Integrand size = 47, antiderivative size = 520 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\frac {-2 x \sqrt {1+x^4} \left (2+3 x^2+2 x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}-2 x \left (1+3 x^2+3 x^4+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}}{10 x^2 \sqrt {1+x^4} \left (-1+x^2+x^4\right )+5 \left (-1+x^2+x^4\right ) \left (1+2 x^4\right )}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {\text {RootSum}\left [1+2 \text {$\#$1}^2-6 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right )+7 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-7 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-3 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}-6 \text {$\#$1}^3-3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{5 \sqrt {2}} \]
Time = 1.05 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=-\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (1+2 x^6+2 \sqrt {1+x^4}+3 x^2 \left (1+\sqrt {1+x^4}\right )+x^4 \left (3+2 \sqrt {1+x^4}\right )\right )}{5 \left (-1+x^2+x^4\right ) \left (1+2 x^4+2 x^2 \sqrt {1+x^4}\right )}+\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+2 \sqrt {2} \text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 \log \left (1+x^2+\sqrt {1+x^4}\right )+8 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]+\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {163 \log \left (1+x^2+\sqrt {1+x^4}\right )-163 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )-59 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+59 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+13 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-13 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{5 \sqrt {2}} \]
(-2*x*Sqrt[x^2 + Sqrt[1 + x^4]]*(1 + 2*x^6 + 2*Sqrt[1 + x^4] + 3*x^2*(1 + Sqrt[1 + x^4]) + x^4*(3 + 2*Sqrt[1 + x^4])))/(5*(-1 + x^2 + x^4)*(1 + 2*x^ 4 + 2*x^2*Sqrt[1 + x^4])) + Sqrt[2]*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sq rt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] + 2*Sqrt[2]*RootSum[1 - 2*#1^2 - 6*#1^ 4 + 2*#1^6 + #1^8 & , (-8*Log[1 + x^2 + Sqrt[1 + x^4]] + 8*Log[Sqrt[2]*x*S qrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] + 3*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 - 3*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^2 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 + Log[ Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^4 )/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ] + RootSum[1 - 2*#1^2 - 6*#1^4 + 2*# 1^6 + #1^8 & , (163*Log[1 + x^2 + Sqrt[1 + x^4]] - 163*Log[Sqrt[2]*x*Sqrt[ x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1] - 59*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^2 + 59*Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^ 2*#1 - Sqrt[1 + x^4]*#1]*#1^2 + 13*Log[1 + x^2 + Sqrt[1 + x^4]]*#1^4 - 13* Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]* #1^4 - Log[1 + x^2 + Sqrt[1 + x^4]]*#1^6 + Log[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]] - #1 - x^2*#1 - Sqrt[1 + x^4]*#1]*#1^6)/(-#1 - 6*#1^3 + 3*#1^5 + 2*#1^7) & ]/(5*Sqrt[2])
Result contains complex when optimal does not.
Time = 9.56 (sec) , antiderivative size = 2475, normalized size of antiderivative = 4.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (x^4+x^2+1\right )^2 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1} \left (x^4+x^2-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}}+\frac {4 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1} \left (x^4+x^2-1\right )}+\frac {4 \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1} \left (x^4+x^2-1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 i \arctan \left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {i x^2+1}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {2 i \arctan \left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {i x^2+1}}\right )}{5 \left (1-\sqrt {5}\right ) \sqrt {(3+i)-(1+i) \sqrt {5}}}+\frac {6 i \arctan \left (\frac {(1+i) \left (i \sqrt {-1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {i x^2+1}}\right )}{5 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {2 i \arctan \left (\frac {(1+i) \left (i \sqrt {-1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {i x^2+1}}\right )}{5 \left (1-\sqrt {5}\right ) \sqrt {(3+i)-(1+i) \sqrt {5}}}+\frac {2 i \sqrt {\frac {2}{-1+\sqrt {5}}} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left ((-1-2 i)+\sqrt {5}\right )^{3/2}}-\frac {2 i \sqrt {\frac {2}{-1+\sqrt {5}}} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {5}\right )} x+2\right )}{\sqrt {(-1-2 i)+\sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left ((-1-2 i)+\sqrt {5}\right )^{3/2}}+\frac {2 \arctan \left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left (1+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {6 \arctan \left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \arctan \left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left (1+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}+\frac {6 \arctan \left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {2 \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (2-\sqrt {2 \left (1+\sqrt {5}\right )} x\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {2 \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\sqrt {2 \left (1+\sqrt {5}\right )} x+2\right )}{\sqrt {(-1-2 i)-\sqrt {5}} \sqrt {i x^2+1}}\right )}{5 \left ((1+2 i)+\sqrt {5}\right ) \sqrt {(-1-i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}-\frac {i \sqrt {(2+2 i) \left ((-2-i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{(35+5 i)-(15+5 i) \sqrt {5}}+\frac {\left (\frac {3}{5}-\frac {3 i}{5}\right ) \text {arctanh}\left (\frac {i \sqrt {-1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )}}+\frac {i \sqrt {(2+2 i) \left ((-2-i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {i \sqrt {-1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{(35+5 i)-(15+5 i) \sqrt {5}}+\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\left (\frac {1}{25}-\frac {3 i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-2-i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {i \sqrt {2 \left (-1+\sqrt {5}\right )} x+2}{\sqrt {(4+2 i)-2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3-\sqrt {5}}-\frac {\sqrt {(2+2 i) \left ((2+i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{(35+5 i)+(15+5 i) \sqrt {5}}+\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}+\frac {\sqrt {(2+2 i) \left ((2+i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{(35+5 i)+(15+5 i) \sqrt {5}}-\frac {\left (\frac {3}{5}+\frac {3 i}{5}\right ) \text {arctanh}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {\left (\frac {5}{2}+\frac {5 i}{2}\right ) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\left (\frac {3}{25}+\frac {i}{25}\right ) \sqrt {\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((2+i)+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {5}\right )} x+2}{\sqrt {(4+2 i)+2 i \sqrt {5}} \sqrt {1-i x^2}}\right )}{3+\sqrt {5}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {x^4+1}}}\right )}{\sqrt {2}}+\frac {16 i \sqrt {1-i x^2}}{5 \left ((16+8 i)-8 \sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}+\frac {2 i \sqrt {1-i x^2}}{5 \left ((-2-i)+\sqrt {5}\right ) \left (2 x+\sqrt {2 \left (-1+\sqrt {5}\right )}\right )}-\frac {2 i \sqrt {1-i x^2}}{5 \left ((2+i)+\sqrt {5}\right ) \left (2 x+i \sqrt {2 \left (1+\sqrt {5}\right )}\right )}+\frac {2 i \sqrt {i x^2+1}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {i x^2+1}}{5 \left ((2-i)+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )}-\frac {2 i \sqrt {i x^2+1}}{5 \left ((-2+i)+\sqrt {5}\right ) \left (2 x+\sqrt {2 \left (-1+\sqrt {5}\right )}\right )}+\frac {2 i \sqrt {i x^2+1}}{5 \left ((2-i)+\sqrt {5}\right ) \left (2 x+i \sqrt {2 \left (1+\sqrt {5}\right )}\right )}\) |
(((16*I)/5)*Sqrt[1 - I*x^2])/(((16 + 8*I) - 8*Sqrt[5])*(Sqrt[2*(-1 + Sqrt[ 5])] - 2*x)) + (((2*I)/5)*Sqrt[1 - I*x^2])/(((2 + I) + Sqrt[5])*(I*Sqrt[2* (1 + Sqrt[5])] - 2*x)) + (((2*I)/5)*Sqrt[1 - I*x^2])/(((-2 - I) + Sqrt[5]) *(Sqrt[2*(-1 + Sqrt[5])] + 2*x)) - (((2*I)/5)*Sqrt[1 - I*x^2])/(((2 + I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])] + 2*x)) + (((2*I)/5)*Sqrt[1 + I*x^2])/( ((-2 + I) + Sqrt[5])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x)) - (((2*I)/5)*Sqrt[1 + I*x^2])/(((2 - I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])] - 2*x)) - (((2*I)/5 )*Sqrt[1 + I*x^2])/(((-2 + I) + Sqrt[5])*(Sqrt[2*(-1 + Sqrt[5])] + 2*x)) + (((2*I)/5)*Sqrt[1 + I*x^2])/(((2 - I) + Sqrt[5])*(I*Sqrt[2*(1 + Sqrt[5])] + 2*x)) - (((2*I)/5)*ArcTan[((1 + I)*(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x))/ (Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/((1 - Sqrt[5])*Sqrt[(3 + I) - (1 + I)*Sqrt[5]]) - (((6*I)/5)*ArcTan[((1 + I)*(Sqrt[2] - I*Sqrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/Sqrt[(- 5 - 5*I)*((-2 + I) + Sqrt[5])] + (((2*I)/5)*ArcTan[((1 + I)*(Sqrt[2] + I*S qrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt[1 + I*x^2])])/( (1 - Sqrt[5])*Sqrt[(3 + I) - (1 + I)*Sqrt[5]]) + (((6*I)/5)*ArcTan[((1 + I )*(Sqrt[2] + I*Sqrt[-1 + Sqrt[5]]*x))/(Sqrt[2*((-1 - 2*I) + Sqrt[5])]*Sqrt [1 + I*x^2])])/Sqrt[(-5 - 5*I)*((-2 + I) + Sqrt[5])] + (((2*I)/5)*Sqrt[2/( -1 + Sqrt[5])]*ArcTan[((1/2 + I/2)*(2 - I*Sqrt[2*(-1 + Sqrt[5])]*x))/(Sqrt [(-1 - 2*I) + Sqrt[5]]*Sqrt[1 + I*x^2])])/((-1 - 2*I) + Sqrt[5])^(3/2) ...
3.31.85.3.1 Defintions of rubi rules used
Not integrable
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.08
\[\int \frac {\left (x^{4}+x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )^{2}}d x\]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 24.93 (sec) , antiderivative size = 10224, normalized size of antiderivative = 19.66 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
integrate((x^4+x^2+1)^2*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1 )^2,x, algorithm="fricas")
Timed out. \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\text {Timed out} \]
Not integrable
Time = 0.34 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}} \,d x } \]
integrate((x^4+x^2+1)^2*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1 )^2,x, algorithm="maxima")
Not integrable
Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}} \,d x } \]
integrate((x^4+x^2+1)^2*(x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2)/(x^4+x^2-1 )^2,x, algorithm="giac")
Not integrable
Time = 8.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}\,{\left (x^4+x^2+1\right )}^2}{\sqrt {x^4+1}\,{\left (x^4+x^2-1\right )}^2} \,d x \]