Integrand size = 37, antiderivative size = 248 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=2 \sqrt [4]{2} \text {RootSum}\left [1-4 \text {$\#$1}^2+12 \sqrt {2} \text {$\#$1}^2+70 \text {$\#$1}^4-24 \sqrt {2} \text {$\#$1}^4-4 \text {$\#$1}^6+12 \sqrt {2} \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}^3+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}^3}{-1+3 \sqrt {2}+35 \text {$\#$1}^2-12 \sqrt {2} \text {$\#$1}^2-3 \text {$\#$1}^4+9 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]
Time = 0.95 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=\arctan \left (\frac {x}{\sqrt {2+x^4}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {2+x^4}}\right ) \]
Result contains complex when optimal does not.
Time = 2.09 (sec) , antiderivative size = 1404, normalized size of antiderivative = 5.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, 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.
| \(\displaystyle \int \frac {\left (x^4-2\right ) \sqrt {x^4+2}}{\left (x^4+x^2+2\right ) \left (x^4+2 x^2+2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (2 x^2+1\right ) \sqrt {x^4+2}}{x^4+x^2+2}-\frac {2 \left (x^2+1\right ) \sqrt {x^4+2}}{x^4+2 x^2+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (4 i-\sqrt {2} \left (i+\sqrt {7}\right )\right ) \sqrt {x^4+2} x}{\left (4 i-i \sqrt {2}-\sqrt {14}\right ) \left (x^2+\sqrt {2}\right )}+\frac {\left (4 i-\sqrt {2} \left (i-\sqrt {7}\right )\right ) \sqrt {x^4+2} x}{\left (4 i-i \sqrt {2}+\sqrt {14}\right ) \left (x^2+\sqrt {2}\right )}-\frac {2 \sqrt {x^4+2} x}{x^2+\sqrt {2}}+\frac {\left (4 i-\sqrt {2} \left (i-\sqrt {7}\right )\right ) \arctan \left (\frac {x}{\sqrt {x^4+2}}\right )}{2 \left (4 i-i \sqrt {2}+\sqrt {14}\right )}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^4+2}}\right )-\frac {\left ((-1+i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+2}}\right )}{2-(1-i) \sqrt {2}}-\frac {\left ((-1-i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+2}}\right )}{2-(1+i) \sqrt {2}}+\frac {2^{3/4} \left (1-2 \sqrt {2}-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\left (4-\sqrt {2}+i \sqrt {14}\right ) \sqrt {x^4+2}}+\frac {2^{3/4} \left (1-2 \sqrt {2}+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\left (4-\sqrt {2}-i \sqrt {14}\right ) \sqrt {x^4+2}}+\frac {2^{3/4} \left ((-1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\left (2-(1-i) \sqrt {2}\right ) \sqrt {x^4+2}}+\frac {2^{3/4} \left ((-1-i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\left (2-(1+i) \sqrt {2}\right ) \sqrt {x^4+2}}-\frac {2^{3/4} \left (i-\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\left (4 i-i \sqrt {2}+\sqrt {14}\right ) \sqrt {x^4+2}}-\frac {2^{3/4} \left (i+\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\left (4 i-i \sqrt {2}-\sqrt {14}\right ) \sqrt {x^4+2}}-\frac {2^{3/4} \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\left ((-1+i)+\sqrt {2}\right ) \sqrt {x^4+2}}-\frac {2^{3/4} \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4+2}}-\frac {\left ((1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt {2}\right ),2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2^{3/4} \left (2-(1+i) \sqrt {2}\right ) \sqrt {x^4+2}}-\frac {\left ((1-i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt {2}\right ),2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2^{3/4} \left (2-(1-i) \sqrt {2}\right ) \sqrt {x^4+2}}+\frac {\left (1+2 \sqrt {2}-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\sqrt {2}\right ),2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2\ 2^{3/4} \left (4-\sqrt {2}+i \sqrt {14}\right ) \sqrt {x^4+2}}+\frac {\left (1+2 \sqrt {2}+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (4-\sqrt {2}\right ),2 \arctan \left (\frac {x}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2\ 2^{3/4} \left (4-\sqrt {2}-i \sqrt {14}\right ) \sqrt {x^4+2}}\) |
