\(\int \frac {(1+x^6) (-1-x^3+x^6)^{2/3}}{x^3 (-1+x^6)} \, dx\) [1767]

Optimal result

Integrand size = 30, antiderivative size = 119 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {\left (-1-x^3+x^6\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1-x^3+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1-x^3+x^6}+\left (-1-x^3+x^6\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

((-1 - x^3 + x^6)^(2/3)*AppellF1[-2/3, -2/3, -2/3, 1/3, (2*x^3)/(1 + Sqrt[5]), (2*x^3)/(1 - Sqrt[5])])/(2*x^2*
(1 - (2*x^3)/(1 - Sqrt[5]))^(2/3)*(1 - (2*x^3)/(1 + Sqrt[5]))^(2/3)) + Defer[Int][(-1 - x^3 + x^6)^(2/3)/(-1 +
x), x]/3 + Defer[Int][(-1 - x^3 + x^6)^(2/3)/(1 + x), x]/3 - ((1 + I*Sqrt[3])*Defer[Int][(-1 - x^3 + x^6)^(2/
3)/(-1 - I*Sqrt[3] + 2*x), x])/3 - ((1 - I*Sqrt[3])*Defer[Int][(-1 - x^3 + x^6)^(2/3)/(1 - I*Sqrt[3] + 2*x), x
])/3 - ((1 - I*Sqrt[3])*Defer[Int][(-1 - x^3 + x^6)^(2/3)/(-1 + I*Sqrt[3] + 2*x), x])/3 - ((1 + I*Sqrt[3])*Def
er[Int][(-1 - x^3 + x^6)^(2/3)/(1 + I*Sqrt[3] + 2*x), x])/3

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (-1-x^3+x^6\right )^{2/3}}{x^3}+\frac {2 x \left (-1-x^3+x^6\right )^{2/3}}{3 \left (-1+x^2\right )}+\frac {(2-x) \left (-1-x^3+x^6\right )^{2/3}}{3 \left (1-x+x^2\right )}+\frac {(-2-x) \left (-1-x^3+x^6\right )^{2/3}}{3 \left (1+x+x^2\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {(2-x) \left (-1-x^3+x^6\right )^{2/3}}{1-x+x^2} \, dx+\frac {1}{3} \int \frac {(-2-x) \left (-1-x^3+x^6\right )^{2/3}}{1+x+x^2} \, dx+\frac {2}{3} \int \frac {x \left (-1-x^3+x^6\right )^{2/3}}{-1+x^2} \, dx-\int \frac {\left (-1-x^3+x^6\right )^{2/3}}{x^3} \, dx \\ & = \frac {1}{3} \int \left (\frac {\left (-1-i \sqrt {3}\right ) \left (-1-x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x}+\frac {\left (-1+i \sqrt {3}\right ) \left (-1-x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{3} \int \left (\frac {\left (-1+i \sqrt {3}\right ) \left (-1-x^3+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-i \sqrt {3}\right ) \left (-1-x^3+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {2}{3} \int \left (\frac {\left (-1-x^3+x^6\right )^{2/3}}{2 (-1+x)}+\frac {\left (-1-x^3+x^6\right )^{2/3}}{2 (1+x)}\right ) \, dx-\frac {\left (-1-x^3+x^6\right )^{2/3} \int \frac {\left (1+\frac {2 x^3}{-1-\sqrt {5}}\right )^{2/3} \left (1+\frac {2 x^3}{-1+\sqrt {5}}\right )^{2/3}}{x^3} \, dx}{\left (1+\frac {2 x^3}{-1-\sqrt {5}}\right )^{2/3} \left (1+\frac {2 x^3}{-1+\sqrt {5}}\right )^{2/3}} \\ & = \frac {\left (-1-x^3+x^6\right )^{2/3} \operatorname {AppellF1}\left (-\frac {2}{3},-\frac {2}{3},-\frac {2}{3},\frac {1}{3},\frac {2 x^3}{1+\sqrt {5}},\frac {2 x^3}{1-\sqrt {5}}\right )}{2 x^2 \left (1-\frac {2 x^3}{1-\sqrt {5}}\right )^{2/3} \left (1-\frac {2 x^3}{1+\sqrt {5}}\right )^{2/3}}+\frac {1}{3} \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{-1+x} \, dx+\frac {1}{3} \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{1+x} \, dx+\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (-1+i \sqrt {3}\right ) \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (-1+i \sqrt {3}\right ) \int \frac {\left (-1-x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {1}{6} \left (\frac {3 \left (-1-x^3+x^6\right )^{2/3}}{x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1-x^3+x^6}}\right )+2 \log \left (x+\sqrt [3]{-1-x^3+x^6}\right )-\log \left (x^2-x \sqrt [3]{-1-x^3+x^6}+\left (-1-x^3+x^6\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 14.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}-\ln \left (\frac {x^{2}-x \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}+\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+2 \ln \left (\frac {x +\left (x^{6}-x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{6 x^{2}}\)	\(119\)
risch	\(\frac {\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (-\frac {x^{6}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-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 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x +\left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-x^{3}-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(556\)
trager	\(\text {Expression too large to display}\)	\(710\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 9.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25 \[ \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {37791663946489640698390389259748112672665344841760398436632573406805797258440392514 \, \sqrt {3} {\left (x^{6} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 42616282523552719904247910491772924807300791980535303720609605641285532900565158554 \, \sqrt {3} {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18323047168343312092760155949313307647509257018220563551640555707801529868232673857 \, x^{6} + 2412309288531539602928760616012406067317723569387452641988516117239867821062383020 \, x^{3} - 18323047168343312092760155949313307647509257018220563551640555707801529868232673857\right )}}{71058247355948940593342690344230822422479089551095495524443013398313353987294270891 \, x^{6} - 120611919705063540903957449627281556219949205233443553235863268572136995238508326602 \, x^{3} - 71058247355948940593342690344230822422479089551095495524443013398313353987294270891}\right ) + x^{2} \log \left (\frac {x^{6} + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - 1}\right ) + 3 \, {\left (x^{6} - x^{3} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]

[In]

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

[Out]

1/6*(2*sqrt(3)*x^2*arctan((37791663946489640698390389259748112672665344841760398436632573406805797258440392514
*sqrt(3)*(x^6 - x^3 - 1)^(1/3)*x^2 + 4261628252355271990424791049177292480730079198053530372060960564128553290
0565158554*sqrt(3)*(x^6 - x^3 - 1)^(2/3)*x + sqrt(3)*(18323047168343312092760155949313307647509257018220563551
640555707801529868232673857*x^6 + 2412309288531539602928760616012406067317723569387452641988516117239867821062
383020*x^3 - 18323047168343312092760155949313307647509257018220563551640555707801529868232673857))/(7105824735
5948940593342690344230822422479089551095495524443013398313353987294270891*x^6 - 120611919705063540903957449627
281556219949205233443553235863268572136995238508326602*x^3 - 7105824735594894059334269034423082242247908955109
5495524443013398313353987294270891)) + x^2*log((x^6 + 3*(x^6 - x^3 - 1)^(1/3)*x^2 + 3*(x^6 - x^3 - 1)^(2/3)*x
- 1)/(x^6 - 1)) + 3*(x^6 - x^3 - 1)^(2/3))/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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