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

Optimal. Leaf size=119 \[ \frac {1}{3} \log \left (\sqrt [3]{x^6-x^3-1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^3-1}-x}\right )}{\sqrt {3}}+\frac {\left (x^6-x^3-1\right )^{2/3}}{2 x^2}-\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6-x^3-1} x+\left (x^6-x^3-1\right )^{2/3}\right ) \]

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Rubi [F]  time = 1.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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} F_1\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*}

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Mathematica [F]  time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1-x^3+x^6\right )^{2/3}}{x^3 \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 119, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\sqrt [3]{x^6-x^3-1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^3-1}-x}\right )}{\sqrt {3}}+\frac {\left (x^6-x^3-1\right )^{2/3}}{2 x^2}-\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6-x^3-1} x+\left (x^6-x^3-1\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 17.56, size = 149, normalized size = 1.25 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.84, size = 573, normalized size = 4.82 \begin {gather*} \frac {\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+9 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x +\left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (x^{2}-x +1\right )}\right )-\frac {\ln \left (-\frac {-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}-9 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+3 \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 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}-9 \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}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 \left (x^{6}-x^{3}-1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+3 \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 ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^6-x^3-1)^(2/3)/x^2+RootOf(9*_Z^2+3*_Z+1)*ln((-3*RootOf(9*_Z^2+3*_Z+1)*x^6+9*RootOf(9*_Z^2+3*_Z+1)^2*x^3
+3*(x^6-x^3-1)^(2/3)*RootOf(9*_Z^2+3*_Z+1)*x-3*RootOf(9*_Z^2+3*_Z+1)*(x^6-x^3-1)^(1/3)*x^2+6*RootOf(9*_Z^2+3*_
Z+1)*x^3-(x^6-x^3-1)^(2/3)*x+(x^6-x^3-1)^(1/3)*x^2+3*RootOf(9*_Z^2+3*_Z+1))/(-1+x)/(x^2+x+1)/(1+x)/(x^2-x+1))-
1/3*ln(-(-3*RootOf(9*_Z^2+3*_Z+1)*x^6-x^6-9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+3*(x^6-x^3-1)^(2/3)*RootOf(9*_Z^2+3*_Z
+1)*x-3*RootOf(9*_Z^2+3*_Z+1)*(x^6-x^3-1)^(1/3)*x^2+2*(x^6-x^3-1)^(2/3)*x-2*(x^6-x^3-1)^(1/3)*x^2+x^3+3*RootOf
(9*_Z^2+3*_Z+1)+1)/(-1+x)/(x^2+x+1)/(1+x)/(x^2-x+1))-ln(-(-3*RootOf(9*_Z^2+3*_Z+1)*x^6-x^6-9*RootOf(9*_Z^2+3*_
Z+1)^2*x^3+3*(x^6-x^3-1)^(2/3)*RootOf(9*_Z^2+3*_Z+1)*x-3*RootOf(9*_Z^2+3*_Z+1)*(x^6-x^3-1)^(1/3)*x^2+2*(x^6-x^
3-1)^(2/3)*x-2*(x^6-x^3-1)^(1/3)*x^2+x^3+3*RootOf(9*_Z^2+3*_Z+1)+1)/(-1+x)/(x^2+x+1)/(1+x)/(x^2-x+1))*RootOf(9
*_Z^2+3*_Z+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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