(-2*x*Sqrt[2 + x^4])/(Sqrt[2] + x^2) + ((4*I - Sqrt[2]*(I - Sqrt[7]))*x*Sq rt[2 + x^4])/((4*I - I*Sqrt[2] + Sqrt[14])*(Sqrt[2] + x^2)) + ((4*I - Sqrt [2]*(I + Sqrt[7]))*x*Sqrt[2 + x^4])/((4*I - I*Sqrt[2] - Sqrt[14])*(Sqrt[2] + x^2)) + ArcTan[x/Sqrt[2 + x^4]]/2 + ((4*I - Sqrt[2]*(I - Sqrt[7]))*ArcT an[x/Sqrt[2 + x^4]])/(2*(4*I - I*Sqrt[2] + Sqrt[14])) - (((-1 - I) + Sqrt[ 2])*ArcTan[(Sqrt[2]*x)/Sqrt[2 + x^4]])/(2 - (1 + I)*Sqrt[2]) - (((-1 + I) + Sqrt[2])*ArcTan[(Sqrt[2]*x)/Sqrt[2 + x^4]])/(2 - (1 - I)*Sqrt[2]) + (2^( 3/4)*((-1 - I) + Sqrt[2])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2 ]*EllipticE[2*ArcTan[x/2^(1/4)], 1/2])/((2 - (1 + I)*Sqrt[2])*Sqrt[2 + x^4 ]) + (2^(3/4)*((-1 + I) + Sqrt[2])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticE[2*ArcTan[x/2^(1/4)], 1/2])/((2 - (1 - I)*Sqrt[2])*Sqr t[2 + x^4]) + (2^(3/4)*(1 - 2*Sqrt[2] + I*Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticE[2*ArcTan[x/2^(1/4)], 1/2])/((4 - Sqrt [2] - I*Sqrt[14])*Sqrt[2 + x^4]) + (2^(3/4)*(1 - 2*Sqrt[2] - I*Sqrt[7])*(S qrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticE[2*ArcTan[x/2^(1/ 4)], 1/2])/((4 - Sqrt[2] + I*Sqrt[14])*Sqrt[2 + x^4]) - (2^(3/4)*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/ 2])/(((-1 - I) + Sqrt[2])*Sqrt[2 + x^4]) - (2^(3/4)*(Sqrt[2] + x^2)*Sqrt[( 2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(((-1 + I ) + Sqrt[2])*Sqrt[2 + x^4]) - (2^(3/4)*(I + Sqrt[7])*(Sqrt[2] + x^2)*Sq...
3.28.10.3.1 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 = 5.88 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.17
| method | result | size |
| elliptic | \(\frac {\left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )+2 \arctan \left (\frac {\sqrt {x^{4}+2}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(42\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}+2}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{4}+2 x^{2}+2}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}+2}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{2}\) | \(126\) |
| default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (\left (x^{2}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}+2 x \sqrt {2}\right )}{2 \sqrt {x^{4}+2}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (\left (x^{2}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}-2 x \sqrt {2}\right ) \sqrt {2}}{2 \sqrt {x^{4}+2}}\right )}{2}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {2 \sqrt {2}-1}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{2}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {2 \sqrt {2}-1}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{2}\) | \(156\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (\left (x^{2}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}+2 x \sqrt {2}\right )}{2 \sqrt {x^{4}+2}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (\left (x^{2}+\sqrt {2}\right ) \sqrt {-2+2 \sqrt {2}}-2 x \sqrt {2}\right ) \sqrt {2}}{2 \sqrt {x^{4}+2}}\right )}{2}+\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {2 \sqrt {2}-1}+2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{2}-\frac {\arctan \left (\frac {\left (x^{2}+\sqrt {2}\right ) \sqrt {2 \sqrt {2}-1}-2 x \sqrt {2}}{\sqrt {x^{4}+2}}\right )}{2}\) | \(156\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} + 2} x}{x^{4} - 2 \, x^{2} + 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\right ) \]
-1/2*sqrt(2)*arctan(2*sqrt(2)*sqrt(x^4 + 2)*x/(x^4 - 2*x^2 + 2)) + 1/2*arc tan(2*sqrt(x^4 + 2)*x/(x^4 - x^2 + 2))
Timed out. \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{{\left (x^{4} + 2 \, x^{2} + 2\right )} {\left (x^{4} + x^{2} + 2\right )}} \,d x } \]
Not integrable
Time = 2.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} + 2} {\left (x^{4} - 2\right )}}{{\left (x^{4} + 2 \, x^{2} + 2\right )} {\left (x^{4} + x^{2} + 2\right )}} \,d x } \]
Not integrable
Time = 6.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx=\int \frac {\left (x^4-2\right )\,\sqrt {x^4+2}}{\left (x^4+x^2+2\right )\,\left (x^4+2\,x^2+2\right )} \,d x \]