# Symbolic integration with SymPy.jl

**URL:** <https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624>\
**Category:** General Usage\
**Tags:** integral, sympy\
**Created:** [March 28, 2022, 3:24pm UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624 "2022-03-28T15:24:14Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![F-YF](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/f-yf/32/17363_2.png) [@F-YF](https://discourse.julialang.org/u/F-YF)\
**Post date:** [March 28, 2022, 3:24pm UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/1 "2022-03-28T15:24:14Z")

</div>

I ran into a problem with SymPy.jl for symbolic integration,

> using SymPy  
> m = symbols(“m”, integer=true)  
> n = symbols(“n”, integer=true)  
> x = symbols(“x”, integer=true)  
> integrate(diff(x^4,(x,2)),x)

The output is 4\*x^3

When I actually need the results of this expression：

> integrate(diff(x^m,(x,n)),x)

An error is thrown：

```julia
ERROR: PyError ($(Expr(:escape, :(ccall(#= /home/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:43 =# @pysym(:PyObject_Call), PyPtr, (PyPtr, PyPtr, PyPtr), o, pyargsptr, kw))))) <class 'TypeError'>
TypeError('\nCannot give expansion for symbolic count. If you just want a list of\nall variables of differentiation, use _wrt_variables.')
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/integrals.py", line 1566, in integrate
    return integral.doit(**doit_flags)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/integrals.py", line 613, in doit
    antideriv = self._eval_integral(
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/integrals.py", line 1098, in _eval_integral
    h = meijerint_indefinite(g, x)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/meijerint.py", line 1672, in meijerint_indefinite
    for a in sorted(_find_splitting_points(f, x) | {S.Zero}, key=default_sort_key):
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/meijerint.py", line 424, in _find_splitting_points
    compute_innermost(expr, innermost)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/integrals/meijerint.py", line 415, in compute_innermost
    m = expr.match(p*x + q)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/core/basic.py", line 1702, in match
    m = canonical(pattern).matches(canonical(self), old=old)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/core/add.py", line 553, in matches
    return self._matches_commutative(expr, repl_dict, old)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/core/operations.py", line 336, in _matches_commutative
    expr = collect(expr, free)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/simplify/radsimp.py", line 391, in collect
    terms = [parse_term(i) for i in args]
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/simplify/radsimp.py", line 391, in <listcomp>
    terms = [parse_term(i) for i in args]
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/simplify/radsimp.py", line 281, in parse_term
    sexpr, deriv = parse_derivative(expr)
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/simplify/radsimp.py", line 211, in parse_derivative
    expr, sym, order = deriv.expr, deriv.variables[0], 1
  File "/home/.julia/conda/3/lib/python3.9/site-packages/sympy/core/function.py", line 1664, in variables
    raise TypeError(filldedent('''

Stacktrace:
  [1] pyerr_check
    @ ~/.julia/packages/PyCall/7a7w0/src/exception.jl:62 [inlined]
  [2] pyerr_check
    @ ~/.julia/packages/PyCall/7a7w0/src/exception.jl:66 [inlined]
  [3] _handle_error(msg::String)
    @ PyCall ~/.julia/packages/PyCall/7a7w0/src/exception.jl:83
  [4] macro expansion
    @ ~/.julia/packages/PyCall/7a7w0/src/exception.jl:97 [inlined]
  [5] #107
    @ ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:43 [inlined]
  [6] disable_sigint
    @ ./c.jl:458 [inlined]
  [7] __pycall!
    @ ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:42 [inlined]
  [8] _pycall!(ret::PyCall.PyObject, o::PyCall.PyObject, args::Tuple{Sym, Sym}, nargs::Int64, kw::Ptr{Nothing})
    @ PyCall ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:29
  [9] _pycall!(ret::PyCall.PyObject, o::PyCall.PyObject, args::Tuple{Sym, Sym}, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ PyCall ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:11
 [10] (::PyCall.PyObject)(::Sym, ::Vararg{Sym}; kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ PyCall ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:86
 [11] (::PyCall.PyObject)(::Sym, ::Vararg{Sym})
    @ PyCall ~/.julia/packages/PyCall/7a7w0/src/pyfncall.jl:86
 [12] integrate(ex::Sym, args::Sym; kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
    @ SymPy ~/.julia/packages/SymPy/ycPbL/src/importexport.jl:102
 [13] integrate(ex::Sym, args::Sym)
    @ SymPy ~/.julia/packages/SymPy/ycPbL/src/importexport.jl:102
 [14] top-level scope
    @ REPL[6]:1

