+
    iZ-                        ^ RI HtHt ^ RIHtHt ^ RIHt ^ RIH	t	 ^ RI
Ht ^ RIHt ^ RIHt ^ RIHtHtHtHtHtHt ]t]t]t]t]t]t^ R	IHtHt ]! R
4      w  t t!t"]! RRR7      w  t#t$t%t&t't(]! RRR7      t)]! RRR7      t*]! RRRRR7      w  t+t,]! RRRRRR7      w  t-t.]! RRR7      t/R t0R t1]! RRR7      w  t2t!]]2^,
          ,          ^],           ]2) ]!,
          ,          ,          ]! ]2]!,           4      ,          ]! ]24      ,          ]! ]!4      ,          t3]! RRRR7      t4^^]4^,          ,
          ,          ]]4^,
          ,          ,          ]! ]^,          ) ^,          4      ,          ]! ]4^,          4      ,          t5^])) ^,          ,          ]! ])^,          4      ,          ]])^,          ^,
          ,          ,          ]! ]) ^,          4      ,          t6]#](,          ],          ]]$,          ]#](,          ,          ,          ^]]#,          ]$]#,          ,          ,           ](^,           ,          ,          t7]! RRR7      w  t8t9]	! ]8],          ]8,          ]9]9,          ,          ]8],          ]9,           ]8]9,           ,          ,          4      ],          ]! ]8^,          4      ,          ]! ]9^,          4      ,          ]! ]8]9,           ^,          4      ,          t:]! RRR7      w  t;t<]]<^,          ,          ]! ]^,          ];^,          ,           ) ^,          ]<^,          ,          4      ,          ]! ^ ]];,          ]<^,          ,          4      ,          t=]! RRR7      t>]! ]?! ]]>,
          4      ) ]$,          4      ^,          ]$,          t@]! RRR7      tA]! RRR7      tBR  tC. R!NR"NR#NR$NR%NR&NR'NR(NR)NR*NR+NR,NR-NR.NR/NR0NR1NR2NR3NR4NR5NR6NR7NR8NR9NR:NR;NR<NR=NR>NR?NR@NRANRBNRCNRDNRENRFNRGNRHNRINRJNRKNRLNRMNRNNRONRPNRQNRRNRSNRTNRUNRVNRWNRXNRYNRZNR[NR\NR]NR^NR_NR`NRaNRbNRcNRdNReNRfNRgNRhNRiNRjNRkNRlNRmNRnNRoNRpNRqNRrNRsNRtNRuNRvNRwNRxNRyNRzNR{NR|NR}NR~NRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNtD^ RIEHEtE ^ RIFHGtG ^ RIHtH. tI]JR8X  Ed   ]K! ]D4       F  w  tLtM]G! 4        ]E! 4       tN]O! ]M4       ]E! 4       ]N,
          tN]I]N]M3.,          tI]HP                  P                  R4       ]HP                  P                  4        ]L]S! ]D4      ^
,          ,          ^ 8X  g   K  ]HP                  P                  R^
]L,          ]S! ]D4      ,          ,          4       K  	  ]T! 4        ]IP                  R R7       ]I F  w  tVtM]T! R]V]M3,          4       K  	  R# R# )    )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                     ^\        ^\        ,          V^,          ,          4      ,          \        W,
          ^,          ) ^,          V^,          ,          4      ,          # )   )r	   r   r   )r   musigmas   &&&ڀ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/benchmarks/bench_meijerint.pynormalr!   !   s=    T!B$uax-  qvk\!^E1H%<!===    c                 6    V\        V) V ,          4      ,          # )Nr   )r   rates   &&r    exponentialr%   %   s    TE!Gr"   z
alpha beta)integerr   zd1 d2znu sigmar   u)polartc                    \        V \        \        \        4      ,          \	        \
        \        \        4      ,          \        ^ \        3\
        \        ) \        3RR7       \        V \        \        \        4      ,          \	        \
        \        \        4      ,          \
        \        ) \        3\        ^ \        3RR7       R# )r   T)meijergN)	r   r%   r   r$   r!   r   mu1sigma1r   )exprs   &r    Er/   ;   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)zIintegrate(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)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                     V ^ ,          ) # )r    )r   s   &r    <lambda>r6     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
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