Integrand size = 34, antiderivative size = 109 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (2+5 x^4+65 x^8\right )}{9 x^9}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \] Output:
1/9*(x^4-1)^(1/4)*(65*x^8+5*x^4+2)/x^9+2*2^(1/2)*arctan(2^(1/2)*x*(x^4-1)^ (1/4)/(-x^2+(x^4-1)^(1/2)))-2*2^(1/2)*arctanh(2^(1/2)*x*(x^4-1)^(1/4)/(x^2 +(x^4-1)^(1/2)))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (2+5 x^4+65 x^8\right )}{9 x^9}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \] Input:
Integrate[((-1 + x^4)^(1/4)*(2 - x^4 + 2*x^8))/(x^10*(-1 + 2*x^4)),x]
Output:
((-1 + x^4)^(1/4)*(2 + 5*x^4 + 65*x^8))/(9*x^9) + 2*Sqrt[2]*ArcTan[(Sqrt[2 ]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^4])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2 ]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7276, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-1} \left (2 x^8-x^4+2\right )}{x^{10} \left (2 x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {2 \sqrt [4]{x^4-1}}{x^{10}}-\frac {3 \sqrt [4]{x^4-1}}{x^6}+\frac {16 \sqrt [4]{x^4-1} x^2}{2 x^4-1}-\frac {8 \sqrt [4]{x^4-1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,2 x^4\right )}{3 \sqrt [4]{1-x^4}}+4 \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {8 \sqrt [4]{x^4-1}}{x}-\frac {2 \left (x^4-1\right )^{5/4}}{9 x^9}-\frac {7 \left (x^4-1\right )^{5/4}}{9 x^5}\) |
Input:
Int[((-1 + x^4)^(1/4)*(2 - x^4 + 2*x^8))/(x^10*(-1 + 2*x^4)),x]
Output:
(8*(-1 + x^4)^(1/4))/x - (2*(-1 + x^4)^(5/4))/(9*x^9) - (7*(-1 + x^4)^(5/4 ))/(9*x^5) - (16*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, 2*x ^4])/(3*(1 - x^4)^(1/4)) + 4*ArcTan[x/(-1 + x^4)^(1/4)] - 4*ArcTanh[x/(-1 + x^4)^(1/4)]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 6.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18
| method | result | size |
| pseudoelliptic | \(\frac {-9 x^{9} \left (2 \arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}-1}}{-\left (x^{4}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}-1}}\right )\right ) \sqrt {2}+\left (65 x^{8}+5 x^{4}+2\right ) \left (x^{4}-1\right )^{\frac {1}{4}}}{9 x^{9}}\) | \(129\) |
| trager | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (65 x^{8}+5 x^{4}+2\right )}{9 x^{9}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}-1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{2 x^{4}-1}\right )\) | \(180\) |
| risch | \(\frac {65 x^{12}-60 x^{8}-3 x^{4}-2}{9 x^{9} \left (x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x^{9}-x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{4}-1\right ) \left (-1+x \right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )^{2}}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x^{9}-x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {3}{4}} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} x +2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{4}-1\right ) \left (-1+x \right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )^{2}}\right )\right ) {\left (\left (x^{4}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(514\) |
Input:
int((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x,method=_RETURNVERBOSE)
Output:
1/9*(-9*x^9*(2*arctan(((x^4-1)^(1/4)*2^(1/2)-x)/x)+2*arctan(((x^4-1)^(1/4) *2^(1/2)+x)/x)+ln(((x^4-1)^(1/4)*x*2^(1/2)+x^2+(x^4-1)^(1/2))/(-(x^4-1)^(1 /4)*x*2^(1/2)+x^2+(x^4-1)^(1/2))))*2^(1/2)+(65*x^8+5*x^4+2)*(x^4-1)^(1/4)) /x^9
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (90) = 180\).
