<s>[INST] antiderivative involving modified bessel function This little integral has been holding me up for weeks. Has anyone come across a similar integral in their work. 
$\int {\frac{d x}{c-I_0(x)}} $
 [/INST] I doubt that there is a closed-form formula.  Neither Maple nor Mathematica could find one.  They don't even get formulas for the case $c=0$.  Would a series expansion in $x$ be of any use?  It is
$$
\frac{1}{c-1} x + \frac{1}{12 (c-1)^2} x^{3} + \frac{c+3}{320 (c-1)^3} x^{5} +
$$
$$
+ \frac{c^2+16 c+19}{16128 (c-1)^4} x^{7} + \frac{c^3+65 c^2+299 c+211}{1327104 (c-1)^5} x^{9} + 
$$
$$
+ \frac{c^4+246 c^3+3156 c^2+7346 c+3651}{162201600 (c-1)^6} x^{11} + O(x^{13})$$
</s>