How do you integrate #(tan(x))/x#?

Question

Scarlett Allison · Accepted Answer

I don't believe there is an intrinsic function that is the anti-derivative

The power series solution is:
# int tanx/xdx = x+1/9x^3+2/75x^5-17/2205x^7+62/25515x^9+... # I believe the only way to handle this integral is to use the Maclaurin power series for #tanx#; as follows; # int tanx/xdx = int (x+1/3x^3+2/15x^5-17/315x^7+62/2835x^9+... )/xdx# # :. int tanx/xdx = int 1+1/3x^2+2/15x^4-17/315x^6+62/2835x^8+... # # :. int tanx/xdx = x+1/3x^3/3+2/15x^5/5-17/315x^7/7+62/2835x^9/9+... # # :. int tanx/xdx = x+1/9x^3+2/75x^5-17/2205x^7+62/25515x^9+... #

Isabella Ackerman · Answer

undefined The integral of (tan(x))/x with respect to x is not expressible in terms of elementary functions. It is a special function called the dilogarithm function, denoted as Li2(x), and can be expressed in terms of it as -Li2(-e^(2i*x)).

Elijah Abram · Answer

undefined The integral of $\frac{\tan(x)}{x}$ does not have an elementary antiderivative. However, it can be expressed in terms of special functions. The integral is denoted by $\text{Ci}(x)$, the cosine integral function. Therefore, the integral can be written as:

$$\int \frac{\tan(x)}{x} dx = \text{Ci}(x) + C$$

Where $C$ is the constant of integration.