Integrand size = 56, antiderivative size = 110 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=-\frac {x}{\sqrt {1+x^2+x^4}}+\sqrt {\frac {1}{3} \left (-2+2 i \sqrt {3}\right )} \arctan \left (\frac {2 x}{\left (-i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}\right )+\sqrt {\frac {1}{3} \left (-2-2 i \sqrt {3}\right )} \arctan \left (\frac {2 x}{\left (i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}\right ) \]
-x/(x^4+x^2+1)^(1/2)+1/3*(-6+6*I*3^(1/2))^(1/2)*arctan(2*x/(-I+3^(1/2))/(x ^4+x^2+1)^(1/2))+1/3*(-6-6*I*3^(1/2))^(1/2)*arctan(2*x/(3^(1/2)+I)/(x^4+x^ 2+1)^(1/2))
Time = 0.83 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=-\frac {x}{\sqrt {1+x^2+x^4}}+\frac {\arctan \left (\frac {\sqrt {3} x \sqrt {1+x^2+x^4}}{1+x^4}\right )}{\sqrt {3}}-\text {arctanh}\left (\frac {x \sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2}\right ) \]
Integrate[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2 )*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)),x]
-(x/Sqrt[1 + x^2 + x^4]) + ArcTan[(Sqrt[3]*x*Sqrt[1 + x^2 + x^4])/(1 + x^4 )]/Sqrt[3] - ArcTanh[(x*Sqrt[1 + x^2 + x^4])/(1 + x^2)^2]
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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \left (x^8+x^6+3 x^4+x^2+1\right )}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^4}{\left (x^4+x^2+1\right )^{3/2}}-\frac {2 x^2}{\left (x^4+x^2+1\right )^{3/2}}+\frac {3}{\left (x^4+x^2+1\right )^{3/2}}-\frac {2 \left (x^6+6 x^4+4 x^2+2\right )}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \int \frac {1}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )}dx-8 \int \frac {x^2}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )}dx-12 \int \frac {x^4}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )}dx-2 \int \frac {x^6}{\left (x^4+x^2+1\right )^{3/2} \left (x^8+3 x^6+5 x^4+3 x^2+1\right )}dx+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}-\frac {8 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {x^4+x^2+1}}+\frac {8 \sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {\left (1-x^2\right ) x}{\sqrt {x^4+x^2+1}}-\frac {\left (x^2+2\right ) x}{3 \sqrt {x^4+x^2+1}}-\frac {2 \left (2 x^2+1\right ) x}{3 \sqrt {x^4+x^2+1}}\) |
Int[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)),x]
3.17.21.3.1 Defintions of rubi rules used
Time = 2.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \(-\frac {x}{\sqrt {x^{4}+x^{2}+1}}-\frac {\ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {2}\, \sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {2}\, \sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\) | \(152\) |
| elliptic | \(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{3}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{3}-\frac {\sqrt {2}\, x}{\sqrt {x^{4}+x^{2}+1}}\right ) \sqrt {2}}{2}\) | \(160\) |
| trager | \(-\frac {x}{\sqrt {x^{4}+x^{2}+1}}+2 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {-x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-2 \sqrt {x^{4}+x^{2}+1}\, x +x^{2}-1}{-x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-3 x^{2}-1}\right )-2 \ln \left (-\frac {x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-4 \sqrt {x^{4}+x^{2}+1}\, x +2 x^{2}+1}{x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+1}\right ) \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+\ln \left (-\frac {x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-4 \sqrt {x^{4}+x^{2}+1}\, x +2 x^{2}+1}{x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+1}\right )\) | \(314\) |
