1/(sinx + secx)

1/(sinx+secx)  =  cosx/(sinxcosx+1) = 2cosx/(2sinxcosx+2)

                                                      =(cosx +sinx)/(3-(sinx-cosx)²) + (cosx-sinx)/((cosx+sinx2)² +1)

 

substitute sinx-cosx and t in first term and cosx-sinx as z in second term so that the integral reduces to

 

I = dt/(3-t² ) + dz/(z² +1)

 

then integrate using standard formulae ..