9 ∫
                (∘  ---------)
   sin(x)arctan     sec(x ) − 1 dx



optimal
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12 cos(x )∘sec-(x) −-1 + 12 tan− 1(∘sec-(x)-−-1) − cos(x )tan−1 (∘sec-(x)-−-1)



M 9.01

click to view



Rubi 4.1

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Maple 18

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Sage 5.4

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Fricas 1.2

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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

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Axiom 5/12

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Maxima 5.28.02

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reduce 2008

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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

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                                                                                                                                                                    (            ∘--------------               ∘ --------------   )
       ( ∘ −(− cos(x))2+cos(x)sign(cos(x))cos(x))                                                                                                                             (sign(cos(x)))2(2 −(− cos(x))2
−  atan  -----------------2------------  cos (x) + 2 (− sign(cos (x)) ⋅----1---2-- ⋅ 12 ⋅ 12asin(2 ⋅ − cos(x ) + 1) + 2(sign (cos (x)))3 ⋅-1---2-- ⋅ 14 ⋅abs(sig1n(cos&#---
                      cos(x)                                          (sign(cos(x))) −1                                                (sign(cos(x))) −1