Integrand size = 38, antiderivative size = 68 \[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=-\frac {2 \sqrt [4]{-1-x^2+x^4} \left (1+x^2+4 x^4\right )}{5 x^5}-\arctan \left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right ) \]
-2/5*(x^4-x^2-1)^(1/4)*(4*x^4+x^2+1)/x^5-arctan(x/(x^4-x^2-1)^(1/4))+arcta nh(x/(x^4-x^2-1)^(1/4))
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=-\frac {2 \sqrt [4]{-1-x^2+x^4} \left (1+x^2+4 x^4\right )}{5 x^5}-\arctan \left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1-x^2+x^4}}\right ) \]
(-2*(-1 - x^2 + x^4)^(1/4)*(1 + x^2 + 4*x^4))/(5*x^5) - ArcTan[x/(-1 - x^2 + x^4)^(1/4)] + ArcTanh[x/(-1 - x^2 + x^4)^(1/4)]
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.
| \(\displaystyle \int \frac {\left (x^2+2\right ) \sqrt [4]{x^4-x^2-1} \left (x^4+x^2+1\right )}{x^6 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 2250 |
| \(\displaystyle \int \frac {\left (x^2+2\right ) \sqrt [4]{x^4-x^2-1} \left (x^4+x^2+1\right )}{x^6 \left (x^2+1\right )}dx\) |
3.9.99.3.1 Defintions of rubi rules used
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol yQ[Px, x]
Time = 8.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53
| method | result | size |
| pseudoelliptic | \(\frac {-5 \ln \left (\frac {\left (x^{4}-x^{2}-1\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}+5 \ln \left (\frac {\left (x^{4}-x^{2}-1\right )^{\frac {1}{4}}+x}{x}\right ) x^{5}+10 \arctan \left (\frac {\left (x^{4}-x^{2}-1\right )^{\frac {1}{4}}}{x}\right ) x^{5}+\left (-16 x^{4}-4 x^{2}-4\right ) \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}}}{10 x^{5}}\) | \(104\) |
| trager | \(-\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} \left (4 x^{4}+x^{2}+1\right )}{5 x^{5}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x -2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{2}-\frac {\ln \left (\frac {2 \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}} x -2 \sqrt {x^{4}-x^{2}-1}\, x^{2}+2 \left (x^{4}-x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 x^{4}+x^{2}+1}{x^{2}+1}\right )}{2}\) | \(199\) |
| risch | \(-\frac {2 \left (4 x^{8}-3 x^{6}-4 x^{4}-2 x^{2}-1\right )}{5 x^{5} \left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (\frac {\ln \left (-\frac {2 x^{12}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}-5 x^{10}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}-4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}-x^{8}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}+7 x^{6}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}+4 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}+x^{4}+2 \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x -3 x^{2}-1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 x^{12}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+5 x^{10}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{6}+x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {3}{4}} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{4}-7 x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1}\, x^{2}-x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{10}+5 x^{6}-3 x^{2}-1\right )^{\frac {1}{4}} x +3 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}-1\right )^{2}}\right )}{2}\right ) {\left (\left (x^{4}-x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}-x^{2}-1\right )^{\frac {3}{4}}}\) | \(701\) |
1/10*(-5*ln(((x^4-x^2-1)^(1/4)-x)/x)*x^5+5*ln(((x^4-x^2-1)^(1/4)+x)/x)*x^5 +10*arctan(1/x*(x^4-x^2-1)^(1/4))*x^5+(-16*x^4-4*x^2-4)*(x^4-x^2-1)^(1/4)) /x^5
Exception generated. \[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
\[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=\int \frac {\left (x^{2} + 2\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \sqrt [4]{x^{4} - x^{2} - 1}}{x^{6} \left (x^{2} + 1\right )}\, dx \]
Integral((x**2 + 2)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 - 1)**(1/4) /(x**6*(x**2 + 1)), x)
\[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )} {\left (x^{4} - x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} + 2\right )}}{{\left (x^{2} + 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (2+x^2\right ) \sqrt [4]{-1-x^2+x^4} \left (1+x^2+x^4\right )}{x^6 \left (1+x^2\right )} \, dx=\int \frac {\left (x^2+2\right )\,\left (x^4+x^2+1\right )\,{\left (x^4-x^2-1\right )}^{1/4}}{x^6\,\left (x^2+1\right )} \,d x \]