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Integrate[x + Sin[x*((I*Pi)/2 + x)], x] ==
x^2/2 + (Sqrt[Pi/2]* (-(Cos[Pi^2/16]*FresnelC[(Pi - (4*I)*x)/ (2*Sqrt[2*Pi])]) - I*Cos[Pi^2/16]* FresnelC[(I*Pi + 4*x)/(2*Sqrt[2*Pi])] - I*Cos[Pi^2/16]*FresnelS[(Pi - (4*I)*x)/ (2*Sqrt[2*Pi])] + Cos[Pi^2/16]* FresnelS[(I*Pi + 4*x)/(2*Sqrt[2*Pi])] + I*FresnelC[(Pi - (4*I)*x)/(2*Sqrt[2*Pi])]* Sin[Pi^2/16] + FresnelC[(I*Pi + 4*x)/ (2*Sqrt[2*Pi])]*Sin[Pi^2/16] - FresnelS[(Pi - (4*I)*x)/(2*Sqrt[2*Pi])]* Sin[Pi^2/16] + I*FresnelS[(I*Pi + 4*x)/ (2*Sqrt[2*Pi])]*Sin[Pi^2/16]))/2



                     I Pi
Integrate[x + Sin[x (---- + x)], x] == 
                      2

 2                      2
x          Pi         Pi            Pi - (4 I) x
-- + (Sqrt[--] (-(Cos[---] FresnelC[------------]) - 
2          2          16            2 Sqrt[2 Pi]
 
               2
             Pi             I Pi + 4 x
       I Cos[---] FresnelC[------------] - 
             16            2 Sqrt[2 Pi]
 
               2
             Pi            Pi - (4 I) x
       I Cos[---] FresnelS[------------] + 
             16            2 Sqrt[2 Pi]
 
             2
           Pi             I Pi + 4 x
       Cos[---] FresnelS[------------] + 
           16            2 Sqrt[2 Pi]
 
                                      2
                  Pi - (4 I) x      Pi
       I FresnelC[------------] Sin[---] + 
                  2 Sqrt[2 Pi]      16
 
                                    2
                 I Pi + 4 x       Pi
       FresnelC[------------] Sin[---] - 
                2 Sqrt[2 Pi]      16
 
                                    2
                Pi - (4 I) x      Pi
       FresnelS[------------] Sin[---] + 
                2 Sqrt[2 Pi]      16
 
                                      2
                   I Pi + 4 x       Pi
       I FresnelS[------------] Sin[---])) / 2
                  2 Sqrt[2 Pi]      16

Time to compute: 0.51 second

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