(self, x, u)
return self.func(newfunc, *newlimits)
r"""
        Performs a change of variables from `x` to `u` using the relationship
        given by `x` and `u` which will define the transformations `f` and `F`
        (which are inverses of each other) as follows:

        1) If `x` is a Symbol (which is a variable of integration) then `u`
           will be interpreted as some function, f(u), with inverse F(u).
           This, in effect, just makes the substitution of x with f(x).

        2) If `u` is a Symbol then `x` will be interpreted as some function,
           F(x), with inverse f(u). This is commonly referred to as
           u-substitution.

        Once f and F have been identified, the transformation is made as
        follows:

        .. math:: \int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x)
                  \frac{\mathrm{d}}{\mathrm{d}x}

        where `F(x)` is the inverse of `f(x)` and the limits and integrand have
        been corrected so as to retain the same value after integration.

        Notes
        =====

        The mappings, F(x) or f(u), must lead to a unique integral. Linear
        or rational linear expression, `2*x`, `1/x` and `sqrt(x)`, will
        always work; quadratic expressions like `x**2 - 1` are acceptable
        as long as the resulting integrand does not depend on the sign of
        the solutions (see examples).

        The integral will be returned unchanged if `x` is not a variable of
        integration.

        `x` must be (or contain) only one of of the integration variables. If
        `u` has more than one free symbol then it should be sent as a tuple
        (`u`, `uvar`) where `uvar` identifies which variable is replacing
        the integration variable.
        XXX can it contain another integration variable?

        Examples
        ========

        >>> from sympy.abc import a, b, c, d, x, u, y
        >>> from sympy import Integral, S, cos, sqrt

        >>> i = Integral(x*cos(x**2 - 1), (x, 0, 1))

        transform can change the variable of integration

        >>> i.transform(x, u)
        Integral(u*cos(u**2 - 1), (u, 0, 1))

        transform can perform u-substitution as long as a unique
        integrand is obtained:

        >>> i.transform(x**2 - 1, u)
        Integral(cos(u)/2, (u, -1, 0))

        This attempt fails because x = +/-sqrt(u + 1) and the
        sign does not cancel out of the integrand:

        >>> Integral(cos(x**2 - 1), (x, 0, 1)).transform(x**2 - 1, u)
        Traceback (most recent call last):
        ...
        ValueError:
        The mapping between F(x) and f(u) did not give a unique integrand.

        transform can do a substitution. Here, the previous
        result is transformed back into the original expression
        using "u-substitution":

        >>> ui = _
        >>> _.transform(sqrt(u + 1), x) == i
        True

        We can accomplish the same with a regular substitution:

        >>> ui.transform(u, x**2 - 1) == i
        True

        If the `x` does not contain a symbol of integration then
        the integral will be returned unchanged. Integral `i` does
        not have an integration variable `a` so no change is made:

        >>> i.transform(a, x) == i
        True

        When `u` has more than one free symbol the symbol that is
        replacing `x` must be identified by passing `u` as a tuple:

        >>> Integral(x, (x, 0, 1)).transform(x, (u + a, u))
        Integral(a + u, (u, -a, -a + 1))
        >>> Integral(x, (x, 0, 1)).transform(x, (u + a, a))
        Integral(a + u, (a, -u, -u + 1))

        See Also
        ========

        variables : Lists the integration variables
        as_dummy : Replace integration variables with dummy ones
r"""
        Performs a change of variables from `x` to `u` using the relationship
        given by `x` and `u` which will define the transformations `f` and `F`
        (which are inverses of each other) as follows:
r Performs a change of variables from x to u using the relationship given by x and u which will define the transformations f and F ( which are inverses of each other ) as follows :
def transform(self, x, u):
        r"""
        Performs a change of variables from `x` to `u` using the relationship
        given by `x` and `u` which will define the transformations `f` and `F`
        (which are inverses of each other) as follows:

        1) If `x` is a Symbol (which is a variable of integration) then `u`
           will be interpreted as some function, f(u), with inverse F(u).
           This, in effect, just makes the substitution of x with f(x).

        2) If `u` is a Symbol then `x` will be interpreted as some function,
           F(x), with inverse f(u). This is commonly referred to as
           u-substitution.

        Once f and F have been identified, the transformation is made as
        follows:

        .. math:: \int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x)
                  \frac{\mathrm{d}}{\mathrm{d}x}

        where `F(x)` is the inverse of `f(x)` and the limits and integrand have
        been corrected so as to retain the same value after integration.

        Notes
        =====

        The mappings, F(x) or f(u), must lead to a unique integral. Linear
        or rational linear expression, `2*x`, `1/x` and `sqrt(x)`, will
        always work; quadratic expressions like `x**2 - 1` are acceptable
        as long as the resulting integrand does not depend on the sign of
        the solutions (see examples).

        The integral will be returned unchanged if `x` is not a variable of
        integration.

