\(\int \frac {(2+x^6) (-1-x^4+x^6)}{x^6 (-1+x^6)^{3/4}} \, dx\) [291]

Optimal result

Integrand size = 28, antiderivative size = 26 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6} \left (-1-5 x^4+x^6\right )}{5 x^5} \]

[Out]

2/5*(x^6-1)^(1/4)*(x^6-5*x^4-1)/x^5

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1847, 1598, 460, 1492} \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \left (x^6-1\right )^{5/4}}{5 x^5}-\frac {2 \sqrt [4]{x^6-1}}{x} \]

[In]

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

[Out]

(-2*(-1 + x^6)^(1/4))/x + (2*(-1 + x^6)^(5/4))/(5*x^5)

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]

Rule 1492

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Sym
bol] :> Int[(f*x)^m*(d + e*x^n)^(q + p)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-2 x^3-x^9}{x^5 \left (-1+x^6\right )^{3/4}}+\frac {-2+x^6+x^{12}}{x^6 \left (-1+x^6\right )^{3/4}}\right ) \, dx \\ & = \int \frac {-2 x^3-x^9}{x^5 \left (-1+x^6\right )^{3/4}} \, dx+\int \frac {-2+x^6+x^{12}}{x^6 \left (-1+x^6\right )^{3/4}} \, dx \\ & = \int \frac {-2-x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx+\int \frac {\sqrt [4]{-1+x^6} \left (2+x^6\right )}{x^6} \, dx \\ & = -\frac {2 \sqrt [4]{-1+x^6}}{x}+\frac {2 \left (-1+x^6\right )^{5/4}}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6} \left (-1-5 x^4+x^6\right )}{5 x^5} \]

[In]

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

[Out]

(2*(-1 + x^6)^(1/4)*(-1 - 5*x^4 + x^6))/(5*x^5)

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}-1\right )}{5 x^{5}}\)	\(23\)
pseudoelliptic	\(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}-1\right )}{5 x^{5}}\)	\(23\)
risch	\(\frac {\frac {2}{5} x^{12}-\frac {4}{5} x^{6}+\frac {2}{5}-2 x^{10}+2 x^{4}}{x^{5} \left (x^{6}-1\right )^{\frac {3}{4}}}\)	\(33\)
gosper	\(\frac {2 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}-5 x^{4}-1\right )}{5 x^{5} \left (x^{6}-1\right )^{\frac {3}{4}}}\)	\(43\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{7} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {7}{6}\right ], \left [\frac {13}{6}\right ], x^{6}\right )}{7 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {7}{6}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {5}{6}, \frac {3}{4}\right ], \left [\frac {1}{6}\right ], x^{6}\right )}{5 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x^{5}}+\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\)	\(159\)

[In]

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

[Out]

2/5*(x^6-1)^(1/4)*(x^6-5*x^4-1)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{6} - 5 \, x^{4} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]

[In]

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

[Out]

2/5*(x^6 - 5*x^4 - 1)*(x^6 - 1)^(1/4)/x^5

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.46 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {x^{7} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {13}{6}\right )} - \frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{4} \\ \frac {7}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} + \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{5} \Gamma \left (\frac {1}{6}\right )} \]

[In]

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

[Out]

x**7*exp(-3*I*pi/4)*gamma(7/6)*hyper((3/4, 7/6), (13/6,), x**6)/(6*gamma(13/6)) - x**5*exp(-3*I*pi/4)*gamma(5/
6)*hyper((3/4, 5/6), (11/6,), x**6)/(6*gamma(11/6)) + x*exp(-3*I*pi/4)*gamma(1/6)*hyper((1/6, 3/4), (7/6,), x*
*6)/(6*gamma(7/6)) + exp(I*pi/4)*gamma(-1/6)*hyper((-1/6, 3/4), (5/6,), x**6)/(3*x*gamma(5/6)) + exp(I*pi/4)*g
amma(-5/6)*hyper((-5/6, 3/4), (1/6,), x**6)/(3*x**5*gamma(1/6))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{12} - 5 \, x^{10} - 2 \, x^{6} + 5 \, x^{4} + 1\right )}}{5 \, {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}} x^{5}} \]

[In]

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

[Out]

2/5*(x^12 - 5*x^10 - 2*x^6 + 5*x^4 + 1)/((x^2 + x + 1)^(3/4)*(x^2 - x + 1)^(3/4)*(x + 1)^(3/4)*(x - 1)^(3/4)*x
^5)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^6 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2\,{\left (x^6-1\right )}^{5/4}-10\,x^4\,{\left (x^6-1\right )}^{1/4}}{5\,x^5} \]

[In]

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

[Out]

(2*(x^6 - 1)^(5/4) - 10*x^4*(x^6 - 1)^(1/4))/(5*x^5)