From: edgar@math.ohio-state.edu.invalid   
      
   I think you are right.  Taking the two solutions, finding corresponding   
   constants, and solving I get   
      
   int(exp(-cos(t)^2)/cos(t)^2, t = Pi/4 .. x)   
      
   equal to   
      
   -(2*sin(2*x)*cos(2*x)*cos(x)^2*2^(1/2)*exp(-1/2)*HeunCPrime(1, 1/2,   
   -1/2, -1, 7/8, (1/2)*cos(2*x)+1/2)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)+2*sin(2*x)*cos(x)^2*2^(1/2)*exp(-1/2)*HeunCPrime(1, 1/2, -1/2, -1,   
   7/8, (1/2)*cos(2*x)+1/2)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)+2*sin(2*x)*cos(x)^2*2^(1/2)*exp(-1/2)*HeunCPrime(1, -1/2, -1/2,   
   -1, 7/8, 1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)-2*sin(2*x)*cos(x)^2*HeunCPrime(1, -1/2, -1/2, -1,   
   7/8, (1/2)*cos(2*x)+1/2)*(2*cos(2*x)+2)^(1/2)*HeunCPrime(1, 1/2, -1/2,   
   -1, 7/8, 1/2)*exp(-1/2)-2*sin(2*x)*cos(x)^2*HeunCPrime(1, -1/2, -1/2,   
   -1, 7/8, (1/2)*cos(2*x)+1/2)*(2*cos(2*x)+2)^(1/2)*HeunC(1, 1/2, -1/2,   
   -1, 7/8,   
   1/2)*exp(-1/2)-2*cos(2*x)*cos(x)*2^(1/2)*sin(x)*exp(-1/2)*HeunCPrime(1,   
   -1/2, -1/2, -1, 7/8, 1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)-2*cos(x)*2^(1/2)*sin(x)*exp(-1/2)*HeunCPrime(1,   
   -1/2, -1/2, -1, 7/8, 1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)+2*cos(x)*(2*cos(2*x)+2)^(1/2)*HeunCPrime(1, 1/2,   
   -1/2, -1, 7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*sin(x)*exp(-1/2)+2*cos(x)*(2*cos(2*x)+2)^(1/2)*HeunC   
   (1, 1/2, -1/2, -1, 7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*sin(x)*exp(-1/2)+cos(2*x)*2^(1/2)*exp(-cos(x)^2)*Heu   
   nCPrime(1, -1/2, -1/2, -1, 7/8, 1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)+2^(1/2)*exp(-cos(x)^2)*HeunCPrime(1, -1/2, -1/2,   
   -1, 7/8, 1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)-(2*cos(2*x)+2)^(1/2)*HeunCPrime(1, 1/2, -1/2, -1,   
   7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*exp(-cos(x)^2)-(2*cos(2*x)+2)^(1/2)*HeunC(1, 1/2,   
   -1/2, -1, 7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*exp(-cos(x)^2))/(cos(x)*(sin(2*x)*cos(2*x)*HeunCPrim   
   e(1, 1/2, -1/2, -1, 7/8, (1/2)*cos(2*x)+1/2)*HeunCPrime(1, -1/2, -1/2,   
   -1, 7/8, 1/2)*2^(1/2)*cos(x)+sin(2*x)*HeunCPrime(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)*2^(1/2)*cos(x)+sin(2*x)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)*2^(1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*cos(x)-sin(2*x)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*(2*cos(2*x)+2)^(1/2)*HeunCPrime(1, 1/2, -1/2, -1,   
   7/8, 1/2)*cos(x)-sin(2*x)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*(2*cos(2*x)+2)^(1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   1/2)*cos(x)-cos(2*x)*HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)*2^(1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*sin(x)-HeunCPrime(1, -1/2, -1/2, -1, 7/8,   
   1/2)*2^(1/2)*HeunC(1, 1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*sin(x)+(2*cos(2*x)+2)^(1/2)*HeunCPrime(1, 1/2,   
   -1/2, -1, 7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8,   
   (1/2)*cos(2*x)+1/2)*sin(x)+(2*cos(2*x)+2)^(1/2)*HeunC(1, 1/2, -1/2, -1,   
   7/8, 1/2)*HeunC(1, -1/2, -1/2, -1, 7/8, (1/2)*cos(2*x)+1/2)*sin(x)))   
      
   A big answer!  Graphically, these two functions agree perfectly in the   
   interval (0 , Pi/2) .   
      
   --   
   G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/   
      
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