\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{a}+\frac {\left (1+a^{4}+\sqrt {a^{4}+1}\right ) \arctan \left (\frac {a \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {-1-\sqrt {a^{4}+1}}}\right )}{a^{2} \sqrt {a^{4}+1}\, \sqrt {-1-\sqrt {a^{4}+1}}}+\frac {\left (-1-a^{4}+\sqrt {a^{4}+1}\right ) \arctan \left (\frac {a \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {-1+\sqrt {a^{4}+1}}}\right )}{a^{2} \sqrt {a^{4}+1}\, \sqrt {-1+\sqrt {a^{4}+1}}}+\frac {\left (\sqrt {2}\, \sqrt {-a^{2}-\sqrt {a^{4}+1}}-\sqrt {2}\, a^{2} \sqrt {-a^{2}-\sqrt {a^{4}+1}}+\sqrt {2}\, \sqrt {a^{4}+1}\, \sqrt {-a^{2}-\sqrt {a^{4}+1}}\right ) \arctan \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {-a^{2}-\sqrt {a^{4}+1}}\, \left (-1+x^{2}+\sqrt {x^{4}+1}\right )}\right )}{2 a^{2}}+\frac {\left (-\sqrt {2}\, \sqrt {-a^{2}+\sqrt {a^{4}+1}}+\sqrt {2}\, a^{2} \sqrt {-a^{2}+\sqrt {a^{4}+1}}+\sqrt {2}\, \sqrt {a^{4}+1}\, \sqrt {-a^{2}+\sqrt {a^{4}+1}}\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-a^{2}+\sqrt {a^{4}+1}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{2 a^{2}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{a^{2}} \]
command
Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x),x]
Mathematica 13.1 output
\[ \frac {2 a \sqrt {x^2+\sqrt {1+x^4}}+\frac {2 \left (1+a^4+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {2 \left (-1-a^4+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}} \left (1-a^2+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )+\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} \left (-1+a^2+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \]________________________________________________________________________________________