        `x` must be (or contain) only one of of the integration variables. If
        `u` has more than one free symbol then it should be sent as a tuple
        (`u`, `uvar`) where `uvar` identifies which variable is replacing
        the integration variable.
        XXX can it contain another integration variable?

        Examples
        ========

        >>> from sympy.abc import a, b, c, d, x, u, y
        >>> from sympy import Integral, S, cos, sqrt

        >>> i = Integral(x*cos(x**2 - 1), (x, 0, 1))

        transform can change the variable of integration

        >>> i.transform(x, u)
        Integral(u*cos(u**2 - 1), (u, 0, 1))

        transform can perform u-substitution as long as a unique
        integrand is obtained:

        >>> i.transform(x**2 - 1, u)
        Integral(cos(u)/2, (u, -1, 0))

        This attempt fails because x = +/-sqrt(u + 1) and the
        sign does not cancel out of the integrand:

        >>> Integral(cos(x**2 - 1), (x, 0, 1)).transform(x**2 - 1, u)
        Traceback (most recent call last):
        ...
        ValueError:
        The mapping between F(x) and f(u) did not give a unique integrand.

        transform can do a substitution. Here, the previous
        result is transformed back into the original expression
        using "u-substitution":

        >>> ui = _
        >>> _.transform(sqrt(u + 1), x) == i
        True

        We can accomplish the same with a regular substitution:

        >>> ui.transform(u, x**2 - 1) == i
        True

        If the `x` does not contain a symbol of integration then
        the integral will be returned unchanged. Integral `i` does
        not have an integration variable `a` so no change is made:

        >>> i.transform(a, x) == i
        True

        When `u` has more than one free symbol the symbol that is
        replacing `x` must be identified by passing `u` as a tuple:

        >>> Integral(x, (x, 0, 1)).transform(x, (u + a, u))
        Integral(a + u, (u, -a, -a + 1))
        >>> Integral(x, (x, 0, 1)).transform(x, (u + a, a))
        Integral(a + u, (a, -u, -u + 1))

        See Also
        ========

        variables : Lists the integration variables
        as_dummy : Replace integration variables with dummy ones
        """

        d = Dummy('d')

        xfree = x.free_symbols.intersection(self.variables)
        if len(xfree) > 1:
            raise ValueError(
                'F(x) can only contain one of: %s' % self.variables)
        xvar = xfree.pop() if xfree else d

        if xvar not in self.variables:
            return self

        u = sympify(u)
        if isinstance(u, Expr):
            ufree = u.free_symbols
            if len(ufree) != 1:
                raise ValueError(filldedent('''
                When f(u) has more than one free symbol, the one replacing x
                must be identified: pass f(u) as (f(u), u)'''))
            uvar = ufree.pop()
        else:
            u, uvar = u
            if uvar not in u.free_symbols:
                raise ValueError(filldedent('''
                Expecting a tuple (expr, symbol) where symbol identified
                a free symbol in expr, but symbol is not in expr's free
                symbols.'''))
            if not isinstance(uvar, Symbol):
                raise ValueError(filldedent('''
                Expecting a tuple (expr, symbol) but didn't get
                a symbol; got %s''' % uvar))

        if x.is_Symbol and u.is_Symbol:
            return self.xreplace({x: u})

        if not x.is_Symbol and not u.is_Symbol:
            raise ValueError('either x or u must be a symbol')

        if uvar == xvar:
            return self.transform(x, u.subs(uvar, d)).xreplace({d: uvar})

        if uvar in self.limits:
            raise ValueError(filldedent('''
            u must contain the same variable as in x
            or a variable that is not already an integration variable'''))

        if not x.is_Symbol:
            F = [x.subs(xvar, d)]
            soln = solve(u - x, xvar, check=False)
            if not soln:
                raise ValueError('no solution for solve(F(x) - f(u), x)')
            f = [fi.subs(uvar, d) for fi in soln]
        else:
            f = [u.subs(uvar, d)]
            pdiff, reps = posify(u - x)
            puvar = uvar.subs([(v, k) for k, v in reps.items()])
            soln = [s.subs(reps) for s in solve(pdiff, puvar)]
            if not soln:
                raise ValueError('no solution for solve(F(x) - f(u), u)')
            F = [fi.subs(xvar, d) for fi in soln]

        newfuncs = set([(self.function.subs(xvar, fi)*fi.diff(d)
                        ).subs(d, uvar) for fi in f])
        if len(newfuncs) > 1:
            raise ValueError(filldedent('''
            The mapping between F(x) and f(u) did not give
            a unique integrand.'''))
        newfunc = newfuncs.pop()

        def _calc_limit_1(F, a, b):
            """
            replace d with a, using subs if possible, otherwise limit
            where sign of b is considered
            """
            wok = F.subs(d, a)
            if wok is S.NaN or wok.is_bounded is False and a.is_bounded:
                return limit(sign(b)*F, d, a)
            return wok

