\[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate((a*x^2+1)/(a*x^2-1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} \arctan \left ({\left (a^{4} \sqrt {\frac {a + 1}{a^{6}}} + a\right )} \sqrt {{\left (2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2\right )} \sqrt {-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}} + x + \sqrt {x^{2} + 1}} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {3}{4}} - {\left (a^{4} \sqrt {\frac {a + 1}{a^{6}}} + a\right )} \sqrt {x + \sqrt {x^{2} + 1}} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {3}{4}}\right ) - 4 \, \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {1}{8} \, {\left ({\left (a^{4} \sqrt {\frac {a + 1}{a^{6}}} - a\right )} \sqrt {-64 \, {\left (2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2\right )} \sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}} + 64 \, x + 64 \, \sqrt {x^{2} + 1}} \sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}} - 8 \, {\left (a^{4} \sqrt {\frac {a + 1}{a^{6}}} - a\right )} \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}}\right )} \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}}\right ) + \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]