Integrand size = 31, antiderivative size = 31 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6} \left (1-2 x^6+9 x^8+x^{12}\right )}{9 x^9} \] Output:
2/9*(x^6-1)^(1/4)*(x^12+9*x^8-2*x^6+1)/x^9
Time = 6.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6} \left (1-2 x^6+9 x^8+x^{12}\right )}{9 x^9} \] Input:
Integrate[((2 + x^6)*(1 - 2*x^6 + x^8 + x^12))/(x^10*(-1 + x^6)^(3/4)),x]
Output:
(2*(-1 + x^6)^(1/4)*(1 - 2*x^6 + 9*x^8 + x^12))/(9*x^9)
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2372, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^6+2\right ) \left (x^{12}+x^8-2 x^6+1\right )}{x^{10} \left (x^6-1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 2372 |
\(\displaystyle \int \left (\frac {x^{18}-3 x^6+2}{x^{10} \left (x^6-1\right )^{3/4}}+\frac {x^{12}+2 x^6}{x^8 \left (x^6-1\right )^{3/4}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt [4]{x^6-1}}{x}+\frac {2 \left (x^6-1\right )^{9/4}}{9 x^9}\) |
Input:
Int[((2 + x^6)*(1 - 2*x^6 + x^8 + x^12))/(x^10*(-1 + x^6)^(3/4)),x]
Output:
(2*(-1 + x^6)^(1/4))/x + (2*(-1 + x^6)^(9/4))/(9*x^9)
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{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)]
Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90
method | result | size |
trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}} \left (x^{12}+9 x^{8}-2 x^{6}+1\right )}{9 x^{9}}\) | \(28\) |
pseudoelliptic | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}} \left (x^{12}+9 x^{8}-2 x^{6}+1\right )}{9 x^{9}}\) | \(28\) |
risch | \(\frac {\frac {2}{9} x^{18}-\frac {2}{3} x^{12}+\frac {2}{3} x^{6}-\frac {2}{9}+2 x^{14}-2 x^{8}}{\left (x^{6}-1\right )^{\frac {3}{4}} x^{9}}\) | \(38\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{12}+9 x^{8}-2 x^{6}+1\right )}{9 x^{9} \left (x^{6}-1\right )^{\frac {3}{4}}}\) | \(48\) |
orering | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{12}+9 x^{8}-2 x^{6}+1\right )}{9 x^{9} \left (x^{6}-1\right )^{\frac {3}{4}}}\) | \(48\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{9} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], x^{6}\right )}{9 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x^{3}}-\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {3}{4}\right ], \left [-\frac {1}{2}\right ], x^{6}\right )}{9 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x^{9}}-\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\) | \(161\) |
Input:
int((x^6+2)*(x^12+x^8-2*x^6+1)/x^10/(x^6-1)^(3/4),x,method=_RETURNVERBOSE)
Output:
2/9*(x^6-1)^(1/4)*(x^12+9*x^8-2*x^6+1)/x^9
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{12} + 9 \, x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{9 \, x^{9}} \] Input:
integrate((x^6+2)*(x^12+x^8-2*x^6+1)/x^10/(x^6-1)^(3/4),x, algorithm="fric as")
Output:
2/9*(x^12 + 9*x^8 - 2*x^6 + 1)*(x^6 - 1)^(1/4)/x^9
Result contains complex when optimal does not.
Time = 3.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.48 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {x^{9} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {x^{6}} \right )}}{9} + \frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} - \frac {e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{x^{3}} + \frac {2 e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{9 x^{9}} \] Input:
integrate((x**6+2)*(x**12+x**8-2*x**6+1)/x**10/(x**6-1)**(3/4),x)
Output:
x**9*exp(-3*I*pi/4)*hyper((3/4, 3/2), (5/2,), x**6)/9 + 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)*hyper((-1/6, 3/4), (5/6,), x**6)/(3*x*gamma(5/6)) - exp(I*pi /4)*hyper((-1/2, 3/4), (1/2,), x**6)/x**3 + 2*exp(I*pi/4)*hyper((-3/2, 3/4 ), (-1/2,), x**6)/(9*x**9)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{18} + 9 \, x^{14} - 3 \, x^{12} - 9 \, x^{8} + 3 \, x^{6} - 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}} x^{9}} \] Input:
integrate((x^6+2)*(x^12+x^8-2*x^6+1)/x^10/(x^6-1)^(3/4),x, algorithm="maxi ma")
Output:
2/9*(x^18 + 9*x^14 - 3*x^12 - 9*x^8 + 3*x^6 - 1)/((x^2 + x + 1)^(3/4)*(x^2 - x + 1)^(3/4)*(x + 1)^(3/4)*(x - 1)^(3/4)*x^9)
\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\int { \frac {{\left (x^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{10}} \,d x } \] Input:
integrate((x^6+2)*(x^12+x^8-2*x^6+1)/x^10/(x^6-1)^(3/4),x, algorithm="giac ")
Output:
integrate((x^12 + x^8 - 2*x^6 + 1)*(x^6 + 2)/((x^6 - 1)^(3/4)*x^10), x)
Time = 7.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\frac {2\,{\left (x^6-1\right )}^{1/4}}{x}-\frac {4\,{\left (x^6-1\right )}^{1/4}}{9\,x^3}+\frac {2\,x^3\,{\left (x^6-1\right )}^{1/4}}{9}+\frac {2\,{\left (x^6-1\right )}^{1/4}}{9\,x^9} \] Input:
int(((x^6 + 2)*(x^8 - 2*x^6 + x^12 + 1))/(x^10*(x^6 - 1)^(3/4)),x)
Output:
(2*(x^6 - 1)^(1/4))/x - (4*(x^6 - 1)^(1/4))/(9*x^3) + (2*x^3*(x^6 - 1)^(1/ 4))/9 + (2*(x^6 - 1)^(1/4))/(9*x^9)
\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx=\int \frac {x^{8}}{\left (x^{6}-1\right )^{\frac {3}{4}}}d x +\int \frac {x^{4}}{\left (x^{6}-1\right )^{\frac {3}{4}}}d x +2 \left (\int \frac {1}{\left (x^{6}-1\right )^{\frac {3}{4}} x^{10}}d x \right )-3 \left (\int \frac {1}{\left (x^{6}-1\right )^{\frac {3}{4}} x^{4}}d x \right )+2 \left (\int \frac {1}{\left (x^{6}-1\right )^{\frac {3}{4}} x^{2}}d x \right ) \] Input:
int((x^6+2)*(x^12+x^8-2*x^6+1)/x^10/(x^6-1)^(3/4),x)
Output:
int(x**8/(x**6 - 1)**(3/4),x) + int(x**4/(x**6 - 1)**(3/4),x) + 2*int(1/(( x**6 - 1)**(3/4)*x**10),x) - 3*int(1/((x**6 - 1)**(3/4)*x**4),x) + 2*int(1 /((x**6 - 1)**(3/4)*x**2),x)