Integrand size = 25, antiderivative size = 105 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\frac {1}{3} \sqrt [4]{2} \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}} \]
1/3*2^(1/4)*arctan(1/2*x*2^(3/4)/(x^4+1)^(1/4))+1/12*arctan(2^(1/4)*x/(x^4 +1)^(1/4))*2^(3/4)+1/3*2^(1/4)*arctanh(1/2*x*2^(3/4)/(x^4+1)^(1/4))+1/12*a rctanh(2^(1/4)*x/(x^4+1)^(1/4))*2^(3/4)
Time = 0.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\frac {4 \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+4 \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4}} \]
(4*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))] + Sqrt[2]*ArcTan[(2^(1/4)*x)/(1 + x ^4)^(1/4)] + 4*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))] + Sqrt[2]*ArcTanh[(2^( 1/4)*x)/(1 + x^4)^(1/4)])/(6*2^(3/4))
Time = 0.41 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7279, 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.
| \(\displaystyle \int \frac {x^4-2}{\sqrt [4]{x^4+1} \left (x^8+x^4-2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {8}{3 \sqrt [4]{x^4+1} \left (2 x^4+4\right )}-\frac {2}{3 \sqrt [4]{x^4+1} \left (2 x^4-2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \sqrt [4]{2} \arctan \left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}\) |
(2^(1/4)*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))])/3 + ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) + (2^(1/4)*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))])/3 + ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4))
3.16.15.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 2.36 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {1}{4}} \left (\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) \sqrt {2}-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{2 x}\right ) \sqrt {2}+4 \ln \left (\frac {-x 2^{\frac {3}{4}}-2 \left (x^{4}+1\right )^{\frac {1}{4}}}{x 2^{\frac {3}{4}}-2 \left (x^{4}+1\right )^{\frac {1}{4}}}\right )-8 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right )\right )}{24}\) | \(121\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{x^{4}+2}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{4}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{x^{4}+2}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{4}+1}\, x^{2}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} \ln \left (-\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}\) | \(460\) |
1/24*2^(1/4)*(ln((-2^(1/4)*x-(x^4+1)^(1/4))/(2^(1/4)*x-(x^4+1)^(1/4)))*2^( 1/2)-2*arctan(1/2*2^(3/4)/x*(x^4+1)^(1/4))*2^(1/2)+4*ln((-x*2^(3/4)-2*(x^4 +1)^(1/4))/(x*2^(3/4)-2*(x^4+1)^(1/4)))-8*arctan(1/x*2^(1/4)*(x^4+1)^(1/4) ))
Result contains complex when optimal does not.
Time = 6.48 (sec) , antiderivative size = 539, normalized size of antiderivative = 5.13 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\frac {1}{48} \cdot 8^{\frac {3}{4}} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + \frac {1}{48} i \cdot 8^{\frac {3}{4}} \log \left (-\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 i \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 i \, x^{4} + 2 i\right )} - 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - \frac {1}{48} i \cdot 8^{\frac {3}{4}} \log \left (-\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 i \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (-3 i \, x^{4} - 2 i\right )} - 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - \frac {1}{48} \cdot 8^{\frac {3}{4}} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + \frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) + \frac {1}{48} i \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{4} + i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \frac {1}{48} i \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{4} - i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) \]
1/48*8^(3/4)*log((8*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 8*8^(1/4)*sqrt(x^4 + 1)* x^2 + 8^(3/4)*(3*x^4 + 2) + 16*(x^4 + 1)^(3/4)*x)/(x^4 + 2)) + 1/48*I*8^(3 /4)*log(-(8*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 8*I*8^(1/4)*sqrt(x^4 + 1)*x^2 - 8^(3/4)*(3*I*x^4 + 2*I) - 16*(x^4 + 1)^(3/4)*x)/(x^4 + 2)) - 1/48*I*8^(3/4 )*log(-(8*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 8*I*8^(1/4)*sqrt(x^4 + 1)*x^2 - 8^ (3/4)*(-3*I*x^4 - 2*I) - 16*(x^4 + 1)^(3/4)*x)/(x^4 + 2)) - 1/48*8^(3/4)*l og((8*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 8*8^(1/4)*sqrt(x^4 + 1)*x^2 - 8^(3/4)* (3*x^4 + 2) + 16*(x^4 + 1)^(3/4)*x)/(x^4 + 2)) + 1/48*2^(3/4)*log((4*sqrt( 2)*(x^4 + 1)^(1/4)*x^3 + 4*2^(1/4)*sqrt(x^4 + 1)*x^2 + 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) + 1/48*I*2^(3/4)*log(-(4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 4*I*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2^(3/4)*(3*I*x^4 + I) - 4* (x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 1/48*I*2^(3/4)*log(-(4*sqrt(2)*(x^4 + 1)^( 1/4)*x^3 - 4*I*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2^(3/4)*(-3*I*x^4 - I) - 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 1/48*2^(3/4)*log((4*sqrt(2)*(x^4 + 1)^(1/4)*x^ 3 - 4*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)* x)/(x^4 - 1))
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\int \frac {x^{4} - 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \left (x^{4} + 2\right )}\, dx \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx=\int \frac {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (x^8+x^4-2\right )} \,d x \]