Integrand size = 28, antiderivative size = 38 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (1+x^6\right )^{3/4} \left (7-11 x^4+14 x^6-11 x^{10}+7 x^{12}\right )}{77 x^{11}} \] Output:
2/77*(x^6+1)^(3/4)*(7*x^12-11*x^10+14*x^6-11*x^4+7)/x^11
Time = 2.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (1+x^6\right )^{7/4} \left (7-11 x^4+7 x^6\right )}{77 x^{11}} \] Input:
Integrate[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]
Output:
(2*(1 + x^6)^(7/4)*(7 - 11*x^4 + 7*x^6))/(77*x^11)
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
\(\displaystyle \int \frac {\left (x^6-2\right ) \left (x^6+1\right )^{3/4} \left (x^6-x^4+1\right )}{x^{12}} \, dx\) |
\(\Big \downarrow \) 2372 |
\(\displaystyle \int \left (\frac {\left (x^6+1\right )^{3/4} \left (x^{12}-x^6-2\right )}{x^{12}}+\frac {\left (x^6+1\right )^{3/4} \left (2 x^3-x^9\right )}{x^{11}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (x^6+1\right )^{11/4}}{11 x^{11}}-\frac {2 \left (x^6+1\right )^{7/4}}{7 x^7}\) |
Input:
Int[((-2 + x^6)*(1 + x^6)^(3/4)*(1 - x^4 + x^6))/x^12,x]
Output:
(-2*(1 + x^6)^(7/4))/(7*x^7) + (2*(1 + x^6)^(11/4))/(11*x^11)
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 = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {2 \left (x^{6}+1\right )^{\frac {7}{4}} \left (7 x^{6}-11 x^{4}+7\right )}{77 x^{11}}\) | \(25\) |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (7 x^{12}-11 x^{10}+14 x^{6}-11 x^{4}+7\right )}{77 x^{11}}\) | \(35\) |
gosper | \(\frac {2 \left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right ) \left (7 x^{6}-11 x^{4}+7\right ) \left (x^{6}+1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(40\) |
orering | \(\frac {2 \left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right ) \left (7 x^{6}-11 x^{4}+7\right ) \left (x^{6}+1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(40\) |
risch | \(\frac {\frac {2}{11} x^{18}+\frac {6}{11} x^{12}+\frac {6}{11} x^{6}+\frac {2}{11}-\frac {2}{7} x^{16}-\frac {4}{7} x^{10}-\frac {2}{7} x^{4}}{x^{11} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(45\) |
meijerg | \(\frac {\operatorname {hypergeom}\left (\left [-\frac {5}{6}, -\frac {3}{4}\right ], \left [\frac {1}{6}\right ], -x^{6}\right )}{5 x^{5}}-\frac {2 \operatorname {hypergeom}\left (\left [-\frac {7}{6}, -\frac {3}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}+\frac {2 \operatorname {hypergeom}\left (\left [-\frac {11}{6}, -\frac {3}{4}\right ], \left [-\frac {5}{6}\right ], -x^{6}\right )}{11 x^{11}}+x \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{6}\right ], \left [\frac {7}{6}\right ], -x^{6}\right )+\frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{6}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}\) | \(78\) |
Input:
int((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x,method=_RETURNVERBOSE)
Output:
2/77*(x^6+1)^(7/4)*(7*x^6-11*x^4+7)/x^11
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} + 14 \, x^{6} - 11 \, x^{4} + 7\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \] Input:
integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="fricas")
Output:
2/77*(7*x^12 - 11*x^10 + 14*x^6 - 11*x^4 + 7)*(x^6 + 1)^(3/4)/x^11
Result contains complex when optimal does not.
Time = 3.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 4.66 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} - \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} + \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {3}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} - \frac {\Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, - \frac {3}{4} \\ - \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{11} \Gamma \left (- \frac {5}{6}\right )} \] Input:
integrate((x**6-2)*(x**6+1)**(3/4)*(x**6-x**4+1)/x**12,x)
Output:
x*gamma(1/6)*hyper((-3/4, 1/6), (7/6,), x**6*exp_polar(I*pi))/(6*gamma(7/6 )) - gamma(-1/6)*hyper((-3/4, -1/6), (5/6,), x**6*exp_polar(I*pi))/(6*x*ga mma(5/6)) - gamma(-5/6)*hyper((-5/6, -3/4), (1/6,), x**6*exp_polar(I*pi))/ (6*x**5*gamma(1/6)) + gamma(-7/6)*hyper((-7/6, -3/4), (-1/6,), x**6*exp_po lar(I*pi))/(3*x**7*gamma(-1/6)) - gamma(-11/6)*hyper((-11/6, -3/4), (-5/6, ), x**6*exp_polar(I*pi))/(3*x**11*gamma(-5/6))
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} + 14 \, x^{6} - 11 \, x^{4} + 7\right )} {\left (x^{4} - x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \] Input:
integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="maxima")
Output:
2/77*(7*x^12 - 11*x^10 + 14*x^6 - 11*x^4 + 7)*(x^4 - x^2 + 1)^(3/4)*(x^2 + 1)^(3/4)/x^11
\[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{6} - 2\right )}}{x^{12}} \,d x } \] Input:
integrate((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x, algorithm="giac")
Output:
integrate((x^6 - x^4 + 1)*(x^6 + 1)^(3/4)*(x^6 - 2)/x^12, x)
Time = 7.77 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2\,x\,{\left (x^6+1\right )}^{3/4}}{11}-\frac {2\,{\left (x^6+1\right )}^{3/4}}{7\,x}+\frac {4\,{\left (x^6+1\right )}^{3/4}}{11\,x^5}-\frac {2\,{\left (x^6+1\right )}^{3/4}}{7\,x^7}+\frac {2\,{\left (x^6+1\right )}^{3/4}}{11\,x^{11}} \] Input:
int(((x^6 + 1)^(3/4)*(x^6 - 2)*(x^6 - x^4 + 1))/x^12,x)
Output:
(2*x*(x^6 + 1)^(3/4))/11 - (2*(x^6 + 1)^(3/4))/(7*x) + (4*(x^6 + 1)^(3/4)) /(11*x^5) - (2*(x^6 + 1)^(3/4))/(7*x^7) + (2*(x^6 + 1)^(3/4))/(11*x^11)
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4} \left (1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (7 x^{12}-11 x^{10}+14 x^{6}-11 x^{4}+7\right )}{77 x^{11}} \] Input:
int((x^6-2)*(x^6+1)^(3/4)*(x^6-x^4+1)/x^12,x)
Output:
(2*(x**6 + 1)**(3/4)*(7*x**12 - 11*x**10 + 14*x**6 - 11*x**4 + 7))/(77*x** 11)