What is the integral of e^{tan(x)} ?

The integral of e^{tan(x)} is -i 1/2 (cos(1)-isin(1))((cos(2)+isin(2))\Ei(tan(x)-i)-\Ei(tan(x)+i))+C

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integral of e^{tan(x)}

\int e^{t a n (x)} d x

- i \frac{1}{2} (cos (1) - i sin (1)) ((cos (2) + i sin (2)) Ei (tan (x) - i) - Ei (tan (x) + i)) + C

Solution steps

\int e^{t a n (x)} d x

Use the nonelementary integral:

\int e^{t a n (x)} d x = - \frac{1}{2} i e^{- i} (e^{2 i} Ei (tan (x) - i) - Ei (tan (x) + i))

= - \frac{1}{2} i e^{- i} (e^{2 i} Ei (tan (x) - i) - Ei (tan (x) + i))

Simplify

- \frac{1}{2} i e^{- i} (e^{2 i} Ei (tan (x) - i) - Ei (tan (x) + i)) : - i \frac{1}{2} (cos (1) - i sin (1)) ((cos (2) + i sin (2)) Ei (tan (x) - i) - Ei (tan (x) + i))

= - i \frac{1}{2} (cos (1) - i sin (1)) ((cos (2) + i sin (2)) Ei (tan (x) - i) - Ei (tan (x) + i))

Add a constant to the solution

= - i \frac{1}{2} (cos (1) - i sin (1)) ((cos (2) + i sin (2)) Ei (tan (x) - i) - Ei (tan (x) + i)) + C

What is the integral of e^{tan(x)} ?
The integral of e^{tan(x)} is -i 1/2 (cos(1)-isin(1))((cos(2)+isin(2))\Ei(tan(x)-i)-\Ei(tan(x)+i))+C