Trigonometric Integral

with(student);
[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint, changevar, 

  completesquare, distance, equate, integrand, intercept, intparts, leftbox, 

  leftsum, makeproc, middlebox, middlesum, midpoint, powsubs, rightbox, 

  rightsum, showtangent, simpson, slope, summand, trapezoid]
J:=changevar(cos(x)=y,Int(1/(1+a*cos(x)),x=0..Pi/2),y);

                         /1                           
                        |              1              
                        |   ----------------------- dy
                        |                     (1/2)   
                       /0             /     2\        
                            (1 + a y) \1 - y /        

value(J);

                     /                     /            (1/2)  \\
                     |                     |    / 2    \       ||
                     |                     |    |a  - 1|       ||
                     |                   ln|a + |------|      a||
                     |                     |    |   2  |       ||
                     |                     \    \  a   /       /|
            piecewise|a <>