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

Optimal. Leaf size=35 \[ \frac{1}{5} x \left (1-x^6\right )^{2/3}-\frac{\left (1-x^6\right )^{2/3}}{5 x^5} \]

[Out]

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

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Rubi [C]  time = 0.0111609, antiderivative size = 36, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {245, 364} \[ x \, _2F_1\left (-\frac{2}{3},\frac{1}{6};\frac{7}{6};x^6\right )-\frac{\, _2F_1\left (-\frac{5}{6},-\frac{2}{3};\frac{1}{6};x^6\right )}{5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Hypergeometric2F1[-5/6, -2/3, 1/6, x^6]/(5*x^5) + x*Hypergeometric2F1[-2/3, 1/6, 7/6, x^6]

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.007662, size = 18, normalized size = 0.51 \[ -\frac{\left (1-x^6\right )^{5/3}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - x^6)^(5/3)/(5*x^5)

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Maple [A]  time = 0.008, size = 35, normalized size = 1. \begin{align*}{\frac{ \left ( x-1 \right ) \left ( 1+x \right ) \left ({x}^{2}+x+1 \right ) \left ({x}^{2}-x+1 \right ) }{5\,{x}^{5}} \left ( -{x}^{6}+1 \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.68922, size = 51, normalized size = 1.46 \begin{align*} \frac{{\left (x^{6} - 1\right )}{\left (x^{2} + x + 1\right )}^{\frac{2}{3}}{\left (-x^{2} + x - 1\right )}^{\frac{2}{3}}{\left (x + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}^{\frac{2}{3}}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6479, size = 49, normalized size = 1.4 \begin{align*} \frac{{\left (x^{6} - 1\right )}{\left (-x^{6} + 1\right )}^{\frac{2}{3}}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(x^6 - 1)*(-x^6 + 1)^(2/3)/x^5

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Sympy [C]  time = 1.15054, size = 68, normalized size = 1.94 \begin{align*} \frac{x \Gamma \left (\frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{6} \\ \frac{7}{6} \end{matrix}\middle |{x^{6} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac{7}{6}\right )} + \frac{\Gamma \left (- \frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, - \frac{2}{3} \\ \frac{1}{6} \end{matrix}\middle |{x^{6} e^{2 i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac{1}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*gamma(1/6)*hyper((-2/3, 1/6), (7/6,), x**6*exp_polar(2*I*pi))/(6*gamma(7/6)) + gamma(-5/6)*hyper((-5/6, -2/3
), (1/6,), x**6*exp_polar(2*I*pi))/(6*x**5*gamma(1/6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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