\(\int \frac {(-1+x^3)^{2/3} (2+x^6)}{x^6} \, dx\) [1578]

Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-6+6 x^3+5 x^6\right )}{15 x^5}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{9} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/15*(x^3-1)^(2/3)*(5*x^6+6*x^3-6)/x^5-2/9*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3)))*3^(1/2)+2/9*ln(-x+(x^3-1)^(1/
3))-1/9*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1501, 462, 283, 245} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=-\frac {2 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{5/3}}{3 x^2}+\frac {\left (x^3-1\right )^{2/3}}{3 x^2} \]

[In]

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

[Out]

(-1 + x^3)^(2/3)/(3*x^2) + (2*(-1 + x^3)^(5/3))/(5*x^5) + (-1 + x^3)^(5/3)/(3*x^2) - (2*ArcTan[(1 + (2*x)/(-1
+ x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + Log[-x + (-1 + x^3)^(1/3)]/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 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 462

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

Rule 1501

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[c^p*
(f*x)^(m + 2*n*p - n + 1)*((d + e*x^n)^(q + 1)/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1))), x] + Dist[1/(e*(m
+ 2*n*p + n*q + 1)), Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + c*x^(2*n))^p - c^p*x
^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2,
2*n] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] &&  !IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=\frac {1}{45} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (-6+6 x^3+5 x^6\right )}{x^5}-10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+10 \log \left (-x+\sqrt [3]{-1+x^3}\right )-5 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((3*(-1 + x^3)^(2/3)*(-6 + 6*x^3 + 5*x^6))/x^5 - 10*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] + 10*
Log[-x + (-1 + x^3)^(1/3)] - 5*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/45

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.61 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {5 x^{9}+x^{6}-12 x^{3}+6}{15 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(59\)
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}-\frac {2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{5}}\)	\(63\)
pseudoelliptic	\(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (15 x^{6}+18 x^{3}-18\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{45 x^{5} \left (\left (x^{3}-1\right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right ) \left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}\)	\(140\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (5 x^{6}+6 x^{3}-6\right )}{15 x^{5}}+\frac {2 \ln \left (-69686010880 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-50451363776 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}[phone] x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-262015609 x^{3}[phone] \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )[phone]\right )}{9}-\frac {2 \ln \left (-759808 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2} x^{3}-310272 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -222816 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+509344 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) x^{3}+6963 x \left (x^{3}-1\right )^{\frac {2}{3}}-16659 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+8954 x^{3}+6078464 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2}-62016 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )-2849\right )}{9}-\frac {64 \ln \left (-759808 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2} x^{3}-310272 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -222816 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+509344 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right ) x^{3}+6963 x \left (x^{3}-1\right )^{\frac {2}{3}}-16659 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+8954 x^{3}+6078464 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )^{2}-62016 \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )-2849\right ) \operatorname {RootOf}\left (1024 \textit {\_Z}^{2}+32 \textit {\_Z} +1\right )}{9}\)	\(451\)

[In]

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

[Out]

1/15*(5*x^9+x^6-12*x^3+6)/x^5/(x^3-1)^(1/3)-2/3/signum(x^3-1)^(1/3)*(-signum(x^3-1))^(1/3)*x*hypergeom([1/3,1/
3],[4/3],x^3)

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 5 \, x^{5} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (5 \, x^{6} + 6 \, x^{3} - 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{45 \, x^{5}} \]

[In]

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

[Out]

-1/45*(10*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)) - 5*x^5*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) - 3*(5*x^6
+ 6*x^3 - 6)*(x^3 - 1)^(2/3))/x^5

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=- \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

-x*exp(-I*pi/3)*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), x**3)/(3*gamma(4/3)) + 2*Piecewise(((-1 + x**(-3))**(2/3
)*exp(-I*pi/3)*gamma(-5/3)/(3*gamma(-2/3)) - (-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-5/3)/(3*x**3*gamma(-2/3
)), 1/Abs(x**3) > 1), (-(1 - 1/x**3)**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) + (1 - 1/x**3)**(2/3)*gamma(-5/3)/(3*x
**3*gamma(-2/3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {2 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]

[In]

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

[Out]

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - 1/3*(x^3 - 1)^(2/3)/(x^2*((x^3 - 1)/x^3 - 1)) + 2/
5*(x^3 - 1)^(5/3)/x^5 - 1/9*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/3)/x^2 + 1) + 2/9*log((x^3 - 1)^(1/3)/x - 1)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.72 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx=\frac {x\,{\left (x^3-1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right )}{{\left (1-x^3\right )}^{2/3}}-\frac {2\,{\left (x^3-1\right )}^{2/3}-2\,x^3\,{\left (x^3-1\right )}^{2/3}}{5\,x^5} \]

[In]

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

[Out]

(x*(x^3 - 1)^(2/3)*hypergeom([-2/3, 1/3], 4/3, x^3))/(1 - x^3)^(2/3) - (2*(x^3 - 1)^(2/3) - 2*x^3*(x^3 - 1)^(2
/3))/(5*x^5)