
    \hZ-                     f   S SK JrJr  S SKJrJr  S SKJr  S SKJ	r	  S SK
Jr  S SKJr  S SKJr  S SKJrJrJrJrJrJr  \r\r\r\r\r\rS S	KJrJr  \" S
5      u  r r!r"\" SSS9u  r#r$r%r&r'r(\" SSS9r)\" SSS9r*\" SSSSS9u  r+r,\" SSSSSS9u  r-r.\" SSS9r/S r0S r1\" SSS9u  r2r!\\2S-
  -  S\-   \2* \!-
  -  -  \" \2\!-   5      -  \" \25      -  \" \!5      -  r3\" SSSS9r4SS\4S-  -
  -  \\4S-
  -  -  \" \S-  * S-  5      -  \" \4S-  5      -  r5S\)* S-  -  \" \)S-  5      -  \\)S-  S-
  -  -  \" \* S-  5      -  r6\#\(-  \-  \\$-  \#\(-  -  -  S\\#-  \$\#-  -  -   \(S-   -  -  r7\" SSS9u  r8r9\	" \8\-  \8-  \9\9-  -  \8\-  \9-   \8\9-   -  -  5      \-  \" \8S-  5      -  \" \9S-  5      -  \" \8\9-   S-  5      -  r:\" SSS9u  r;r<\\<S-  -  \" \S-  \;S-  -   * S-  \<S-  -  5      -  \" S \\;-  \<S-  -  5      -  r=\" SSS9r>\" \?" \\>-
  5      * \$-  5      S-  \$-  r@\" SSS 9rA\" S!SS9rBS" rC/ S#QrDS S$KEJErE  S S%KFJGrG  S S&KHrH/ rI\JS':X  a  \K" \D5       H  u  rLrM\G" 5         \E" 5       rN\O" \M5        \E" 5       \N-
  rN\I\N\M4/-  rI\HR                  R                  S(5        \HR                  R                  5         \L\S" \D5      S)-  -  S :X  d  Mw  \HR                  R                  S*S)\L-  \S" \D5      -  -  5        M     \T" 5         \IR                  S+ S,9  \I H  u  rVrM\T" S-\V\M4-  5        M     g&g&).    )oopi)Symbolsymbolsexp)sqrt)besseli)gamma)	integrate)mellin_transforminverse_fourier_transforminverse_mellin_transformlaplace_transforminverse_laplace_transformfourier_transform)xyznu beta rhoza b c d k pT)positivek)real)negativezmu1 mu2)r   nonzerofinitezsigma1 sigma2)r   r   r   r   lambdac                 n    S[        S[        -  US-  -  5      -  [        X-
  S-  * S-  US-  -  5      -  $ )N      )r	   r   r   )r   musigmas      X/var/www/auris/envauris/lib/python3.13/site-packages/sympy/benchmarks/bench_meijerint.pynormalr"   !   s=    T!B$uax-  qvk\!^E1H%<!===    c                 &    U[        U* U -  5      -  $ )Nr   )r   rates     r!   exponentialr&   %   s    TE!Gr#   z
alpha betar   )integerr   r   zd1 d2znu sigmar   u)polartc                 \   [        U [        [        [        5      -  [	        [
        [        [        5      -  [        S[        4[
        [        * [        4SS9  [        U [        [        [        5      -  [	        [
        [        [        5      -  [
        [        * [        4[        S[        4SS9  g )Nr   T)meijerg)	r   r&   r   r%   r"   r   mu1sigma1r   )exprs    r!   Er0   ;   st    d;q$''q#v(>>BZ!bS"t=d;q$''q#v(>>"b\Aq":t=r#   )z MT(x**nu*Heaviside(x - 1), x, s)z MT(x**nu*Heaviside(1 - x), x, s)z*MT((1-x)**(beta - 1)*Heaviside(1-x), x, s)z*MT((x-1)**(beta - 1)*Heaviside(x-1), x, s)zMT((1+x)**(-rho), x, s)zMT(abs(1-x)**(-rho), x, s)zKMT((1-x)**(beta-1)*Heaviside(1-x) + a*(x-1)**(beta-1)*Heaviside(x-1), x, s)zMT((x**a-b**a)/(x-b), x, s)z!MT((x**a-bpos**a)/(x-bpos), x, s)zMT(exp(-x), x, s)zMT(exp(-1/x), x, s)z"MT(log(x)**4*Heaviside(1-x), x, s)z"MT(log(x)**3*Heaviside(x-1), x, s)zMT(log(x + 1), x, s)zMT(log(1/x + 1), x, s)zMT(log(abs(1 - x)), x, s)zMT(log(abs(1 - 1/x)), x, s)zMT(log(x)/(x+1), x, s)zMT(log(x)**2/(x+1), x, s)zMT(log(x)/(x+1)**2, x, s)zMT(erf(sqrt(x)), x, s)zMT(besselj(a, 2*sqrt(x)), x, s)z*MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s)z*MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s)z MT(besselj(a, sqrt(x))**2, x, s)z2MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s)z5MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s)z1MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s)z:MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)zMT(bessely(a, 