Integrand size = 27, antiderivative size = 53 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \] Output:
-1/4*arctan(2^(1/4)*x/(x^4-2)^(1/2))*2^(3/4)-1/4*arctanh(2^(1/4)*x/(x^4-2) ^(1/2))*2^(3/4)
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \] Input:
Integrate[(Sqrt[-2 + x^4]*(2 + x^4))/(4 - 6*x^4 + x^8),x]
Output:
-1/2*(ArcTan[(2^(1/4)*x)/Sqrt[-2 + x^4]] + ArcTanh[(2^(1/4)*x)/Sqrt[-2 + x ^4]])/2^(1/4)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.81 (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}}\) |
Input:
Int[(Sqrt[-2 + x^4]*(2 + x^4))/(4 - 6*x^4 + x^8),x]
Output:
(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 +...
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 = 1.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (\operatorname {arctanh}\left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )-\arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )\right )}{4}\) | \(41\) |
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 | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+8 \sqrt {x^{4}-2}\, x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 x^{4}-4}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )-8 \sqrt {x^{4}-2}\, x -4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}+4}\right )}{8}\) | \(183\) |
Input:
int((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x,method=_RETURNVERBOSE)
Output:
-1/4*2^(3/4)*(arctanh(1/2*2^(3/4)/x*(x^4-2)^(1/2))-arctan(1/2*2^(3/4)/x*(x ^4-2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (37) = 74\).
Time = 0.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=-\frac {1}{8} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2 \, \sqrt {x^{4} - 2} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x\right )}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) \] Input:
integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="fricas")
Output:
-1/8*2^(3/4)*arctan(2*sqrt(x^4 - 2)*(2^(3/4)*x^3 + 2^(1/4)*(x^5 - 2*x))/(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*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))
Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\text {Timed out} \] Input:
integrate((x**4-2)**(1/2)*(x**4+2)/(x**8-6*x**4+4),x)
Output:
Timed out
\[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4} \,d x } \] Input:
integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="maxima")
Output:
integrate((x^4 + 2)*sqrt(x^4 - 2)/(x^8 - 6*x^4 + 4), x)
\[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4} \,d x } \] Input:
integrate((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x, algorithm="giac")
Output:
integrate((x^4 + 2)*sqrt(x^4 - 2)/(x^8 - 6*x^4 + 4), x)
Timed out. \[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=\int \frac {\sqrt {x^4-2}\,\left (x^4+2\right )}{x^8-6\,x^4+4} \,d x \] Input:
int(((x^4 - 2)^(1/2)*(x^4 + 2))/(x^8 - 6*x^4 + 4),x)
Output:
int(((x^4 - 2)^(1/2)*(x^4 + 2))/(x^8 - 6*x^4 + 4), x)
\[ \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx=2 \left (\int \frac {\sqrt {x^{4}-2}}{x^{8}-6 x^{4}+4}d x \right )+\int \frac {\sqrt {x^{4}-2}\, x^{4}}{x^{8}-6 x^{4}+4}d x \] Input:
int((x^4-2)^(1/2)*(x^4+2)/(x^8-6*x^4+4),x)
Output:
2*int(sqrt(x**4 - 2)/(x**8 - 6*x**4 + 4),x) + int((sqrt(x**4 - 2)*x**4)/(x **8 - 6*x**4 + 4),x)