Date: Sep 5, 2013 2:03 PM
Author: Peter Luschny
Subject: Re: The integration test suites for Sage.

M> Your numbers 21, 23, 49 are shown as done while Albert's are not. 
M> And your number 43 is shown as wrong whereas Albert records a success.

Albert used Maxima 5.28 whereas I used Sage 5.10. I do not know which
Maxima version Sage 5.10 uses. They might be different.

So let's check by the output given on my page:

Charlwood_problem(43)
integrand : tan(x)/sqrt(tan(x)^4 + 1)
antideriv : -1/4*sqrt(2)*arctanh(-1/2*(tan(x)^2 - 1)*sqrt(2)/sqrt(tan(x)^4 + 1))
maxima : -1/4*sqrt(2)*arcsinh(2*sin(x)^2 - 1)

Looks like 'antideriv' - 'solution' = 0. Is this ok?
Thus 43 is indeed a success. Wouldn't Maxima's result in this
case not be the 'better' antiderivative for Albert's "Book"?

Charlwood_problem(21)
integrand : x^3*arcsin(x)/sqrt(-x^4 + 1)
antideriv : 1/4*sqrt(x^2 + 1)*x - 1/2*sqrt(-x^4 + 1)*arcsin(x) + 1/4*arcsinh(x)
maxima : 1/4*sqrt(x^2 + 1)*x - 1/2*sqrt(-x^4 + 1)*arcsin(x) + 1/4*arcsinh(x)

That's easy to judge.

Charlwood_problem(23)
integrand : x*log(x + sqrt(x^2 + 1))*arctan(x)/sqrt(x^2 + 1)
antideriv : sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1))*arctan(x) - x*arctan(x) - 1/2*log(x + sqrt(x^2 + 1))^2 + 1/2*log(x^2 + 1)
maxima : (sqrt(x^2 + 1)*log(x + sqrt(x^2 + 1)) - x)*arctan(x) + 1/2*log(x + sqrt(x^2 + 1))^2 - log(x + sqrt(x^2 + 1))*arcsinh(x) + 1/2*log(x^2 + 1)

What did I overlook?

Charlwood_problem(49)
integrand : arcsin(x/sqrt(-x^2 + 1))
antideriv : x*arcsin(x/sqrt(-x^2 + 1)) + arctan(sqrt(-2*x^2 + 1))
maxima : x*arcsin(x/sqrt(-x^2 + 1)) - 1/2*(-2*I*x^2 + I)/sqrt(2*x^2 - 1) - 1/2*I*sqrt(2*x^2 - 1) - 1/2*I*log(sqrt(2*x^2 - 1) - 1) + 1/2*I*log(sqrt(2*x^2 - 1) + 1)

Maxima uses 'I' here. I think Albert rates this as an error. And he
is right. Charlwood demanded only real solutions, if I remember right.
So I will classify this as deficient.

Summary: 43+, 21+, 23+, 49-

Peter