Integrand size = 27, antiderivative size = 53 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \]
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.92 (sec) , antiderivative size = 1437, normalized size of antiderivative = 27.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \frac {\sqrt {x^4-2} \left (x^4+2\right )}{x^8-6 x^4+4} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (1+\sqrt {5}\right ) \sqrt {x^4-2}}{2 x^4-2 \sqrt {5}-6}+\frac {\left (1-\sqrt {5}\right ) \sqrt {x^4-2}}{2 x^4+2 \sqrt {5}-6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\sqrt {2}-x^2} \sqrt {x^2+\sqrt {2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{\sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}+\frac {\sqrt {\sqrt {2}-x^2} \sqrt {x^2+\sqrt {2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{\sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}+\frac {\sqrt {\sqrt {2}-x^2} \sqrt {x^2+\sqrt {2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {x^4-2}}+\frac {\sqrt {\sqrt {2}-x^2} \sqrt {x^2+\sqrt {2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {x^4-2}}-\frac {\sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (-\sqrt {\frac {2}{3+\sqrt {5}}},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{4 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (\sqrt {\frac {2}{3+\sqrt {5}}},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{4 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \operatorname {EllipticPi}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )},\arcsin \left (\frac {x}{\sqrt [4]{2}}\right ),-1\right )}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}\) |
(Sqrt[Sqrt[2] - x^2]*Sqrt[Sqrt[2] + x^2]*EllipticF[ArcSin[x/2^(1/4)], -1]) /(2*2^(1/4)*(1 - Sqrt[2/(3 + Sqrt[5])])*Sqrt[-2 + x^4]) + (Sqrt[Sqrt[2] - x^2]*Sqrt[Sqrt[2] + x^2]*EllipticF[ArcSin[x/2^(1/4)], -1])/(2*2^(1/4)*(1 + Sqrt[2/(3 + Sqrt[5])])*Sqrt[-2 + x^4]) + (Sqrt[Sqrt[2] - x^2]*Sqrt[Sqrt[2 ] + x^2]*EllipticF[ArcSin[x/2^(1/4)], -1])/(2^(1/4)*(2 - Sqrt[2*(3 + Sqrt[ 5])])*Sqrt[-2 + x^4]) + (Sqrt[Sqrt[2] - x^2]*Sqrt[Sqrt[2] + x^2]*EllipticF [ArcSin[x/2^(1/4)], -1])/(2^(1/4)*(2 + Sqrt[2*(3 + Sqrt[5])])*Sqrt[-2 + x^ 4]) + ((1 - Sqrt[5])*Sqrt[(Sqrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[ 2]*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/(4*2^( 1/4)*Sqrt[(2 - Sqrt[2]*x^2)^(-1)]*Sqrt[-2 + x^4]) + ((1 + Sqrt[5])*Sqrt[(S qrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[2]*x^2]*EllipticF[ArcSin[(2^ (3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/(4*2^(1/4)*Sqrt[(2 - Sqrt[2]*x^2)^ (-1)]*Sqrt[-2 + x^4]) - (Sqrt[(Sqrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + S qrt[2]*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/(4 *2^(1/4)*(1 - Sqrt[2/(3 + Sqrt[5])])*Sqrt[(2 - Sqrt[2]*x^2)^(-1)]*Sqrt[-2 + x^4]) - (Sqrt[(Sqrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[2]*x^2]*El lipticF[ArcSin[(2^(3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/(4*2^(1/4)*(1 + Sqrt[2/(3 + Sqrt[5])])*Sqrt[(2 - Sqrt[2]*x^2)^(-1)]*Sqrt[-2 + x^4]) - (Sqr t[(Sqrt[2] + x^2)/(Sqrt[2] - x^2)]*Sqrt[-2 + Sqrt[2]*x^2]*EllipticF[ArcSin [(2^(3/4)*x)/Sqrt[-2 + Sqrt[2]*x^2]], 1/2])/(2*2^(1/4)*(2 - Sqrt[2*(3 +...
3.7.84.3.1 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 = 4.69 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (\arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\operatorname {arctanh}\left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )\right )}{4}\) | \(41\) |
| default | \(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}\) | \(73\) |
| elliptic | \(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}\) | \(73\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+8 \sqrt {x^{4}-2}\, x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 x^{4}-4}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-2 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+8 \sqrt {x^{4}-2}\, x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}+4}\right )}{8}\) | \(184\) |
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 296, normalized size of antiderivative = 5.58 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (i \, x^{8} - 2 i \, x^{4} + 4 i\right )} + 4 \, {\left (x^{5} - \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} - 4 \cdot 2^{\frac {1}{4}} {\left (i \, x^{6} - 2 i \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-i \, x^{8} + 2 i \, x^{4} - 4 i\right )} + 4 \, {\left (x^{5} - \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} - 4 \cdot 2^{\frac {1}{4}} {\left (-i \, x^{6} + 2 i \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) \]
-1/16*2^(3/4)*log((2^(3/4)*(x^8 - 2*x^4 + 4) + 4*(x^5 + sqrt(2)*x^3 - 2*x) *sqrt(x^4 - 2) + 4*2^(1/4)*(x^6 - 2*x^2))/(x^8 - 6*x^4 + 4)) + 1/16*2^(3/4 )*log(-(2^(3/4)*(x^8 - 2*x^4 + 4) - 4*(x^5 + sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) + 4*2^(1/4)*(x^6 - 2*x^2))/(x^8 - 6*x^4 + 4)) - 1/16*I*2^(3/4)*log((2^ (3/4)*(I*x^8 - 2*I*x^4 + 4*I) + 4*(x^5 - sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) - 4*2^(1/4)*(I*x^6 - 2*I*x^2))/(x^8 - 6*x^4 + 4)) + 1/16*I*2^(3/4)*log((2^ (3/4)*(-I*x^8 + 2*I*x^4 - 4*I) + 4*(x^5 - sqrt(2)*x^3 - 2*x)*sqrt(x^4 - 2) - 4*2^(1/4)*(-I*x^6 + 2*I*x^2))/(x^8 - 6*x^4 + 4))
Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4} \,d x } \]
\[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int \frac {\sqrt {x^4-2}\,\left (x^4+2\right )}{x^8-6\,x^4+4} \,d x \]