\(\int ((1-x^6)^{2/3}+\frac {(1-x^6)^{2/3}}{x^6}) \, dx\) [941]

Optimal result

Integrand size = 27, antiderivative size = 35 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {\left (1-x^6\right )^{2/3}}{5 x^5}+\frac {1}{5} x \left (1-x^6\right )^{2/3} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {251, 371} \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5} \]

[In]

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

[Out]

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

Rule 251

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.51 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {\left (1-x^6\right )^{5/3}}{5 x^5} \]

[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

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57

method	result	size
trager	\(\frac {\left (x^{6}-1\right ) \left (-x^{6}+1\right )^{\frac {2}{3}}}{5 x^{5}}\)	\(20\)
risch	\(-\frac {x^{12}-2 x^{6}+1}{5 x^{5} \left (-x^{6}+1\right )^{\frac {1}{3}}}\)	\(25\)
meijerg	\(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};x^{6}\right )-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};x^{6}\right )}{5 x^{5}}\)	\(27\)
gosper	\(\frac {\left (-x^{6}+1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right ) \left (x +1\right ) \left (x -1\right )}{5 x^{5}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {{\left (x^{6} - 1\right )} {\left (-x^{6} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.94 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {2}{3} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{2 i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (-x^{2} + x - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

[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

Giac [F]

\[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=\int { {\left (-x^{6} + 1\right )}^{\frac {2}{3}} + \frac {{\left (-x^{6} + 1\right )}^{\frac {2}{3}}}{x^{6}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 19.88 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.40 \[ \int \left (\left (1-x^6\right )^{2/3}+\frac {\left (1-x^6\right )^{2/3}}{x^6}\right ) \, dx=-\frac {{\left (1-x^6\right )}^{5/3}}{5\,x^5} \]

[In]

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

[Out]

-(1 - x^6)^(5/3)/(5*x^5)