        def _calc_limit(a, b):
            """
            replace d with a, using subs if possible, otherwise limit
            where sign of b is considered
            """
            avals = list(set([_calc_limit_1(Fi, a, b) for Fi in F]))
            if len(avals) > 1:
                raise ValueError(filldedent('''
                The mapping between F(x) and f(u) did not
                give a unique limit.'''))
            return avals[0]

        newlimits = []
        for xab in self.limits:
            sym = xab[0]
            if sym == xvar:
                if len(xab) == 3:
                    a, b = xab[1:]
                    a, b = _calc_limit(a, b), _calc_limit(b, a)
                    if a > b:
                        a, b = b, a
                        newfunc = -newfunc
                    newlimits.append((uvar, a, b))
                elif len(xab) == 2:
                    a = _calc_limit(xab[1], 1)
                    newlimits.append((uvar, a))
                else:
                    newlimits.append(uvar)
            else:
                newlimits.append(xab)

        return self.func(newfunc, *newlimits)
def transform ( self , x , u ) : d = Dummy ( 'd' ) xfree = x . free_symbols . intersection ( self . variables ) if len ( xfree ) > 1 : raise ValueError ( 'F(x) can only contain one of: %s' % self . variables ) xvar = xfree . pop ( ) if xfree else d if xvar not in self . variables : return self u = sympify ( u ) if isinstance ( u , Expr ) : ufree = u . free_symbols if len ( ufree ) != 1 : raise ValueError ( filldedent ( '''
                When f(u) has more than one free symbol, the one replacing x
                must be identified: pass f(u) as (f(u), u)''' ) ) uvar = ufree . pop ( ) else : u , uvar = u if uvar not in u . free_symbols : raise ValueError ( filldedent ( '''
                Expecting a tuple (expr, symbol) where symbol identified
                a free symbol in expr, but symbol is not in expr's free
                symbols.''' ) ) if not isinstance ( uvar , Symbol ) : raise ValueError ( filldedent ( '''
                Expecting a tuple (expr, symbol) but didn't get
                a symbol; got %s''' % uvar ) ) if x . is_Symbol and u . is_Symbol : return self . xreplace ( { x : u } ) if not x . is_Symbol and not u . is_Symbol : raise ValueError ( 'either x or u must be a symbol' ) if uvar == xvar : return self . transform ( x , u . subs ( uvar , d ) ) . xreplace ( { d : uvar } ) if uvar in self . limits : raise ValueError ( filldedent ( '''
            u must contain the same variable as in x
            or a variable that is not already an integration variable''' ) ) if not x . is_Symbol : F = [ x . subs ( xvar , d ) ] soln = solve ( u - x , xvar , check = False ) if not soln : raise ValueError ( 'no solution for solve(F(x) - f(u), x)' ) f = [ fi . subs ( uvar , d ) for fi in soln ] else : f = [ u . subs ( uvar , d ) ] pdiff , reps = posify ( u - x ) puvar = uvar . subs ( [ ( v , k ) for k , v in reps . items ( ) ] ) soln = [ s . subs ( reps ) for s in solve ( pdiff , puvar ) ] if not soln : raise ValueError ( 'no solution for solve(F(x) - f(u), u)' ) F = [ fi . subs ( xvar , d ) for fi in soln ] newfuncs = set ( [ ( self . function . subs ( xvar , fi ) * fi . diff ( d ) ) . subs ( d , uvar ) for fi in f ] ) if len ( newfuncs ) > 1 : raise ValueError ( filldedent ( '''
            The mapping between F(x) and f(u) did not give
            a unique integrand.''' ) ) newfunc = newfuncs . pop ( ) def _calc_limit_1 ( F , a , b ) : """
            replace d with a, using subs if possible, otherwise limit
            where sign of b is considered
            """ wok = F . subs ( d , a ) if wok is S . NaN or wok . is_bounded is False and a . is_bounded : return limit ( sign ( b ) * F , d , a ) return wok def _calc_limit ( a , b ) : """
            replace d with a, using subs if possible, otherwise limit
            where sign of b is considered
            """ avals = list ( set ( [ _calc_limit_1 ( Fi , a , b ) for Fi in F ] ) ) if len ( avals ) > 1 : raise ValueError ( filldedent ( '''
                The mapping between F(x) and f(u) did not
                give a unique limit.''' ) ) return avals [ 0 ] newlimits = [ ] for xab in self . limits : sym = xab [ 0 ] if sym == xvar : if len ( xab ) == 3 : a , b = xab [ 1 : ] a , b = _calc_limit ( a , b ) , _calc_limit ( b , a ) if a > b : a , b = b , a newfunc = - newfunc newlimits . append ( ( uvar , a , b ) ) elif len ( xab ) == 2 : a = _calc_limit ( xab [ 1 ] , 1 ) newlimits . append ( ( uvar , a ) ) else : newlimits . append ( uvar ) else : newlimits . append ( xab ) return self . func ( newfunc , * newlimits )