O14-912
> restart:with(student);

  [D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint,

        changevar, combine, completesquare, distance, equate, extrema,

        integrand, intercept, intparts, isolate, leftbox, leftsum,

        makeproc, maximize, middlebox, middlesum, midpoint, minimize,

        powsubs, rightbox, rightsum, showtangent, simpson, slope,

        summand, trapezoid, value]

> f:=1/(x-cos(t));

                                    1
                           f := ----------
                                x - cos(t)

> assume (a>1);
> g:=Doubleint(f,x=a..b,t=0..Pi);value(g);

                           Pi    b
                          /     /
                         |     |       1
                   g :=  |     |   ---------- dx dt
                         |     |   x - cos(t)
                        /     /
                          0     a~


                                                           2
                                           1 + a~ - sqrt(a~  - 1)
  -I ln(a~ - sqrt((a~ - 1) (a~ + 1))) ln(- ----------------------)
                                                     %4

                                2
                1 + a~ + sqrt(a~  - 1)
         + I ln(----------------------) %5 - I
                          %6

                           2
           1 + a~ - sqrt(a~  - 1)                   1
        ln(----------------------) ln(-----------------------------)
                     %4               -a~ + sqrt((a~ - 1) (a~ + 1))

                                       2
                         1 + b + sqrt(b  - 1)
         + I ln(%3) ln(- --------------------) +
                                  %2

                                                            2
                                             -1 - b + sqrt(b  - 1)
        I ln(b - sqrt((b - 1) (b + 1))) ln(- ---------------------)
                                                      %1

                              2
                1 + b + sqrt(b  - 1)        1
         + I ln(--------------------) ln(- ----)
                         %2                 %3

                                                               2
         - I %5 ln(1 + a~ + sqrt((a~ - 1) (a~ + 1))) + 1/3 I Pi

                   2                                       2
         + 1/2 I %5  + 1/2 I ln(-b + sqrt((b - 1) (b + 1)))

                                   2
                   1 + a~ - sqrt(a~  - 1)                2
         - I dilog(----------------------) - 1/2 I ln(%3)
                             %4

                              2
              1 + a~ + sqrt(a~  - 1)
         + ln(----------------------) Pi - I dilog(%2)
                        %6

                                   2
                   1 + a~ + sqrt(a~  - 1)
         - I dilog(----------------------)
                             %6

                                 2
                   1 + b + sqrt(b  - 1)
         + I dilog(--------------------) + I dilog(%6) + I dilog(%1)
                            %2

         - Pi ln(a~ + 1) + Pi ln(b + 1)

                                                 2
         + 1/2 I ln(a~ - sqrt((a~ - 1) (a~ + 1)))

                                 2
                   1 - b + sqrt(b  - 1)
         - I dilog(--------------------) + I dilog(%4)
                            %1

         + I %5 ln(-1 + a~ + sqrt((a~ - 1) (a~ + 1))) + I

        ln(a~ - sqrt((a~ - 1) (a~ + 1)))

                              2
             -1 + a~ - sqrt(a~  - 1)
        ln(- -----------------------) -
                       %4

                           2
             1 - b + sqrt(b  - 1)                   1
        I ln(--------------------) ln(- --------------------------)
                      %1                -b + sqrt((b - 1) (b + 1))

                                        2
                         -1 + b + sqrt(b  - 1)
         - I ln(%3) ln(- ---------------------)
                                  %2

                                                              2
                                                1 - b + sqrt(b  - 1)
         - I ln(b - sqrt((b - 1) (b + 1))) ln(- --------------------)
                                                         %1

                   2
  %1 := -b + sqrt(b  - 1)

                  2
  %2 := b + sqrt(b  - 1)

  %3 := b + sqrt((b - 1) (b + 1))

                    2
  %4 := a~ - sqrt(a~  - 1)

  %5 := ln(a~ + sqrt((a~ - 1) (a~ + 1)))

                    2
  %6 := a~ + sqrt(a~  - 1)

> 
> assume(x>1);h:=int(f,t=0..Pi);

                                    Pi
                          h := -------------
                                      2
                               sqrt(x~  - 1)

> k:=int(h,x=a..b);

  k := ln(b + sqrt((b - 1) (b + 1))) Pi

         - ln(a~ + sqrt((a~ - 1) (a~ + 1))) Pi

>