)abbrev package INTEF ElementaryIntegration
++ Integration of elementary functions
++ Author: Manuel Bronstein
++ Date Created: 1 February 1988
++ Description:
++ This package provides functions for integration, limited integration,
++ extended integration and the risch differential equation for
++ elemntary functions.
++ Keywords: elementary, function, integration.
ElementaryIntegration(R, F) : Exports == Implementation where
  R : Join(GcdDomain, Comparable, CharacteristicZero,
           PolynomialFactorizationExplicit,
           RetractableTo Integer, LinearlyExplicitOver Integer)
  F : Join(AlgebraicallyClosedField, TranscendentalFunctionCategory,
           FunctionSpace R)

  SY  ==> Symbol
  K   ==> Kernel F
  P   ==> SparseMultivariatePolynomial(R, K)
  UP  ==> SparseUnivariatePolynomial F
  RF  ==> Fraction UP
  IR  ==> IntegrationResult F
  IRRF ==> IntegrationResult RF
  FF  ==> Record(ratpart : RF, coeff : RF)
  LLG ==> List Record(coeff : F, logand : F)
  Ext_Rec ==> Record(ratpart:F, coeff:F)
  U2  ==> Union(Ext_Rec, "failed")
  U3  ==> Union(Record(mainpart:F, limitedlogs:LLG), "failed")
  U4  ==> Union(Record(ratpart : F, coeff : F, prim : F), "failed")
  ANS ==> Record(special : F, integrand : F)
  PSOL==> Record(ans : F, right : F, sol? : Boolean)
  PSOL2 ==> Record(ans : F, right : F, primpart : F, sol? : Boolean)
  FAIL==> error "failed - cannot handle that integrand"
  OPDIFF ==> '%diff
  Param_Rec_F ==> Record(ratpart : F, coeffs : Vector F)

  Exports ==> with
    lfextendedint : (F, SY, F) -> U2
       ++ lfextendedint(f, x, g) returns functions \spad{[h, c]} such that
       ++ \spad{dh/dx = f - cg}, if (h, c) exist, "failed" otherwise.
    lfintegrate  : (F, SY) -> IR
       ++ lfintegrate(f, x) = g such that \spad{dg/dx = f}.

  Implementation ==> add
    import from IntegrationTools(R, F)
    import from RationalIntegration(F, UP)
    import from AlgebraicIntegration(R, F)
    import from AlgebraicManipulations(R, F)
    import from TranscendentalIntegration(F, UP)
    import from PureAlgebraicIntegration(R, F, F)
    import from IntegrationResultFunctions2(F, F)
    import from IntegrationResultFunctions2(RF, F)
    import from FunctionSpacePrimitiveElement(R, F)
    import from PolynomialCategoryQuotientFunctions(IndexedExponents K,
                                                             K, R, P, F)
    import from ParametricIntegration(R, F)

    N ==> NonNegativeInteger

    PRIM := 'prim
    ALGOP := '%alg

    alglfint    : (F, K, List K, SY) -> IR
    algprimint  : (F, K, K, SY) -> IR
    algexpint   : (F, K, K, SY) -> IR
    primint     : (F, SY, K) -> IR
    expint      : (F, SY, K) -> IR
    lambint     : (F, SY, K) -> IR
    prim?       : (K, SY)  -> Boolean
    isx?        : (F, SY)  -> Boolean
    addx        : (IR, F) -> IR
    lfintegrate0 : (F, SY) -> IR
    unknownint  : (F, SY) -> IR
    diffint     : (F, SY, K) -> IR
    tryChangeVar : (F, K, SY) -> Union(IR, "failed")

    prim?(k, x)      == is?(k, 'log) or has?(operator k, PRIM)

    lambint(f, x, k) ==
        eta' := differentiate(eta := first argument k, x)
        dfac := (monomial(1, 1)$UP + 1$UP)
        derivative : UP -> UP :=
           (x1 : UP) : UP +-> dfac*differentiate(x1,
              (x2 : F) : F +-> differentiate(x2, x), 0$UP) +
                differentiate(x1, (x2 : F) : F +-> 0,
                  monomial(eta'/eta, 1))
        -- XXX FIXME need proper extended integration function
        extint : F -> U2 := f +-> lfextendedint(f, x, eta'/eta)
        r1 := lambintegrate((monomial(1, 1)$UP + 1$UP)*univariate(f, k),
                      eta'/eta, (x1 : F) : F +-> differentiate(x1, x),
                      derivative, extint, (x2 : F) : IR +-> lfintegrate(x2, x))
        map((x1 : RF) : F +-> multivariate(x1, k), r1.answer) + r1.a0

