I happened upon this in MIT OpenCourseWare:

https://www.youtube.com/watch?v=PNKj529yY5c&feature=youtu.be&t=963

This transform allows converting an integral of the form (tan(x))^n to a polynomial problem.

For example:

Let:

f( tan x ) = ( tan x )^4

and

y = tan x

Then:

f( y ) = ( y )^4

Substituting into the transform:

Integral [tan^4 (x) dx] = Integral [ y^4 / (1 + y^2) dy ]

Using long division:

y^4 / (1 + y^2) = y^2 - 1 + 1 / (1 + y^2)

Substituting into the right-hand-side of the transformed problem above and performing the integration:

Integral [ y^2 - 1 + 1 / (1 + y^2) dy ] = 1/3 * y^3 - y + Arctan(y)

Then returning to the x-world:

Integral [tan^4 (x) dx] = 1/3 * tan^3(x) - tan(x) + x + C

He said in the video: "there's a whole family of things like that," but I don't know where.

My Google searches didn't point me to a list where this transform is included.

Are other members of this family available online somewhere?

Maybe this family could be printed on T-shirts?