2*sqrt(x)), x, s)z*MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s)z*MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s)z1MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s)z1MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s)z MT(bessely(a, sqrt(x))**2, x, s)zMT(besselk(a, 2*sqrt(x)), x, s)zEMT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(a, 2*sqrt(2*sqrt(x))), x, s)z1MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s)z1MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s)z#MT(exp(-x/2)*besselk(a, x/2), x, s)z>LT((t-apos)**bpos*exp(-cpos*(t-apos))*Heaviside(t-apos), t, s)zLT(t**apos, t, s)zLT(Heaviside(t), t, s)zLT(Heaviside(t - apos), t, s)zLT(1 - exp(-apos*t), t, s)zBLT((exp(2*t)-1)*exp(-bpos - t)*Heaviside(t)/2, t, s, noconds=True)zLT(exp(t), t, s)zLT(exp(2*t), t, s)zLT(exp(apos*t), t, s)zLT(log(t/apos), t, s)zLT(erf(t), t, s)zLT(sin(apos*t), t, s)zLT(cos(apos*t), t, s)z"LT(exp(-apos*t)*sin(bpos*t), t, s)z"LT(exp(-apos*t)*cos(bpos*t), t, s)z%LT(besselj(0, t), t, s, noconds=True)z%LT(besselj(1, t), t, s, noconds=True)z&FT(Heaviside(1 - abs(2*apos*x)), x, k)z2FT(Heaviside(1-abs(apos*x))*(1-abs(apos*x)), x, k)z#FT(exp(-apos*x)*Heaviside(x), x, k)z0IFT(1/(apos + 2*pi*I*x), x, posk, noconds=False)z1IFT(1/(apos + 2*pi*I*x), x, -posk, noconds=False)z!IFT(1/(apos + 2*pi*I*x), x, negk)z%FT(x*exp(-apos*x)*Heaviside(x), x, k)z/FT(exp(-apos*x)*sin(bpos*x)*Heaviside(x), x, k)zFT(exp(-apos*x**2), x, k)z-IFT(sqrt(pi/apos)*exp(-(pi*k)**2/apos), k, x)zFT(exp(-apos*abs(x)), x, k)z=integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)z?integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)zBintegrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)zBintegrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)zkintegrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)zmintegrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)zmintegrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)zointegrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)zsintegrate((x+y+1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)z|integrate((x+y-1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),                   (x, -oo, oo), (y, -oo, oo), meijerg=True)zvintegrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),                (x, -oo, oo), (y, -oo, oo), meijerg=True)zpintegrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),          (x, -oo, oo), (y, -oo, oo), meijerg=True)z9integrate(exponential(x, rate), (x, 0, oo), meijerg=True)z;integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True)z>integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True)zE(1)zE(x*y)z	E(x*y**2)zE((x+y+1)**2)zE(x+y+1)zE((x+y-1)**2)z-integrate(betadist, (x, 0, oo), meijerg=True)z/integrate(x*betadist, (x, 0, oo), meijerg=True)z2integrate(x**2*betadist, (x, 0, oo), meijerg=True)z(integrate(chi, (x, 0, oo), meijerg=True)z*integrate(x*chi, (x, 0, oo), meijerg=True)z-integrate(x**2*chi, (x, 0, oo), meijerg=True)z/integrate(chisquared, (x, 0, oo), meijerg=True)z1integrate(x*chisquared, (x, 0, oo), meijerg=True)z4integrate(x**2*chisquared, (x, 0, oo), meijerg=True)zDintegrate(((x-k)/sqrt(2*k))**3*chisquared, (x, 0, oo), meijerg=True)z*integrate(dagum, (x, 0, oo), meijerg=True)z,integrate(x*dagum, (x, 0, oo), meijerg=True)z/integrate(x**2*dagum, (x, 0, oo), meijerg=True)z&integrate(f, (x, 0, oo), meijerg=True)z(integrate(x*f, (x, 0, oo), meijerg=True)z+integrate(x**2*f, (x, 0, oo), meijerg=True)z)integrate(rice, (x, 0, oo), meijerg=True)z.integrate(laplace, (x, -oo, oo), meijerg=True)z0integrate(x*laplace, (x, -oo, oo), meijerg=True)z3integrate(x**2*laplace, (x, -oo, oo), meijerg=True)z=integrate(log(x) * x**(k-1) * exp(-x) / gamma(k), (x, 