-- tries various tricks since the integrand contains something not elementary
    unknownint(f, x) ==
      (da := differentiate(a := denom(f)::F, x)) ~= 0 and
        zero? differentiate(c := numer(f)::F / da, x) => (c * log a)::IR
      mkAnswer(0, empty(), [[f, x::F]])

    LOG ==> Record(scalar : Fraction(Integer), coeff : UP, logand : UP)

    dummy := create()$SingletonAsOrderedSet

    diffint1(f : F, x : SY, k : K) : Union(IR, "failed") ==
        fu := univariate(f, k)
        denom(fu) ~= 1 => "failed"
        nfu := numer(fu)
        degree(nfu) ~= 1 => "failed"
        f1 := leadingCoefficient(nfu)
        args := argument(k)
        #args ~= 3 => error "internal error, k is not a diff"
        arg3 := args(3)
        da3 := differentiate(arg3, x)
        da3 = 0 => "failed"
        k1 := eval(args(1), retract(args(2))@K, arg3)
        dv := new()$Symbol
        f2 := eval(f1/da3, k1, dv::F)
        nres := lfintegrate(f2, dv)
        not(empty?(notelem(nres))) => "failed"
        logs := logpart(nres)
        nlogs : List(LOG) := []
        nrat : F := 0
        for lg in logs repeat
            cpol := lg.coeff
            every?((x1 : F) : Boolean +-> D(x1, x) = 0, coefficients(cpol)) =>
                nlogs := cons(lg, nlogs)
            degree(cpol) ~= 1 =>
                -- print(lg::OutputForm)
                return "failed"
            alpha := -coefficient(cpol, 0)/leadingCoefficient(cpol)
            nrat := nrat + lg.scalar*alpha*
                      log(retract(eval(lg.logand, dummy, alpha))@F)
        nres2 := mkAnswer(ratpart(nres) + nrat, reverse!(nlogs), [])
        nres3 := map((x1 : F) : F+->eval(x1, kernel(dv), k1::F), nres2)
        f3 := f - differentiate(nres3, x)
        -- should not happen in differential transcendental
        -- case, but in principle possible
        member?(k, kernels(f3)) => "failed"
        nres3 + lfintegrate(f3, x)

    diffint(f, x, k) ==
        (r1 := diffint1(f, x, k)) case IR => r1
        mkAnswer(0, empty(), [[f, x::F]])

    isx?(f, x) ==
      (k := retractIfCan(f)@Union(K, "failed")) case "failed" => false
      (r := symbolIfCan(k@K)) case "failed" => false
      r@SY = x

    alglfint(f, k, l, x) ==
      xf := x::F
      symbolIfCan(kx := ksec(k, l, x)) case SY => addx(palgint(f, kx, k), xf)
      is?(kx, 'exp) => addx(algexpint(f, kx, k, x), xf)
      prim?(kx, x)       => addx(algprimint(f, kx, k, x), xf)
      has?(operator kx, ALGOP) =>
        is?(operator k, 'nthRoot) and is?(operator kx, 'nthRoot) and
           not(member?(kx, tower(k::F))) =>
              al :=  alg_split_roots([f], k, kx)$RootFSSplit(R, F)
              -- print(al::OutputForm)
              res := 0$IR
              for rec in al repeat
                  ir1 := lfintegrate(first(rec.funs), x)
                  if rec.npow1 + rec.npow2 > 0 then
                      oroot := (k::F)^rec.npow1*(kx::F)^rec.npow2
                      ir1 := map((x1 : F) : F +->
                                eval(x1, retract(rec.nroot)@K, oroot),
                                ir1)
                  res := res + ir1
              -- print(res::OutputForm)
              res
        rec := primitiveElement(kx::F, k::F)
        y   := rootOf(rec.prim)
        map((x1 : F) : F+->eval(x1, retract(y)@K, rec.primelt),
          lfintegrate(eval(f, [kx, k], [(rec.pol1) y, (rec.pol2) y]), x))
      unknownint(f, x)

