### 9

 optimal M 9.01 Rubi 4.1 Maple 18 Sage 5.4 Fricas 1.2 update: per post on sci.math.symbolic on June 20, 2013 BTW: Current developement FriCAS can also do #9:    (3) -> integrate(sin(x)*atan(sqrt(sec(x) - 1)), x)       (3)                                                    +------------+                                                    |- cos(x) + 1                                             cos(x) |------------                        +----------+               \|   cos(x)         - 2cos(x)atan(\|sec(x) - 1 ) + atan(---------------------)                                                   cos(x) - 1       +                +------------+                |- cos(x) + 1         cos(x) |------------               \|   cos(x)    /       2                                           Type: Union(Expression(Integer),...) sympy 0.7.1 Axiom 5/12 Maxima 5.28.02 reduce 2008 mupad 2013a evalin(symengine,'int(sin(x)*atan(sqrt(sec(x)-1)),x)')    pretty(ans)                                                /              1/2                  1/2 \                                                | 3 asin(cos(x)   )   3 (1 - cos(x))    |             1/2                                         cos(x) | ----------------- - ----------------- | (1 - cos(x))                                                |            3/2          2 cos(x)      |          / /   1        \1/2 \                 \    2 cos(x)                           /    - atan| | ------ - 1 |    | cos(x) - ----------------------------------------------------------------          \ \ cos(x)     /    /                                  /   1        \1/2                                                               3 | ------ - 1 |                                                                 \ cos(x)     / xcas