On yesterday, a now closed question asked for a while how to compute the antiderivative $$I=\int \log\big[1+\sin^2(t)\big]\,dt$$ which, as given by a CAS, is just awful (see @Forester's comment and @Turing's answer). During the time the question was open, I tried on my side and gave up since being unable to find anything simpler.
Then, I tried to approximate the integrand since we just need a good representation of it for $0 \leq t \leq \frac \pi 2$ because of the symmetries and periodicity. What I am proposing is a polynomial function matching the function and its first and second derivative values at a minimum number of data points. For the time being, I chose $t=0$, $t=\frac \pi 4$ and $t=\frac \pi 2$. The final result is $$f(t)=\log\big[1+\sin^2(t)\big]\sim t^2+\frac 19 \sum_{n=3}^8 \frac{a_n}{\pi^n} t^n=g(t)$$ the $a_n$'s being $$\left( \begin{array}{ccc} n & a_n & \frac{a_n}{9\pi^n}\\ 3 & -768 \pi -217 \pi ^2-22608 \log (2)+18432 \log (3) & +0.087806356 \\ 4 & 7680 \pi +1940 \pi ^2+286416 \log (2)-221184 \log (3) & -1.360303642 \\ 5 & -27648 \pi -8500 \pi ^2-1501632 \log (2)+1105920 \log (3) & +1.225383826 \\ 6 & 43008 \pi +19568 \pi ^2+3960576 \log (2)-2801664 \log (3) & -0.512958585 \\ 7 & -24576 \pi -22720 \pi ^2-5170176 \log (2)+3538944 \log (3) & +0.102611702 \\ 8 & 10496 \pi ^2+2654208 \log (2)-1769472 \log (3) & -0.007207813 \end{array} \right)$$
About the norm $$\Phi=\int_0^{\frac \pi 2} \big[f(t)-g(t)\big]^2 \, dt=1.79 \times 10^{-8}$$ while the beautiful approximation $$\sin(t) \simeq \frac{16 (\pi -t) t}{5 \pi ^2-4 (\pi -t) t}\qquad \text{for} \qquad 0\leq t\leq\pi$$ would lead to $\Phi=8.74 \times 10^{-7}$ (almost $50$ times larger).
From this $$J(x)=\int_0^x \log\big[1+\sin^2(t)\big]\,dt\sim \frac 13 x^3+\frac 19 \sum_{n=3}^8 \frac{a_n}{(n+1)\,\pi^n} x^{n+1}$$
As a test $$J\left(\frac{\pi }{2}\right)=\frac{\pi }{181440}\left(432 (128 \log (3)-87 \log (2))-55 \pi ^2 \right)\approx 0.591383$$ to be compared to the exact value $$ \pi \log\left(\frac{1+\sqrt{2}}{2}\right)\approx 0.591331$$ recalled by @projectilemotion in comments.
My questions
Without adding more terms, could it be possible to improve the approximation ? Would changes of base points could have a significant impact (if yes, how to optimize them ?)
Would, by chance, be known (in a simple form) the exact values of the integrals $$K_n=\int_{-\frac \pi n}^{\frac \pi n} \log\big[1+\sin^2(t)\big]\,dt$$ beside the case of $n=2$ ? If there were, I suppose that much better approximations could be done.