Find $ \int \cos(x) \frac {\ln(\sin(x))} {\ln(\tan(x))}dx$

Question

Find $\displaystyle \int \cos(x) \frac {\ln(\sin(x))} {\ln(\tan(x))}dx.$

A friend and I found this problem statement in our high school mathematics textbook under "hard integrals", and after hours trying every trick we've been taught, we can't solve it.

How do you solve this?

Answer 1

According to Maple this antiderivative is not elementary. Since this involves the "purely transcendental" case of the Risch-Norman algorithm, which I believe Maple has implemented completely, I'm confident that this is correct.

Answer 2

$$I=\int\cos(x)\frac{\ln(\sin x)}{\ln(\tan x)}dx$$ we could try and use that: $$\frac{\ln(\sin x)}{\ln(\tan x)}=\frac{\ln(\sin x)}{\ln(\sin x)-\ln(\cos x)}$$ or that $\ln(\tan x)=-2\ln(\csc^2x-1)$ and maybe use $u=\sin(x)$ although with the two logs present this doesn't look easy either.

EDIT: Another thought would be use feynmans rule: $$I(t)=\int\cos(x)\frac{\ln(\sin xt)}{\ln(\tan x)}dx$$ or something similar then differentiate wrt t.

Answer 3

For the fun of it, I tried series expansion of the integrand. We obtain things like $$\cos (x)\frac{ \log (\sin (x))}{\log (\tan (x))}=1-\frac{ (t+1)}{2 t}x^2+\frac{ (t+2)^2}{24 t^2}x^4-\frac{ \left(t^3+t^2+12t+40\right)}{720 t^3}x^6+\frac{\left(9 t^4+576 t^3-568 t^2-2688 t+6720\right)}{362880 t^4}x^8+\cdots$$ where $t=\log(x)$. So, the general form is $$\cos (x)\frac{ \log (\sin (x))}{\log (\tan (x))}=\sum_{n=0}^\infty \frac{P_n(t)}{t^n} x^{2n}$$

This means that we face a bunch of integrals $$I_{m,n}=\int\frac {x^m}{\log^n(x)}\,dx=-\frac{E_n(-(m+1) \log (x)) } {\log^{n-1}(x)}$$

Limited to the truncated expansion given at the beginning, for a limited range, the results do not look too bad for the integral between $0$ and $k$ $$\left( \begin{array}{ccc} k & \text{approximation} & \text{exact} \\ 0.05 & 0.049986 & 0.049986 \\ 0.10 & 0.099897 & 0.099897 \\ 0.15 & 0.149694 & 0.149694 \\ 0.20 & 0.199369 & 0.199369 \\ 0.25 & 0.248956 & 0.248956 \\ 0.30 & 0.298540 & 0.298540 \\ 0.35 & 0.348275 & 0.348275 \\ 0.40 & 0.398411 & 0.398411 \\ 0.45 & 0.449341 & 0.449341 \\ 0.50 & 0.501690 & 0.501692 \\ 0.55 & 0.556477 & 0.556491 \\ 0.60 & 0.615470 & 0.615546 \\ 0.65 & 0.682009 & 0.682434 \\ 0.70 & 0.763211 & 0.765857 \\ 0.75 & 0.876866 & 0.899287 \end{array} \right)$$