I am trying to solve this integral: $$\int \cos\left(\ln\left(e^t+1\right)+at+b\right)dt $$ with $$a, b \in \Re$$ I can solve this: $$\int \cos\left(\ln\left(x\right)\right)dt $$ Defining: $u=\cos(\ln(x))$ and $v = \frac{1}{x}$. Then using the fomular two times: $$\int udv=uv-\int vdu$$ I can get the solution $$\frac {1}{2}x\cos(\ln(x))+ \frac {1}{2}x\sin(\ln(x)) +C$$
I also found the orther topic in our website The integral $\int\ln(x)\cos(1+(\ln(x))^2)\,dx$. But still have no idea with this.
I wonder is this non-integrable function ?!