Integrand size = 50, antiderivative size = 158 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\arctan \left (\frac {x \sqrt {1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right )-i \sqrt {2} \text {arctanh}\left (\frac {-2 i x+2 \sqrt {2} x^3-2 i x \sqrt {1-3 x^2-2 x^4}}{-\sqrt {2}+3 \sqrt {2} x^2+2 \sqrt {2} x^4-\sqrt {2} \sqrt {1-3 x^2-2 x^4}-2 i x^2 \sqrt {1-3 x^2-2 x^4}}\right ) \]
arctan(x*(-2*x^4-3*x^2+1)^(1/2)/(2*x^4+3*x^2-1))-I*2^(1/2)*arctanh((-2*I*x +2*x^3*2^(1/2)-2*I*x*(-2*x^4-3*x^2+1)^(1/2))/(-2^(1/2)+3*2^(1/2)*x^2+2*2^( 1/2)*x^4-2^(1/2)*(-2*x^4-3*x^2+1)^(1/2)-2*I*x^2*(-2*x^4-3*x^2+1)^(1/2)))
Time = 0.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {1-3 x^2-2 x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1-3 x^2-2 x^4}}\right ) \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.40 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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.
| \(\displaystyle \int \frac {\sqrt {-2 x^4-3 x^2+1} \left (2 x^4+1\right )}{\left (2 x^4+x^2-1\right ) \left (2 x^4+2 x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {2 \sqrt {-2 x^4-3 x^2+1} \left (2 x^2+1\right )}{2 x^4+2 x^2-1}+\frac {\sqrt {-2 x^4-3 x^2+1}}{x^2+1}+\frac {2 \sqrt {-2 x^4-3 x^2+1}}{2 x^2-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\sqrt {\frac {1}{2} \left (9 \sqrt {17}-37\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {3 \sqrt {17}-5} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )\) |
((1 - 2*Sqrt[3] - Sqrt[17])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], ( -13 + 3*Sqrt[17])/4])/Sqrt[2*(3 + Sqrt[17])] + ((1 + 2*Sqrt[3] - Sqrt[17]) *EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/Sqrt[ 2*(3 + Sqrt[17])] + (Sqrt[-5 + 3*Sqrt[17]]*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/2 - Sqrt[(-37 + 9*Sqrt[17])/2]*Ellipt icF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/4, ArcSin[(2*x)/Sqrt[-3 + Sqrt[17] ]], (-13 + 3*Sqrt[17])/4] - Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17 ])/(2*(1 - Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17] )/4] - Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/(2*(1 + Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(-3 + Sqrt[17])/2, ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]] , (-13 + 3*Sqrt[17])/4]
3.22.59.3.1 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 = 5.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.35
| method | result | size |
| elliptic | \(\frac {\left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )-2 \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(55\) |
| default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 i x \sqrt {2}-2 i x^{2}-\sqrt {2}\, x^{2}-i+\sqrt {2}-3 x \right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {4 i x +2 x^{2}-2+\left (2 i x^{2}-3 x +i\right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}+\arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )\) | \(125\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 i x \sqrt {2}-2 i x^{2}-\sqrt {2}\, x^{2}-i+\sqrt {2}-3 x \right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {4 i x +2 x^{2}-2+\left (2 i x^{2}-3 x +i\right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}+\arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )\) | \(125\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x \sqrt {-2 x^{4}-3 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {-2 x^{4}-3 x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+2 x^{2}-1}\right )}{2}\) | \(148\) |
1/2*(2^(1/2)*arctan((-2*x^4-3*x^2+1)^(1/2)/x)-2*arctan(1/2*(-2*x^4-3*x^2+1 )^(1/2)*2^(1/2)/x))*2^(1/2)
Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 5 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 4 \, x^{2} - 1}\right ) \]
integrate((-2*x^4-3*x^2+1)^(1/2)*(2*x^4+1)/(2*x^4+x^2-1)/(2*x^4+2*x^2-1),x , algorithm="fricas")
-1/2*sqrt(2)*arctan(2*sqrt(2)*sqrt(-2*x^4 - 3*x^2 + 1)*x/(2*x^4 + 5*x^2 - 1)) + 1/2*arctan(2*sqrt(-2*x^4 - 3*x^2 + 1)*x/(2*x^4 + 4*x^2 - 1))
\[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\int \frac {\left (2 x^{4} + 1\right ) \sqrt {- 2 x^{4} - 3 x^{2} + 1}}{\left (x^{2} + 1\right ) \left (2 x^{2} - 1\right ) \left (2 x^{4} + 2 x^{2} - 1\right )}\, dx \]
Integral((2*x**4 + 1)*sqrt(-2*x**4 - 3*x**2 + 1)/((x**2 + 1)*(2*x**2 - 1)* (2*x**4 + 2*x**2 - 1)), x)
\[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + 1\right )} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1}}{{\left (2 \, x^{4} + 2 \, x^{2} - 1\right )} {\left (2 \, x^{4} + x^{2} - 1\right )}} \,d x } \]
integrate((-2*x^4-3*x^2+1)^(1/2)*(2*x^4+1)/(2*x^4+x^2-1)/(2*x^4+2*x^2-1),x , algorithm="maxima")
\[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + 1\right )} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1}}{{\left (2 \, x^{4} + 2 \, x^{2} - 1\right )} {\left (2 \, x^{4} + x^{2} - 1\right )}} \,d x } \]
integrate((-2*x^4-3*x^2+1)^(1/2)*(2*x^4+1)/(2*x^4+x^2-1)/(2*x^4+2*x^2-1),x , algorithm="giac")
Timed out. \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\int \frac {\left (2\,x^4+1\right )\,\sqrt {-2\,x^4-3\,x^2+1}}{\left (2\,x^4+x^2-1\right )\,\left (2\,x^4+2\,x^2-1\right )} \,d x \]