(integrand, symbol, **options)
return result
Returns the steps needed to compute an integral.

    This function attempts to mirror what a student would do by hand as
    closely as possible.

    SymPy Gamma uses this to provide a step-by-step explanation of an
    integral. The code it uses to format the results of this function can be
    found at
    https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.

    Examples
    ========

    >>> from sympy import exp, sin, cos
    >>> from sympy.integrals.manualintegrate import integral_steps
    >>> from sympy.abc import x
    >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    URule(u_var=_u, u_func=exp(x), constant=1,
        substep=ArctanRule(context=1/(_u**2 + 1), symbol=_u),
        context=exp(x)/(exp(2*x) + 1), symbol=x)
    >>> print(repr(integral_steps(sin(x), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
    >>> print(repr(integral_steps((x**2 + 3)**2 , x))) \
    # doctest: +NORMALIZE_WHITESPACE
    RewriteRule(rewritten=x**4 + 6*x**2 + 9,
    substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
        ConstantTimesRule(constant=6, other=x**2,
            substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
                context=6*x**2, symbol=x),
        ConstantRule(constant=9, context=9, symbol=x)],
    context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)


    Returns
    =======
    rule : namedtuple
        The first step; most rules have substeps that must also be
        considered. These substeps can be evaluated using ``manualintegrate``
        to obtain a result.
Returns the steps needed to compute an integral.
Returns the steps needed to compute an integral .
def integral_steps(integrand, symbol, **options):
    """Returns the steps needed to compute an integral.

    This function attempts to mirror what a student would do by hand as
    closely as possible.

    SymPy Gamma uses this to provide a step-by-step explanation of an
    integral. The code it uses to format the results of this function can be
    found at
    https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.

    Examples
    ========

    >>> from sympy import exp, sin, cos
    >>> from sympy.integrals.manualintegrate import integral_steps
    >>> from sympy.abc import x
    >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    URule(u_var=_u, u_func=exp(x), constant=1,
        substep=ArctanRule(context=1/(_u**2 + 1), symbol=_u),
        context=exp(x)/(exp(2*x) + 1), symbol=x)
    >>> print(repr(integral_steps(sin(x), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
    >>> print(repr(integral_steps((x**2 + 3)**2 , x))) \
    # doctest: +NORMALIZE_WHITESPACE
    RewriteRule(rewritten=x**4 + 6*x**2 + 9,
    substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
        ConstantTimesRule(constant=6, other=x**2,
            substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
                context=6*x**2, symbol=x),
        ConstantRule(constant=9, context=9, symbol=x)],
    context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)


    Returns
    =======
    rule : namedtuple
        The first step; most rules have substeps that must also be
        considered. These substeps can be evaluated using ``manualintegrate``
        to obtain a result.

    """
    cachekey = (integrand, symbol)
    if cachekey in _integral_cache:
        if _integral_cache[cachekey] is None:
            # cyclic integral! null_safe will eliminate that path
            return None
        else:
            return _integral_cache[cachekey]
    else:
        _integral_cache[cachekey] = None

    integral = IntegralInfo(integrand, symbol)

    def key(integral):
        integrand = integral.integrand

        if isinstance(integrand, TrigonometricFunction):
            return TrigonometricFunction
        elif isinstance(integrand, sympy.Derivative):
            return sympy.Derivative
        elif symbol not in integrand.free_symbols:
            return sympy.Number
        else:
            for cls in (sympy.Pow, sympy.Symbol, sympy.exp, sympy.log,
                        sympy.Add, sympy.Mul, sympy.atan, sympy.asin, sympy.acos, sympy.Heaviside):
                if isinstance(integrand, cls):
                    return cls

    def integral_is_subclass(*klasses):
        def _integral_is_subclass(integral):
            k = key(integral)
            return k and issubclass(k, klasses)
        return _integral_is_subclass

    result = do_one(
        null_safe(switch(key, {
            sympy.Pow: do_one(null_safe(power_rule), null_safe(inverse_trig_rule)),
            sympy.Symbol: power_rule,
            sympy.exp: exp_rule,
            sympy.Add: add_rule,
            sympy.Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule)),
            sympy.Derivative: derivative_rule,
            TrigonometricFunction: trig_rule,
            sympy.Heaviside: heaviside_rule,
            sympy.Number: constant_rule
        })),
        do_one(
            null_safe(trig_rule),
            null_safe(alternatives(
                rewrites_rule,
                substitution_rule,
                condition(
                    integral_is_subclass(sympy.Mul, sympy.Pow),
                    partial_fractions_rule),
                condition(
                    integral_is_subclass(sympy.Mul, sympy.log, sympy.atan, sympy.asin, sympy.acos),
                    parts_rule),
                condition(
                    integral_is_subclass(sympy.Mul, sympy.Pow),
                    distribute_expand_rule),
                trig_powers_products_rule
            )),
            null_safe(trig_substitution_rule)
        ),
        fallback_rule)(integral)
    del _integral_cache[cachekey]
    return result
def integral_steps ( integrand , symbol , * * options ) : cachekey = ( integrand , symbol ) if cachekey in _integral_cache : if _integral_cache [ cachekey ] is None : # cyclic integral! null_safe will eliminate that path return None else : return _integral_cache [ cachekey ] else : _integral_cache [ cachekey ] = None integral = IntegralInfo ( integrand , symbol ) def key ( integral ) : integrand = integral . integrand if isinstance ( integrand , TrigonometricFunction ) : return TrigonometricFunction elif isinstance ( integrand , sympy . Derivative ) : return sympy . Derivative elif symbol not in integrand . free_symbols : return sympy . Number else : for cls in ( sympy . Pow , sympy . Symbol , sympy . exp , sympy . log , sympy . Add , sympy . Mul , sympy . atan , sympy . asin , sympy . acos , sympy . Heaviside ) : if isinstance ( integrand , cls ) : return cls def integral_is_subclass ( * klasses ) : def _integral_is_subclass ( integral ) : k = key ( integral ) return k and issubclass ( k , klasses ) return _integral_is_subclass result = do_one ( null_safe ( switch ( key , { sympy . Pow : do_one ( null_safe ( power_rule ) , null_safe ( inverse_trig_rule ) ) , sympy . Symbol : power_rule , sympy . exp : exp_rule , sympy . Add : add_rule , sympy . Mul : do_one ( null_safe ( mul_rule ) , null_safe ( trig_product_rule ) ) , sympy . Derivative : derivative_rule , TrigonometricFunction : trig_rule , sympy . Heaviside : heaviside_rule , sympy . Number : constant_rule } ) ) , do_one ( null_safe ( trig_rule ) , null_safe ( alternatives ( rewrites_rule , substitution_rule , condition ( integral_is_subclass ( sympy . Mul , sympy . Pow ) , partial_fractions_rule ) , condition ( integral_is_subclass ( sympy . Mul , sympy . log , sympy . atan , sympy . asin , sympy . acos ) , parts_rule ) , condition ( integral_is_subclass ( sympy . Mul , sympy . Pow ) , distribute_expand_rule ) , trig_powers_products_rule ) ) , null_safe ( trig_substitution_rule ) ) , fallback_rule ) ( integral ) del _integral_cache [ cachekey ] return result