3 - 1)^2*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (4\,x^6-5\,x^3+1\right )}{x^6\,{\left (2\,x^3-1\right )}^2} \,d x \]

[In]

int(((x^3 - 1)^(2/3)*(4*x^6 - 5*x^3 + 1))/(x^6*(2*x^3 - 1)^2),x)

[Out]

int(((x^3 - 1)^(2/3)*(4*x^6 - 5*x^3 + 1))/(x^6*(2*x^3 - 1)^2), x)