```

But this works:

> integrate(diff(x^m,(x,2)),x)

```julia
          ⎛⎧ m ⎞
          ⎜⎪ x ⎟
          ⎜⎪─────── for m ≠ 1⎟
m⋅(m - 1)⋅⎜⎨m⋅x - x ⎟
          ⎜⎪ ⎟
          ⎜⎪log(x) otherwise⎟
          ⎝⎩ ⎠

```

It looks like the problem is on the symbol n. I can’t find a solution, so I hope you guys can help me

---

<div class="post-metadata">

**Author:** ![Palli](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/palli/32/3380_2.png) [@Palli](https://discourse.julialang.org/u/Palli)\
**Post date:** [March 28, 2022, 5:30pm UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/2 "2022-03-28T17:30:53Z")

</div>

While SymPy.jl should (continue to) work (and I don’t know enough to help you), or if not from Julia then directly from Python, FYI I thought we had a replacement in Julia-only packages such as Symbolics.jl.

While I see “the goal is different”, you may want to look into it:  
[https://symbolics.juliasymbolics.org/dev/comparison/](https://symbolics.juliasymbolics.org/dev/comparison/)

> Comparison of Julia’s Symbolics.jl vs SymPy for Symbolic Computation

---

<div class="post-metadata">

**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [March 28, 2022, 6:00pm UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/3 "2022-03-28T18:00:03Z")

</div>

This error message is from the underlying Python library, as it can’t handle the “expansion for symbolic count” (the symbolic `n`). If this is really your problem, you can do the differentiation by hand, and then integrate the resulting `c x^{m-n}`, but I don’t know how to tease that out of SymPy.

---

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**Author:** ![F-YF](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/f-yf/32/17363_2.png) [@F-YF](https://discourse.julialang.org/u/F-YF)\
**Post date:** [March 29, 2022, 2:59am UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/4 "2022-03-29T02:59:58Z")

</div>

> [@Palli](#):
>
> While I see “the goal is different”, you may want to look into it:  
> [https://symbolics.juliasymbolics.org/dev/comparison/](https://symbolics.juliasymbolics.org/dev/comparison/)

Well, the integral in my calculation is also an irreplaceable step, but Symbolics.jl cannot implement the operation of the integral.

---

<div class="post-metadata">

**Author:** ![F-YF](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/f-yf/32/17363_2.png) [@F-YF](https://discourse.julialang.org/u/F-YF)\
**Post date:** [March 29, 2022, 3:08am UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/5 "2022-03-29T03:08:19Z")

</div>

> [@j\_verzani](#):
>
> If this is really your problem, you can do the differentiation by hand, and then integrate the resulting

This is a good idea, but there must be derivatives of order n(symbolic variables) in my problem, so, it is inevitable that similar operations[like this: integrate(diff(x^m,(x,n)),x)] will occur.

---

<div class="post-metadata">

**Author:** ![rocco\_sprmnt21](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rocco_sprmnt21/32/20127_2.png) [@rocco\_sprmnt21](https://discourse.julialang.org/u/rocco_sprmnt21)\
**Post date:** [June 6, 2023, 7:43am UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/6 "2023-06-06T07:43:41Z")

</div>

but

```julia
julia> integrate(diff(x^1,(x,2)),x)
0

```

and also

```julia

julia> integrate(diff(x^m,(x,2)),x)(m=>1)
0

```

which are different from

```julia
          ⎛⎧ m ⎞
          ⎜⎪ x ⎟
          ⎜⎪─────── for m ≠ 1⎟
m⋅(m - 1)⋅⎜⎨m⋅x - x ⎟
          ⎜⎪ ⎟
          ⎜⎪log(x) otherwise⎟
          ⎝⎩ ⎠

```

PS  
Why was this convention chosen?

---

<div class="post-metadata">

**Author:** ![rocco\_sprmnt21](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rocco_sprmnt21/32/20127_2.png) [@rocco\_sprmnt21](https://discourse.julialang.org/u/rocco_sprmnt21)\
**Post date:** [June 6, 2023, 8:14am UTC](https://discourse.julialang.org/t/symbolic-integration-with-sympy-jl/78624/7 "2023-06-06T08:14:57Z")

</div>

I’m not sure how to interpret them and I don’t know if they can help you, but these kind of expressions produce results 🙂

```julia

integrate(diff(x^(n+2), n),x)

integrate(diff(x^(n+m), n),x)

```

if you redefine the old m as n+m (new) you should have your formula