The statement pattern $[(p \rightarrow q) \wedge \sim q] \rightarrow r$ is a tautology when $r$ is equivalent to

If $3 \sin \alpha=5 \sin \beta$, then $\tan \left(\frac{\alpha+\beta}{2}\right)+\tan \left(\frac{\alpha-\beta}{2}\right)=$

$$\int \frac{\mathrm{d} x}{2 \mathrm{e}^{2 x}+3 \mathrm{e}^x+1}=$$

$$\int \frac{\mathrm{e}^{2030 \log x}-\mathrm{e}^{2029 \log x}}{\mathrm{e}^{2028 \log x}-\mathrm{e}^{2027 \log x}} \mathrm{~d} x=\ldots$$