How to evaluate this integral without numerical methods?

Question

$$ \int_{0}^{\pi/3}\frac{\mathrm{e}^x}{\cos\left(x\right)}\,\mathrm{d}x $$

I tried it with Trapezoidal and Simpson's got the correct answer which matches with Wolfram but how to find that form with hypergeometric and polygamma function ?.

Answer 1

You can transform this into a form suitable for the hypergeometric function by choosing $x=\ln t$. We then find,

$$ dx=dt/t\\ e^x=t\\ \cos x=\frac{t^i+t^{-i}}{2}\\ \int\frac{e^x}{\cos x}\,dx=2\int \frac{dt}{t^i+t^{-i}} $$

WolframAlpha gives

$$ 2\int \frac{dt}{t^i+t^{-i}}=(1-i)t^{1+i}\, _2F_1\left(\frac{1-i}{2},1;\frac{3-i}{2};-t^{2i} \right)\color{gray}{+const} $$

Compare that with this, also from WolframAlpha,

$$ \int\frac{e^x}{\cos x}\,dx=(1-i)e^{(1+i)x}\, _2F_1\left(\frac{1-i}{2},1;\frac{3-i}{2};e^{-2ix} \right)\color{gray}{+const} $$

You should be able to take it from here.

Answer 2

This is not an answer since the result just came out from a CAS.

$$\int_{0}^{a}\frac{e^x}{\cos x}\,dx=-\frac{1}{2} i \left((2+2 i) e^{(1+i) a} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i a}\right)-\psi ^{(0)}\left(\frac{3}{4}-\frac{i}{4}\right)+\psi ^{(0)}\left(\frac{1}{4}-\frac{i}{4}\right)\right)$$ provided that $(2 \Re(a)+\pi \geq 0\land 2 \Re(a)\leq \pi )\lor a\notin \mathbb{R}$