Solve $\int e^{\sin x}\frac{x\cos^3x-\sin x}{\cos^2x} \mathrm{d}x$

Question

How to solve$$\int e^{\sin x}\frac{x\cos^3x-\sin x}{\cos^2x} \mathrm{d}x$$ I tried to use integration by part, but it seems become more difficult.

Answer 1

Let $$\mathcal{I}=\int e^{\sin x}\frac{x\cos^3x}{\cos^2x} \mathrm{d}x~~~,~~~\mathcal{J}=\int e^{\sin x}\frac{\sin x}{\cos^2x} \mathrm{d}x$$ use integration by part we get $$\mathcal{I}=\int e^{\sin x}\frac{x\cos^3x}{\cos^2x} \mathrm{d}x=\int xe^{\sin x}\cos x\, \mathrm{d}x=xe^{\sin x}-\int e^{\sin x}\, \mathrm{d}x$$ $$\mathcal{J}=\int e^{\sin x}\frac{\sin x}{\cos^2x} \mathrm{d}x=\frac{e^{\sin x}}{\cos x}-\int e^{\sin x}\, \mathrm{d}x$$ So $$\int e^{\sin x}\frac{x\cos^3x-\sin x}{\cos^2x} \, \mathrm{d}x=\mathcal{I}-\mathcal{J}$$