Define the function $L$ as follows: $$L(a)=\int_0^a \frac{dx}{\sqrt{x^4+1}}$$ I‘ve been trying to calculate the following integral in terms of $L$: $$I(a)=\int_0^a \frac{dx}{\sqrt{x^4+x^2+1}}$$ So far I‘ve tried lots of substitutions in this integral to reduce it to some value of $L$, a constant multiple of a value of $L$, or some sum of values of $L$ multiplied by other functions of $a$, without any luck.
Can anyone find a clever substitution that allows $I(a)$ to be calculated in terms of some combination of values of $L(\cdot)$?
That would imply some identity relating values of the incomplete elliptic integral of the first kind $\text{EllipticF}(z,k)$ (in Maple's notation) at $k=i$ and $k = (1+i\sqrt{3})/2$. I'm not aware of such identities (not that this is necessarily definitive).