Integrand size = 47, antiderivative size = 520 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\frac {-2 x \sqrt {1+x^4} \left (2+3 x^2+2 x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}-2 x \left (1+3 x^2+3 x^4+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}}{10 x^2 \sqrt {1+x^4} \left (-1+x^2+x^4\right )+5 \left (-1+x^2+x^4\right ) \left (1+2 x^4\right )}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {\text {RootSum}\left [1+2 \text {$\#$1}^2-6 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (-1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right )+7 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-7 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-3 \log \left (-1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}+x^2 \text {$\#$1}+\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}-6 \text {$\#$1}^3-3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{5 \sqrt {2}} \]
[Out]
Result contains complex when optimal does not.
Time = 5.95 (sec) , antiderivative size = 2475, normalized size of antiderivative = 4.76, number of steps used = 112, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6874, 2157, 212, 2158, 745, 739, 6857, 6860} \[ \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 {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 )} \]
[In]
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Rule 212
Rule 739
Rule 745
Rule 2157
Rule 2158
Rule 6857
Rule 6860
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )}\right ) \, dx \\ & = 4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx+4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx+\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx \\ & = 4 \int \left (-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+4 \int \left (\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{5 \sqrt {5} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16}{5} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right )^2 \sqrt {1+x^4}} \, dx+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5}} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {16}{5} \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1+\sqrt {5}\right ) \left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+\sqrt {5}\right ) \left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16}{5} \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1+\sqrt {5}\right ) \left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}\right ) \left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {16 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}+\frac {16 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{5 \sqrt {5}}-\frac {8 \int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {-1+\sqrt {5}} \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}}-\frac {8 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx}{\sqrt {5}} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {2 \left (-1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (2 \left (-1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-1+\sqrt {5}\right )}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}-2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {8 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {2 \left (1+\sqrt {5}\right )}+2 x\right )^2 \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}-\frac {16 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-2 \left (1+\sqrt {5}\right )-4 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (1+\sqrt {5}\right )}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {(8 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{5 \sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}-\frac {(4 i) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.00 (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}} \]
[In]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.08
\[\int \frac {\left (x^{4}+x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )^{2}}d x\]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 25.48 (sec) , antiderivative size = 10224, normalized size of antiderivative = 19.66 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} + x^{2} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + x^{2} - 1\right )}^{2} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}\,{\left (x^4+x^2+1\right )}^2}{\sqrt {x^4+1}\,{\left (x^4+x^2-1\right )}^2} \,d x \]
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