\(\int \frac {2+x^6}{x^2 (-1+x^6)^{3/4}} \, dx\) [62]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6}}{x} \]

[Out]

2*(x^6-1)^(1/4)/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{x^6-1}}{x} \]

[In]

Int[(2 + x^6)/(x^2*(-1 + x^6)^(3/4)),x]

[Out]

(2*(-1 + x^6)^(1/4))/x

Rule 460

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt [4]{-1+x^6}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6}}{x} \]

[In]

Integrate[(2 + x^6)/(x^2*(-1 + x^6)^(3/4)),x]

[Out]

(2*(-1 + x^6)^(1/4))/x

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\)	\(13\)
risch	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\)	\(13\)
pseudoelliptic	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\)	\(13\)
gosper	\(\frac {2 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{x \left (x^{6}-1\right )^{\frac {3}{4}}}\)	\(33\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}-\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\)	\(66\)

[In]

int((x^6+2)/x^2/(x^6-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

2*(x^6-1)^(1/4)/x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{x} \]

[In]

integrate((x^6+2)/x^2/(x^6-1)^(3/4),x, algorithm="fricas")

[Out]

2*(x^6 - 1)^(1/4)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} \]

[In]

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

[Out]

x**5*exp(-3*I*pi/4)*gamma(5/6)*hyper((3/4, 5/6), (11/6,), x**6)/(6*gamma(11/6)) - exp(I*pi/4)*gamma(-1/6)*hype
r((-1/6, 3/4), (5/6,), x**6)/(3*x*gamma(5/6))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \]

[In]

integrate((x^6+2)/x^2/(x^6-1)^(3/4),x, algorithm="maxima")

[Out]

2*(x^2 + x + 1)^(1/4)*(x^2 - x + 1)^(1/4)*(x + 1)^(1/4)*(x - 1)^(1/4)/x

Giac [F]

\[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\int { \frac {x^{6} + 2}{{\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^6+2)/x^2/(x^6-1)^(3/4),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2\,{\left (x^6-1\right )}^{1/4}}{x} \]

[In]

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

[Out]

(2*(x^6 - 1)^(1/4))/x