| default | \(-\frac {\left (-i \sqrt {3}+3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+x^{2}+1}+i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}-x^{3}+2 x^{2}-x +2}{x^{2}}\right )+\left (i \sqrt {3}+3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+x^{2}+1}-i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}-x^{3}+2 x^{2}-x +2}{x^{2}}\right )+\left (-i \sqrt {3}-3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+x^{2}+1}+i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}+x^{3}+2 x^{2}+x +2}{x^{2}}\right )+\sqrt {x^{4}+x^{2}+1}\, \left (i \sqrt {3}-3\right ) \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+x^{2}+1}-i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}+x^{3}+2 x^{2}+x +2}{x^{2}}\right )+6 x}{6 \sqrt {x^{4}+x^{2}+1}}\) | \(341\) |
| pseudoelliptic | \(-\frac {\left (-i \sqrt {3}+3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+x^{2}+1}+i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}-x^{3}+2 x^{2}-x +2}{x^{2}}\right )+\left (i \sqrt {3}+3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+x^{2}+1}-i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}-x^{3}+2 x^{2}-x +2}{x^{2}}\right )+\left (-i \sqrt {3}-3\right ) \sqrt {x^{4}+x^{2}+1}\, \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+x^{2}+1}+i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}+x^{3}+2 x^{2}+x +2}{x^{2}}\right )+\sqrt {x^{4}+x^{2}+1}\, \left (i \sqrt {3}-3\right ) \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+x^{2}+1}-i x \left (x^{2}+1\right ) \sqrt {3}+2 x^{4}+x^{3}+2 x^{2}+x +2}{x^{2}}\right )+6 x}{6 \sqrt {x^{4}+x^{2}+1}}\) | \(341\) |
int((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4+3*x^2 +1),x,method=_RETURNVERBOSE)
-x/(x^4+x^2+1)^(1/2)-1/2*ln((x^4+x^2+1)/x^2+1/x*(x^4+x^2+1)^(1/2)+1)-1/6*2 ^(1/2)*6^(1/2)*arctan(1/6*(2*(x^4+x^2+1)^(1/2)*2^(1/2)/x+2^(1/2))*6^(1/2)) +1/2*ln((x^4+x^2+1)/x^2-1/x*(x^4+x^2+1)^(1/2)+1)-1/6*2^(1/2)*6^(1/2)*arcta n(1/6*(2*(x^4+x^2+1)^(1/2)*2^(1/2)/x-2^(1/2))*6^(1/2))
Time = 0.40 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{4} + x^{2} + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )}}{3 \, {\left (x^{5} + x^{3} + x\right )}}\right ) - 3 \, {\left (x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{8} + 5 \, x^{6} + 7 \, x^{4} + 5 \, x^{2} - 2 \, {\left (x^{5} + 2 \, x^{3} + x\right )} \sqrt {x^{4} + x^{2} + 1} + 1}{x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1}\right ) + 6 \, \sqrt {x^{4} + x^{2} + 1} x}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \]
integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4 +3*x^2+1),x, algorithm="fricas")
-1/6*(2*sqrt(3)*(x^4 + x^2 + 1)*arctan(1/3*sqrt(3)*sqrt(x^4 + x^2 + 1)*(x^ 4 + 1)/(x^5 + x^3 + x)) - 3*(x^4 + x^2 + 1)*log((x^8 + 5*x^6 + 7*x^4 + 5*x ^2 - 2*(x^5 + 2*x^3 + x)*sqrt(x^4 + x^2 + 1) + 1)/(x^8 + 3*x^6 + 5*x^4 + 3 *x^2 + 1)) + 6*sqrt(x^4 + x^2 + 1)*x)/(x^4 + x^2 + 1)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=\text {Timed out} \]
integrate((x**4-1)*(x**8+x**6+3*x**4+x**2+1)/(x**4+x**2+1)**(3/2)/(x**8+3* x**6+5*x**4+3*x**2+1),x)
\[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4 +3*x^2+1),x, algorithm="maxima")
integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*(x^4 - 1)/((x^8 + 3*x^6 + 5*x^4 + 3*x^2 + 1)*(x^4 + x^2 + 1)^(3/2)), x)
\[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} {\left (x^{4} - 1\right )}}{{\left (x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4 +3*x^2+1),x, algorithm="giac")
integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*(x^4 - 1)/((x^8 + 3*x^6 + 5*x^4 + 3*x^2 + 1)*(x^4 + x^2 + 1)^(3/2)), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx=\int \frac {\left (x^4-1\right )\,\left (x^8+x^6+3\,x^4+x^2+1\right )}{{\left (x^4+x^2+1\right )}^{3/2}\,\left (x^8+3\,x^6+5\,x^4+3\,x^2+1\right )} \,d x \]
int(((x^4 - 1)*(x^2 + 3*x^4 + x^6 + x^8 + 1))/((x^2 + x^4 + 1)^(3/2)*(3*x^ 2 + 5*x^4 + 3*x^6 + x^8 + 1)),x)