dx/(sin³xsin(x+a))

multiply and divide by sinx

 sinx/ sin²x(sinxsin(x+a))dx

 sin(x+a-a) / sin²x(sinxsin(x+a))dx

 [sin(x+a)cosa - cos(x+a)sina]/ sin²x(sinxsin(x+a))   dx

splitting into 2 integrals

[sin(x+a)cosa/ sin²x(sinxsin(x+a)) dx  .....1st integral

cosa cosec²x (sin(x+a)/sinx) dx

-cosa/sina  -sina cosec²x (cosa+cotx sina)  dx

put cosa+cotx sina = t

-sina cosec²x dx = dt

so we get

-cota t dt

so integrating we get

-2/3 cota [cosa+cotxsina]^3/2  .....X

 

similarly in the second integral ..

we have

-sinacosec²x (cot(x+a) [cotx cosa - sina])  dx

tana -cosec²a cosa (cot(x+a) [cotx cosa - sina])  dx

put cotx cosa - sina = t

-cosec²x cosa dx = dt

tana t/((t+sina)tana+cosa)   dt

 

now apply parts

tanat d(2((t+sina)tana+cosa))) 

tana [ t * (2((t+sina)tana+cosa))) -  (2((t+sina)tana+cosa))) dt ]

tana  [cotx cosa - sina] (2((cotx cosa - sina+sina)tana+cosa))) - 4/3  ((cotx cosa - sina+sina)tana+cosa))) ^3/2  .............Y

 

answer is X+Y+c