0, oo))zEintegrate(sin(z*x)*(x**2-1)**(-(y+S(1)/2)), (x, 1, oo), meijerg=True)Iintegrate(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo), meijerg=True)zKintegrate(besselj(0,x)*besselj(1,x)*besselk(0,x), (x, 0, oo), meijerg=True)r1   z>integrate(besselj(a,x)*besselj(b,x)/x, (x,0,oo), meijerg=True)zhyperexpand(meijerg((-s - a/2 + 1, -s + a/2 + 1), (-a/2 - S(1)/2, -s + a/2 + S(3)/2), (a/2, -a/2), (-a/2 - S(1)/2, -s + a/2 + S(3)/2), 1))a  gammasimp(S('2**(2*s)*(-pi*gamma(-a + 1)*gamma(a + 1)*gamma(-a - s + 1)*gamma(-a + s - 1/2)*gamma(a - s + 3/2)*gamma(a + s + 1)/(a*(a + s)) - gamma(-a - 1/2)*gamma(-a + 1)*gamma(a + 1)*gamma(a + 3/2)*gamma(-s + 3/2)*gamma(s - 1/2)*gamma(-a + s + 1)*gamma(a - s + 1)/(a*(-a + s)))*gamma(-2*s + 1)*gamma(s + 1)/(pi*s*gamma(-a - 1/2)*gamma(a + 3/2)*gamma(-s + 1)*gamma(-s + 3/2)*gamma(s - 1/2)*gamma(-a - s + 1)*gamma(-a + s - 1/2)*gamma(a - s + 1)*gamma(a - s + 3/2))'))zmellin_transform(E1(x), x, s)z3inverse_mellin_transform(gamma(s)/s, s, x, (0, oo))z$mellin_transform(expint(a, x), x, s)zmellin_transform(Si(x), x, s)z^inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0))z#mellin_transform(Ci(sqrt(x)), x, s)zWinverse_mellin_transform(-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S(1)/2)),s, u, (0, 1))zlaplace_transform(Ci(x), x, s)z%laplace_transform(expint(a, x), x, s)z%laplace_transform(expint(1, x), x, s)z%laplace_transform(expint(2, x), x, s)z3inverse_laplace_transform(-log(1 + s**2)/2/s, s, u)z-inverse_laplace_transform(log(s + 1)/s, s, x)z6inverse_laplace_transform((s - log(s + 1))/s**2, s, x)zlaplace_transform(Chi(x), x, s)zlaplace_transform(Shi(x), x, s)z>integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True, conds="none")zAintegrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True, conds="none")z@integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,conds="none")z1integrate(-cos(x)/x, (x, tpos, oo), meijerg=True)z1integrate(-sin(x)/x, (x, tpos, oo), meijerg=True)z,integrate(sin(x)/x, (x, 0, z), meijerg=True)z-integrate(sinh(x)/x, (x, 0, z), meijerg=True)z%integrate(exp(-x)/x, x, meijerg=True)z(integrate(exp(-x)/x**2, x, meijerg=True)z$integrate(cos(u)/u, u, meijerg=True)z%integrate(cosh(u)/u, u, meijerg=True)z(integrate(expint(1, x), x, meijerg=True)z(integrate(expint(2, x), x, meijerg=True)z!integrate(Si(x), x, meijerg=True)z!integrate(Ci(u), u, meijerg=True)z"integrate(Shi(x), x, meijerg=True)z"integrate(Chi(u), u, meijerg=True)z2integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True)z8integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True))time)clear_cacheN__main__.
   z%sc                     U S   * $ )Nr    )r   s    r!   <lambda>r9     s
    !ur#   )keyz%.2fs %s)Wsympy.core.numbersr   r   sympy.core.symbolr   r   &sympy.functions.elementary.exponentialr   (sympy.functions.elementary.miscellaneousr	   sympy.functions.special.besselr
   'sympy.functions.special.gamma_functionsr   sympy.integrals.integralsr   sympy.integrals.transformsr   r   r   r   r   r   LTFTMTIFTILTIMT	sympy.abcr   r   nubetarhoaposbposcposdposposkpr   negkr-   mu2r.   sigma2r%   r"   r&   alphabetadistkintchi
chisquareddagumd1d2fnupossigmaposricer   abslaplacer(   tposr0   benchr2   sympy.core.cacher3   systimings__name__	enumeratenstring_texecstdoutwriteflushlenprintsorttir8   r#   r!   <module>rv      s   ' / 6 9 2 9 /E E  &D#")-$"G dD$a
3TcD!94dCStT $t5h&>lT2tuqy>1q5UFTM2253FF
5\+c4$/!d1f*oa$(m#CAaL0tAv>!AYuQqSz!!acAg,.sA2a4y8
Qq!D&DF##QDt);%;q1u$EE	4	(B2a4"*r2v
1r	R"W5	56q8
2a4[r!t"BGQ;/0*t4x1}S1a4%(?+A-hk9::71aG'{G+ <, ,Dt
s1r6{l4
 
"4
'
3dcD!=l	\  ( 

zu%	6VVVb[RL>!



E
b !Q&JJTRTSZ%789 & 
GLL_L%
FjB<'(  r#   