Integrand size = 29, antiderivative size = 104 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}} \]
1/4*arctan((-1/2*2^(1/2)+1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(-x^4+1)^(1/2) )*2^(1/2)-1/4*arctanh((-1/2*2^(1/2)-1/2*2^(1/2)*x^2+1/2*2^(1/2)*x^4)/x/(-x ^4+1)^(1/2))*2^(1/2)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{2} (-1)^{3/4} \left (\arctan \left (\frac {(1+i) x}{\sqrt {2-2 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-1} x}{\sqrt {1-x^4}}\right )\right ) \]
-1/2*((-1)^(3/4)*(ArcTan[((1 + I)*x)/Sqrt[2 - 2*x^4]] + ArcTanh[((-1)^(1/4 )*x)/Sqrt[1 - x^4]]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int \frac {\sqrt {1-x^4} \left (x^4+1\right )}{x^8-x^4+1} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1-x^4}}{2 x^4-i \sqrt {3}-1}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {1-x^4}}{2 x^4+i \sqrt {3}-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \left (1+i \sqrt {3}\right ) \operatorname {EllipticF}(\arcsin (x),-1)-\frac {1}{2} \left (1-i \sqrt {3}\right ) \operatorname {EllipticF}(\arcsin (x),-1)+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin (x),-1\right )\) |
-1/2*((1 - I*Sqrt[3])*EllipticF[ArcSin[x], -1]) - ((1 + I*Sqrt[3])*Ellipti cF[ArcSin[x], -1])/2 + EllipticPi[(-I - Sqrt[3])/2, ArcSin[x], -1]/2 + Ell ipticPi[(I - Sqrt[3])/2, ArcSin[x], -1]/2 + EllipticPi[1/Sqrt[(1 - I*Sqrt[ 3])/2], ArcSin[x], -1]/2 + EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[x] , -1]/2
3.15.98.3.1 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]
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.53
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\left (-1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )+\left (-1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )\right )}{4}\) | \(55\) |
default | \(\frac {\left (\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(114\) |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(114\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{4}+1}\, x}{\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1\right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1\right )}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \sqrt {-x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +1\right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +1\right )}\right )}{4}\) | \(217\) |
1/4*2^(1/2)*((-1+I)*arctan((1/2-1/2*I)*(-x^4+1)^(1/2)/x*2^(1/2))-(1+I)*arc tan((1/2+1/2*I)*(-x^4+1)^(1/2)/x*2^(1/2)))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.66 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} - \left (3 i + 3\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} + \left (3 i - 3\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} - \left (3 i - 3\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} + \left (3 i + 3\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) \]
(1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*((I + 1)*x^8 + (2*I - 2)*x^6 - (3*I + 3)*x^4 - (2*I - 2)*x^2 + I + 1) - 4*(x^5 + I*x^3 - x)*sqrt(-x^4 + 1))/(x^ 8 - x^4 + 1)) - (1/16*I - 1/16)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x^8 - (2*I + 2)*x^6 + (3*I - 3)*x^4 + (2*I + 2)*x^2 - I + 1) - 4*(x^5 - I*x^3 - x)*sq rt(-x^4 + 1))/(x^8 - x^4 + 1)) + (1/16*I - 1/16)*sqrt(2)*log((sqrt(2)*((I - 1)*x^8 + (2*I + 2)*x^6 - (3*I - 3)*x^4 - (2*I + 2)*x^2 + I - 1) - 4*(x^5 - I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1)) - (1/16*I + 1/16)*sqrt(2)*l og((sqrt(2)*(-(I + 1)*x^8 - (2*I - 2)*x^6 + (3*I + 3)*x^4 + (2*I - 2)*x^2 - I - 1) - 4*(x^5 + I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1))
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{x^{8} - x^{4} + 1}\, dx \]
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1} \,d x } \]
\[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\int \frac {\sqrt {1-x^4}\,\left (x^4+1\right )}{x^8-x^4+1} \,d x \]