Time = 1.56 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {36 \, \sqrt {2} x^{9} \arctan \left (\frac {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2} - \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {5}{4}}}{2 \, {\left (x^{5} - x\right )}}\right ) - 9 \, \sqrt {2} x^{9} \log \left (\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{2 \, x^{4} - 1}\right ) + 9 \, \sqrt {2} x^{9} \log \left (\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{2 \, x^{4} - 1}\right ) + 2 \, {\left (65 \, x^{8} + 5 \, x^{4} + 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{18 \, x^{9}} \] Input:
integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="fricas" )
Output:
1/18*(36*sqrt(2)*x^9*arctan(1/2*(sqrt(2)*(x^4 - 1)^(3/4)*x^2 - sqrt(2)*(x^ 4 - 1)^(5/4))/(x^5 - x)) - 9*sqrt(2)*x^9*log((2*x^4 + 2*sqrt(2)*(x^4 - 1)^ (1/4)*x^3 + 4*sqrt(x^4 - 1)*x^2 + 2*sqrt(2)*(x^4 - 1)^(3/4)*x - 1)/(2*x^4 - 1)) + 9*sqrt(2)*x^9*log((2*x^4 - 2*sqrt(2)*(x^4 - 1)^(1/4)*x^3 + 4*sqrt( x^4 - 1)*x^2 - 2*sqrt(2)*(x^4 - 1)^(3/4)*x - 1)/(2*x^4 - 1)) + 2*(65*x^8 + 5*x^4 + 2)*(x^4 - 1)^(1/4))/x^9
\[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (2 x^{8} - x^{4} + 2\right )}{x^{10} \cdot \left (2 x^{4} - 1\right )}\, dx \] Input:
integrate((x**4-1)**(1/4)*(2*x**8-x**4+2)/x**10/(2*x**4-1),x)
Output:
Integral(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(2*x**8 - x**4 + 2)/(x**10*(2 *x**4 - 1)), x)
\[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{8} - x^{4} + 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} - 1\right )} x^{10}} \,d x } \] Input:
integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="maxima" )
Output:
integrate((2*x^8 - x^4 + 2)*(x^4 - 1)^(1/4)/((2*x^4 - 1)*x^10), x)
Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=-2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}\right ) - 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{x} + \frac {8 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{9 \, x^{9}} \] Input:
integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="giac")
Output:
-2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^4 - 1)^(1/4)/x)) - 2*sqrt(2) *arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(x^4 - 1)^(1/4)/x)) - sqrt(2)*log(sqrt(2 )*(x^4 - 1)^(1/4)/x + sqrt(x^4 - 1)/x^2 + 1) + sqrt(2)*log(-sqrt(2)*(x^4 - 1)^(1/4)/x + sqrt(x^4 - 1)/x^2 + 1) + (x^4 - 1)^(1/4)*(1/x^4 - 1)/x + 8*( x^4 - 1)^(1/4)/x + 2/9*(x^8 - 2*x^4 + 1)*(x^4 - 1)^(1/4)/x^9
Timed out. \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8-x^4+2\right )}{x^{10}\,\left (2\,x^4-1\right )} \,d x \] Input:
int(((x^4 - 1)^(1/4)*(2*x^8 - x^4 + 2))/(x^10*(2*x^4 - 1)),x)
Output:
int(((x^4 - 1)^(1/4)*(2*x^8 - x^4 + 2))/(x^10*(2*x^4 - 1)), x)
\[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {29 \left (x^{4}-1\right )^{\frac {1}{4}} x^{8}+5 \left (x^{4}-1\right )^{\frac {1}{4}} x^{4}+2 \left (x^{4}-1\right )^{\frac {1}{4}}-36 \left (\int \frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{2 x^{10}-3 x^{6}+x^{2}}d x \right ) x^{9}}{9 x^{9}} \] Input:
int((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x)
Output:
(29*(x**4 - 1)**(1/4)*x**8 + 5*(x**4 - 1)**(1/4)*x**4 + 2*(x**4 - 1)**(1/4 ) - 36*int((x**4 - 1)**(1/4)/(2*x**10 - 3*x**6 + x**2),x)*x**9)/(9*x**9)