To consolidate effort, we now write a procedure that automatically separates out the trig and
polynomial parts of the integrand and performs integration by parts with the polynomial term
chosen as the one that is differentiated. Comments in the procedure body explain what's
happening. This procedure is highly optimized for this particular form of integral.
ugh := proc( expr )
local p, q, H, K, locI, locp, T, remdr;
global parts, _k_;
# update the parts index
if not assigned(_k_) then
_k_ := 0;
else
_k_ := _k_ + 1;
fi;
# grab the integral
if type( expr, function ) and op(0,expr)=Int then
H := expr;
else
H := select( has, expr, Int );
fi;
# get the integrand
K := op(1,H);
# grab the polynomial part and store in p
q := remove( has, %, {sin,cos} );
if type(q,`+`) then #already have it
p := q;
else
for p in q do
if type(p,`+`) then
break;
fi;
od:
fi;
if not has(p,{s,g}) then
locp := location( K, p );
debug_print(procname,`polynomial factor p:`,1,p);
debug_print(procname,`location of p in integrand: `,2,locp);
else
locp := [];
debug_print(procname,`no polynomial factor!`,1);
fi;
# now integrate by parts
debug_print(procname,`integrating by parts...`,0);
if nops(locp) > 0 then
q := intparts( expr, p );
debug_print(procname,`entire integrated mess:`,4,q);
else
debug_print(procname,`entire integrated mess:`,4);
RETURN( value( expr ) );
fi;
# store the evaluated part
parts[_k_] := remove( has, q, Int );
debug_print( procname,
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