manualintegrate nested pow

[eagleoflqj]

References to other Issues or PRs

Fixes #23712
Fixes #23505

Brief description of what is fixed or changed

This is an extension to power rule.
Nested pow like sqrt(x**(S(5)/3)) can be treated as x**(S(5)/6), but the structure should be kept in the end.

$$ \begin{align} &\int \sqrt{x^\frac{5}{3}}\mathrm{d}x\\ =&x\sqrt{x^\frac{5}{3}} - \int x\mathrm{d}\sqrt{x^\frac{5}{3}}\\ =&x\sqrt{x^\frac{5}{3}} - \int x\cdot\frac{1}{2}(x^\frac{5}{3})^{-\frac{1}{2}}\cdot\frac{5}{3}x^\frac{2}{3}\mathrm{d}x\\ =&x\sqrt{x^\frac{5}{3}} - \frac{5}{6}\int \sqrt{x^\frac{5}{3}}\mathrm{d}x\\ =&\frac{6}{11}x\sqrt{x^\frac{5}{3}} \end{align} $$

However, if the exponent is equivalent to -1, the integrand should be multiplied by x*log(x) instead of x/(exponent+1)

Other comments

Release Notes

• integrals
  • Support integral steps for nested pow like (c*(a+b*x)**d)**e

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• integrals
  • Support integral steps for nested pow like (c*(a+b*x)**d)**e (#23716 by @eagleoflqj)

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Fixes #23712
Fixes #23505
#### Brief description of what is fixed or changed

This is an extension to power rule.
Nested pow like `sqrt(x**(S(5)/3))` can be treated as `x**(S(5)/6)`, but the structure should be kept in the end.

$$
\begin{align}
&\int \sqrt{x^\frac{5}{3}}\mathrm{d}x\\
=&x\sqrt{x^\frac{5}{3}} - \int x\mathrm{d}\sqrt{x^\frac{5}{3}}\\
=&x\sqrt{x^\frac{5}{3}} - \int x\cdot\frac{1}{2}(x^\frac{5}{3})^{-\frac{1}{2}}\cdot\frac{5}{3}x^\frac{2}{3}\mathrm{d}x\\
=&x\sqrt{x^\frac{5}{3}} - \frac{5}{6}\int \sqrt{x^\frac{5}{3}}\mathrm{d}x\\
=&\frac{6}{11}x\sqrt{x^\frac{5}{3}}
\end{align}
$$

However, if the exponent is equivalent to -1, the integrand should be multiplied by `x*log(x)` instead of `x/(exponent+1)`
#### Other comments


#### Release Notes

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  * Added a new solver for logarithmic equations.

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<!-- BEGIN RELEASE NOTES -->
* integrals
  * Support integral steps for nested pow like (c*(a+b*x)**d)**e
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Update

The release notes on the wiki have been updated.

[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
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beginning with + are slowdowns (the PR is slower then master or
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Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

       before           after         ratio
     [41d90958]       [6afa478c]
     <sympy-1.11.1^0>                 
-         966±3μs          620±2μs     0.64  solve.TimeSparseSystem.time_linear_eq_to_matrix(10)
-     2.80±0.01ms         1.16±0ms     0.41  solve.TimeSparseSystem.time_linear_eq_to_matrix(20)
-     5.67±0.01ms         1.70±0ms     0.30  solve.TimeSparseSystem.time_linear_eq_to_matrix(30)

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[eagleoflqj]

@oscarbenjamin is this good to go?

[oscarbenjamin]

Does this assume that (a**b**c) == a**(b*c)? If so then that identity does not always hold.

[eagleoflqj]

Basic power rule: to calculate integral of x**n, n != -1, we multiply it by x and divide it by n+1, so we get x**(n+1)/(n+1).
Basic log rule: to calculate integral of x**-1, we multiply it by x*log(x), so we get log(x).

Now let's extend these basic rules to more complicated forms like (x**2)**(-1/2) and x*cbrt(1/x).

I know (x**2)**(-1/2) != x**-1, but to calculate the integral, we should decide whether to use power rule or log rule. The decision is made by treating (x**2)**(-1/2) as x**(2*(-1/2)), so the "nominal" exponent is -1, thus we apply log rule: multiply it by x*log(x) to get x*log(x)/sqrt(x**2). The correctness is easily verified: diff(x*log(x)/sqrt(x**2)) == 1/sqrt(x**2).

Similar thing happens to x*cbrt(1/x) as well. The "nominal" exponent is 1+(1/3)*(-1) == 2/3 which is not -1, so we apply power rule: multiply it by x and divide it by 5/3. The result is 3*x**2*cbrt(1/x)/5, and diff(3*x**2*cbrt(1/x)/5) == x*(1/x)**(1/3).

[oscarbenjamin]

Oh, I see. So we just use sub_exp*e in the divisor.

[oscarbenjamin]

It is generally better not to raise and catch ValueError as part of normal flow control because various bugs can result in raising ValueError and then this would obscure those bugs by silencing the exception. Here a special purpose exception should be used like class NoMatch(Exception).

[oscarbenjamin]

I think this looks good.

[jksuom]

I'm not sure that this would work if the base is not positive.

[eagleoflqj]

I'm not sure that this would work if the base is not positive.

This aims to solve negative base issue. See #23712 (comment) which is wrong for negative base.
The result produced here is correct.