    lfintegrate(f, x) ==
      zero? f => 0
      xf := x::F
      empty?(l := varselect(kernels f, x)) => (xf * f)::IR
      symbolIfCan(k := kmax l) case SY =>
        map((x1 : RF) : F+->multivariate(x1, k), integrate univariate(f, k))
      is?(k, 'exp)  => addx(expint(f, x, k), xf)
      is?(k, 'lambertW) => addx(lambint(f, x, k), xf)
      prim?(k, x)        => addx(primint(f, x, k), xf)
      has?(operator k, ALGOP) => alglfint(f, k, l, x)
      is?(k, OPDIFF) => diffint(f, x, k)
      unknownint(f, x)

    addx(i, x) ==
      elem? i => i
      mkAnswer(ratpart i, logpart i,
                                [[ne.integrand, x] for ne in notelem i])

    tryChangeVar(f, t, x) ==
        z := new()$Symbol
        g := subst(f / differentiate(t::F, x), [t], [z::F])
        freeOf?(g, x) =>               -- can we do change of variables?
            map((x1 : F) : F+->eval(x1, kernel z, t::F), lfintegrate(g, z))
        "failed"


    algexpint(f, t, y, x) ==
        (u := tryChangeVar(f, t, x)) case IR => u
        algint(f, t, y,
          (x1 : UP) : UP+->differentiate(x1,
            (x2 : F) : F+->differentiate(x2, x),
                       monomial(differentiate(first argument t, x), 1)),
               f1 +-> lfintegrate(f1, x))

    algprimint(f, t, y, x) ==
        (u := tryChangeVar(f, t, x)) case IR => u
        algint(f, t, y,
          (x1 : UP) : UP+->differentiate(x1,
            (x2 : F) : F+->differentiate(x2, x),
                                            differentiate(t::F, x)::UP),
               f1 +-> lfintegrate(f1, x))

    lfextendedint(f, x, g) ==
      res1 := extendedint(f, x, [g]).particular
      res1 case "failed" => "failed"
      [res1.ratpart, (res1.coeffs)(1)]


    DREC ==> Record(answer : RF, logpart : RF, ir : IRRF)

    denint_dummy(f : RF) : DREC == [0, f, 0]

    DMap ==> (RF, K, SY) -> DREC

    denint_li : DMap :=
        R is Integer and F is Expression(Integer) =>
            (li_int$DenominatorIntegration(R, F))@DMap
        R is Complex(Integer) and F is Expression(Complex(Integer)) =>
            (li_int$DenominatorIntegration(R, F))@DMap
        (rf : RF, k : K, x : SY) : DREC +-> denint_dummy(rf)

    denint_poly : DMap :=
        R is Integer and F is Expression(Integer) =>
            (poly_int$DenominatorIntegration(R, F))@DMap
        R is Complex(Integer) and F is Expression(Complex(Integer)) =>
            (poly_int$DenominatorIntegration(R, F))@DMap
        (rf : RF, k : K, x : SY) : DREC +-> denint_dummy(rf)

    primint(f, x, k) ==
      lk := varselect([a for a in tower f | a ~= k], x)
      denint : RF -> DREC :=
          is?(k, 'log) => (rf : RF) : DREC +-> denint_li(rf, k, x)
          denint_dummy
      dk := differentiate(k::F, x)
      ext := (f1 : F, d : N) : U4 +->
          polylog_int(f1, x, k, d, lk, dk)
      r1 := primintegrate(univariate(f, k),
              (x1 : UP) : UP+->differentiate(x1,
                (x2 : F) : F+->differentiate(x2, x), dk::UP),
                  denint, ext)
      map((x1 : RF) : F+->multivariate(x1, k), r1.answer) + lfintegrate(r1.a0, x)

    risch_de_solver(x3 : Integer, x4 : F, eta : F, x : SY) : PSOL2 ==
        risch_de_ext(x3, eta, x4, x
                    )$ElementaryRischDEX2(R, F)

    expint(f, x, k) ==
      eta := first argument k
      r1  := expintegrate(univariate(f, k),
               (x1 : UP) : UP+->differentiate(x1,
                 (x2 : F) : F+->differentiate(x2, x),
                 monomial(differentiate(eta, x), 1)),
               (rf : RF) : DREC +-> denint_poly(rf, k, x),
               (x3 : Integer, x4 : F) : PSOL2 +->
                    risch_de_solver(x3, x4, eta, x))
      map((x1 : RF) : F+->multivariate(x1, k), r1.answer) + lfintegrate(r1.a0, x)

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