Symbolic integration gives result with wrong sign (integrand is positive):
sage: z = integrate(log(1 + 1/(x**2 + 1)), x, -oo, oo); z
2*pi - 2*sqrt(2)*pi
sage: z.n()
-2.60258056913715
sage: numerical_integral(log(1 + 1/(x**2 + 1)), -oo, oo)
(2.6025805691369728, 5.127078281515988e-10)
Sage 8.3
By default, Sage uses Maxima which indeed gets it wrong.
sage: integrate(log(1 + 1/(x**2 + 1)), x, -oo, oo, algorithm='maxima')
2*pi - 2*sqrt(2)*pi
Nonetheless, Sympy and Giac both get it right:
sage: integrate(log(1 + 1/(x**2 + 1)), x, -oo, oo, algorithm='sympy')
-2*pi + 2*sqrt(2)*pi
sage: integrate(log(1 + 1/(x**2 + 1)), x, -oo, oo, algorithm='giac')
2*pi*(sqrt(2) - 1)
To elaborate Sebastien's answer, which is quite correct :
sage: foo=log(1+(1/(x^2+1))).integrate(x) ; foo
x*log(1/(x^2 + 1) + 1) + 2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*arctan(x)
sage: bar=log(1+(1/(x^2+1))).integrate(x, algorithm="sympy") ; bar
x*log(1/(x^2 + 1) + 1) + 2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*arctan(x)
sage: bool(bar==foo)
True
Maxima and Sympy get the same primitive. But
sage: foo.limit(x=oo)
-pi + sqrt(2)*pi
Correct.
sage: foo.limit(x=-oo)
pi - sqrt(2)*pi
Also correct
sage: foo.limit(x=oo)-foo.limit(x=-oo)
-2*pi + 2*sqrt(2)*pi
Still correct.
sage: log(1+(1/(x^2+1))).integrate(x,-oo,oo)
2*pi - 2*sqrt(2)*pi
Wrong. Maxima's problem isn't about primitive determination, but its use...
So where should we fix this?
In Maxima, it seems...
Asked: 2018-08-27 12:12:20 +0200
Seen: 341 times
Last updated: Aug 28 '18
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