using math, reduce;

// Initialization (this requires the REDUCE image file in the current dir).
reduce::start "C:/Users/scios/Desktop/pure-lang/pure-reduce/reduce.img" {"-v"};

// Verbosity level. 0 means no noise at all.
reduce::verbosity 0;

// Package: DEFINT (definite integrals)
// Calculating integrals by using the Meijer G integration formula.
//http://www.reduce-algebra.com/docs/defint.pdf
//
reduce::load "defint" ;
def exp x = (quote e)^x;

// Some definite integrals
let r1 = simplify $ intg (exp(-x)) x 0 infinity ; // -> 1
let r2 = simplify $ intg (x*sin(1/x)) x 0 infinity ; // -> uneval
let r3 = simplify $ intg (x^2*cos(x)*exp(-2*x)) x 0 infinity ; // -> 4%125
//let r4 = simplify $ intg (x*exp(-1/2x)*Heaviside(1-x)) x 0 infinity ; 
let r4 = simplify $ intg (x*exp(-1/2*x)) x 0 1 ; // -> 2*sqrt e*(2*sqrt e-3)/e
let r5 = simplify $ intg (x*log(1+x)) x 0 1 ; // -> 1%4 ;
let r6 = simplify $ intg (cos(2*x)) x y (2*y); // -> (sin (4*y)-sin (2*y))/2


// Laplace transform
// =================
let r7 = simplify $ laplace_transform (exp(-a*x)) x ; // -> 1/(a+s) 


// Hankel transform
// ================
let r8 = simplify $ hankel_transform (exp(-a*x)) x ; // ->
         //  s^(n/2)*gamma (n/2)*hypergeometric [(n+2)/2] [n+1] 
         // ((-s)/a)*n/(2*a^(n/2)*gamma (n+1)*a)

         
// Y transform
// ===========
let r9 = simplify $ y_transform (exp(-a*x)) x ; // ->         
         // (a^n*gamma (n+1)*gamma ((-n)/2)*gamma ((-2*n-1)/2)*
         // gamma ((2*n+3)/2)*hypergeometric [(-n+2)/2] [-n+1] 
         // ((-s)/a)+s^n*gamma (-n)*gamma (n/2)*hypergeometric [(n+2)/2] [n+1] 
         // ((-s)/a)*n*pi)/(2*s^(n/2)*a^(n/2)*gamma ((-2*n-1)/2)*
         // gamma ((2*n+3)/2)*a*pi)   

         
// K transform
// ===========
let r10 = simplify $ k_transform (exp(-a*x)) x ; // ->
         // (-a^n*gamma (n+1)*gamma ((-n)/2)*hypergeometric [(-n+2)/2] [-n+1] 
         // (s/a)+s^n*gamma (-n)*gamma (n/2)*hypergeometric [(n+2)/2] [n+1] 
         // (s/a)*n)/(4*s^(n/2)*a^(n/2)*a)

         
// StruveH transform
// =================
let r11 = simplify $ struveh_transform (exp(-a*x)) x ; // ->      
         // 2*s^((n+1)/2)*gamma ((n+3)/2)*
         // hypergeometric [1,(n+3)/2] [(2*n+3)/2,1.5] ((-s)/a)/
         // (sqrt pi*a^((n+1)/2)*gamma ((2*n+3)/2)*a)

         
// Fourier sine transform
// ======================
let r12 = simplify $ fourier_sin (exp(-a*x)) x ; // -> s/(a^2+s^2)


// Fourier cosine transform
// ========================
let r13 = simplify $ fourier_cos (exp(-a*x)) x ; // -> a/(a^2+s^2)