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

Optimal. Leaf size=131 \[ \frac {\sqrt [3]{x^6+1}}{x}-\frac {1}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^6+1}+2 x\right )+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+1}-x}\right )}{\sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^6+1} x-\sqrt [3]{2} \left (x^6+1\right )^{2/3}-2 x^2\right )}{3\ 2^{2/3}} \]

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Rubi [F]  time = 0.59, 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 ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.66, size = 131, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^6}}{x}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \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 )}{3\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 33.06, size = 327, normalized size = 2.50 \begin {gather*} \frac {2 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} \left (-2\right )^{\frac {2}{3}} {\left (x^{14} - 14 \, x^{11} + 6 \, x^{8} - 14 \, x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} {\left (x^{13} - 2 \, x^{10} - 6 \, x^{7} - 2 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{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 ) + 2 \, \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {6 \, \left (-2\right )^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - \left (-2\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x^{3} + 1}\right ) - \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {3 \, \left (-2\right )^{\frac {2}{3}} {\left (x^{7} - 4 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \left (-2\right )^{\frac {1}{3}} {\left (x^{12} - 14 \, x^{9} + 6 \, x^{6} - 14 \, x^{3} + 1\right )} - 12 \, {\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 ) + 18 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{18 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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