Some integrals with problematic evaluation

[quinyu]

Code:

int(tan(x)*tan(x-A),x)

This integral is of the tan(a+bx)*tan(c+dx) general form, where IF b^2-d^2 is 0 AND bc-ad isn't 0, we can rewrite with secants into

Code:

-bx/d+(b/d)cos((bc-ad)/d)*int(sec(a+bx)sec(c+dx),x)

So we are now at

Code:

-x+cos(a)*int(sec(a-x)sec(x),x)

which isn't much better. Nevertheless, for an integral of the general form

Code:

int(sec(a+bx)sec(c+dx),x)

it can be split into

Code:

csc((bc-ad)/d)int(tan(a+bx),x)+csc((bc-ad)/b)int(tan(c+dx),x)

Knowing that the integral

Code:

int(tan(c+dx),x)=1/d*ln(cos(c+dx))

we apply this twice, and after the simplification of the trigonometric functions arrive to

Code:

-x+cot(a)ln(cos(a-x))-cot(a)ln(cos(x))

∎