Hi Joseph,

perhaps you need the Normal command:

n=20;
out=Series[Sin[Sqrt[(2+Cos[\[Phi]] Cos[\[Theta]])^2+(Cos[\[Phi]] Sin[\[Theta]])^2]]^2,{\[Phi],0,n},{\[Theta],0,n}];
out=Normal[out];
Integrate[out,{\[Phi],0,Pi/2},{\[Theta],0,Pi/2}]

This works for me just fine for n=20:

(\[Pi]^2 (127104656229455436459374343475349510487418916044800000000 Sin[3]^2+\[Pi]^40 (6671526653329717151169403103811 Cos[6]-488075285077837356731633919062 Sin[6])-107520 \[Pi]^36 (29631816193922255545220087871 Cos[6]-104897152921608864240035512462 Sin[6])+8171520 \[Pi]^34 (41655253524777230571763338363 Cos[6]-14425449735263628689695671622 Sin[6])+358469546803200 \[Pi]^28 (14047776264463421772993963 Cos[6]-7432217112172292535267094 Sin[6])-65040714571972608000 \[Pi]^24 (729523240143939095523 Cos[6]-36755838202049296924822 Sin[6])-7057369202688000 \[Pi]^30 (2418502198994877988743 Cos[6]-35089765902142749351838 Sin[6])+338211715774257561600000 \[Pi]^22 (115389039747970605912 Cos[6]-62114605962136586119 Sin[6])-40500988248653652704624640000 \[Pi]^18 (26868740590357242 Cos[6]-316716580849159003 Sin[6])+277026759620790984499632537600000 \[Pi]^16 (675059805347538 Cos[6]-279399090013967 Sin[6])-976552570874442715279344650944512000000 \[Pi]^12 (11027593158 Cos[6]-41854053445 Sin[6])+1421860543193188593446725811775209472000000 \[Pi]^10 (385324062 Cos[6]-72735673 Sin[6])-14594431610708709543487096684320518087639040000000 \[Pi]^6 (2946 Cos[6]-4679 Sin[6])-8826712238156627531900996074677049339404091392000000000 \[Pi]^2 Sin[6]+2451864510599063203305832242965847038723358720000000 \[Pi]^4 (354 Cos[6]+73 Sin[6])-1126113550209005366009806842925965901824000000 \[Pi]^8 (3549894 Cos[6]+5059067 Sin[6])-2712646030206785320220401808179200000 \[Pi]^14 (840771833154 Cos[6]+723869366249 Sin[6])-1217562176787327221760000 \[Pi]^20 (534486745420139847210 Cos[6]+348015855538169860349 Sin[6])-17034472864088064000 \[Pi]^26 (6088724789183387730849 Cos[6]+3683816581314441744154 Sin[6])-10296115200 \[Pi]^32 (766203463636963251923047245 Cos[6]+575702804649685578979873334 Sin[6])-10080 \[Pi]^38 (20746609616746270904283312087 Cos[6]+25637944308055752336839391682 Sin[6])))/508418624917821745837497373901398041949675664179200000000