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]

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

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Rubi [C]  time = 0.01, 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*}

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Mathematica [A]  time = 0.01, 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]

-1/5*(1 - x^6)^(5/3)/x^5

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fricas [A]  time = 0.51, size = 19, normalized size = 0.54 \[ \frac {{\left (x^{6} - 1\right )} {\left (-x^{6} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

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

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)

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maple [A]  time = 0.01, size = 35, normalized size = 1.00 \[ \frac {\left (-x^{6}+1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right ) \left (x +1\right ) \left (x -1\right )}{5 x^{5}} \]

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+1)/x^5*(x-1)

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maxima [A]  time = 2.15, size = 38, normalized size = 1.09 \[ \frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x^{2} + x - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

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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mupad [B]  time = 3.59, size = 14, normalized size = 0.40 \[ -\frac {{\left (1-x^6\right )}^{5/3}}{5\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1 - x^6)^(5/3)/(5*x^5)

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sympy [C]  time = 1.14, size = 68, normalized size = 1.94 \[ \frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {2}{3} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} \]

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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