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

Optimal. Leaf size=142 \[ -2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^6+1}+2 x\right )-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+1}-x}\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^6+1} x-\sqrt [3]{2} \left (x^6+1\right )^{2/3}-2 x^2\right )}{\sqrt [3]{2}}+\frac {\left (x^6+1\right )^{2/3} \left (2 x^6-15 x^3+2\right )}{10 x^5} \]

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Rubi [F]  time = 0.93, 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^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Hypergeometric2F1[-5/6, -2/3, 1/6, -x^6]/(5*x^5) - (3*Hypergeometric2F1[-2/3, -1/3, 2/3, -x^6])/(2*x^2) + x*Hy
pergeometric2F1[-2/3, 1/6, 7/6, -x^6] - 2*Defer[Int][(1 + x^6)^(2/3)/(1 + x), x] + 2*(1 + I*Sqrt[3])*Defer[Int
][(1 + x^6)^(2/3)/(-1 - I*Sqrt[3] + 2*x), x] + 2*(1 - I*Sqrt[3])*Defer[Int][(1 + x^6)^(2/3)/(-1 + I*Sqrt[3] +
2*x), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx &=\int \left (\left (1+x^6\right )^{2/3}-\frac {\left (1+x^6\right )^{2/3}}{x^6}+\frac {3 \left (1+x^6\right )^{2/3}}{x^3}-\frac {2 \left (1+x^6\right )^{2/3}}{1+x}+\frac {2 (-2+x) \left (1+x^6\right )^{2/3}}{1-x+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\left (1+x^6\right )^{2/3}}{1+x} \, dx\right )+2 \int \frac {(-2+x) \left (1+x^6\right )^{2/3}}{1-x+x^2} \, dx+3 \int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx+\int \left (1+x^6\right )^{2/3} \, dx-\int \frac {\left (1+x^6\right )^{2/3}}{x^6} \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )-2 \int \frac {\left (1+x^6\right )^{2/3}}{1+x} \, dx+2 \int \left (\frac {\left (1+i \sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}-\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )-2 \int \frac {\left (1+x^6\right )^{2/3}}{1+x} \, dx+\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.41, size = 142, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^6\right )^{2/3} \left (2-15 x^3+2 x^6\right )}{10 x^5}-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )-2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^6)^(2/3)*(2 - 15*x^3 + 2*x^6))/(10*x^5) - 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^6)^(
1/3))] - 2^(2/3)*Log[2*x + 2^(2/3)*(1 + x^6)^(1/3)] + Log[-2*x^2 + 2^(2/3)*x*(1 + x^6)^(1/3) - 2^(1/3)*(1 + x^
6)^(2/3)]/2^(1/3)

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fricas [B]  time = 51.28, size = 344, normalized size = 2.42 \begin {gather*} \frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (x^{13} - 2 \, x^{10} - 6 \, x^{7} - 2 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{14} - 14 \, x^{11} + 6 \, x^{8} - 14 \, x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 51 \, x^{12} - 52 \, x^{9} + 51 \, x^{6} - 30 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 93 \, x^{12} + 20 \, x^{9} - 93 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 5 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{7} - 4 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \left (-4\right )^{\frac {2}{3}} {\left (x^{12} - 14 \, x^{9} + 6 \, x^{6} - 14 \, x^{3} + 1\right )} + 24 \, {\left (x^{8} - x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 3 \, {\left (2 \, x^{6} - 15 \, x^{3} + 2\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(10*sqrt(3)*(-4)^(1/3)*x^5*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(x^13 - 2*x^10 - 6*x^7 - 2*x^4 + x)*(x^6 + 1)
^(2/3) + 6*sqrt(3)*(-4)^(1/3)*(x^14 - 14*x^11 + 6*x^8 - 14*x^5 + x^2)*(x^6 + 1)^(1/3) - sqrt(3)*(x^18 - 30*x^1
5 + 51*x^12 - 52*x^9 + 51*x^6 - 30*x^3 + 1))/(x^18 + 6*x^15 - 93*x^12 + 20*x^9 - 93*x^6 + 6*x^3 + 1)) + 10*(-4
)^(1/3)*x^5*log((3*(-4)^(2/3)*(x^6 + 1)^(1/3)*x^2 + 6*(x^6 + 1)^(2/3)*x - (-4)^(1/3)*(x^6 + 2*x^3 + 1))/(x^6 +
2*x^3 + 1)) - 5*(-4)^(1/3)*x^5*log((6*(-4)^(1/3)*(x^7 - 4*x^4 + x)*(x^6 + 1)^(2/3) + (-4)^(2/3)*(x^12 - 14*x^
9 + 6*x^6 - 14*x^3 + 1) + 24*(x^8 - x^5 + x^2)*(x^6 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1)) + 3*(2*x^6
- 15*x^3 + 2)*(x^6 + 1)^(2/3))/x^5

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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 )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} \left (x^{6}-x^{3}+1\right )}{x^{6} \left (x^{3}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^3-1\right )\,{\left (x^6+1\right )}^{2/3}\,\left (x^6-x^3+1\right )}{x^6\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (x^{2} + x + 1\right ) \left (x^{6} - x^{3} + 1\right )}{x^{6} \left (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**3-1)*(x**6+1)**(2/3)*(x**6-x**3+1)/x**6/(x**3+1),x)

[Out]

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

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