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# an integral with arccos wrong

The value of the integral integrate(exp(arccos(x)),x,0,1) is badly return by Sage. However, sympy_integrator or giac_integrator give the exact value (and also the primitive), maxima giving the same wrong value than Sage (the both unable to give a primitive)

 sage: integrate(exp(arccos(x)),x,0,1)
-1/2*e^(-1/2*pi) + 1/2


Calculus gives through the change of variable x=cos t the equality

\int_0^1 exp(arccos(x))dx= \int_0^{pi/2}e^t sin(t)dt
and the second integral is easily computed :

sage: integrate(exp(t)*sin(t),t,0,pi/2)
1/2*e^(1/2*pi) + 1/2

from sage.symbolic.integration.external import sympy_integrator
sympy_integrator(exp(arccos(x)), x, 0, 1)
1/2*e^(1/2*pi) + 1/2
# GOOD

from sage.symbolic.integration.external import maxima_integrator
maxima_integrator(exp(arccos(x)), x, 0, 1)
-1/2*e^(-1/2*pi) + 1/2
# BAD

from sage.symbolic.integration.external import giac_integrator
giac_integrator(exp(arccos(x)), x, 0, 1)
1/2*e^(1/2*pi) + 1/2
# GOOD

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## 1 Answer

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actually when you do integrate(...), Maxima is used: there is no such "Sage integrator". this corresponds to what you observed.

the backend doing the symbolic integrator can be conveniently specified with the algorithm keyword, as in integrate(exp(arccos(x)),x,0,1, algorithm='giac'), etc.

there is a collection of integrals that are wrong in the symbolics wiki. it should be checked if yours is reported there (i didn't check yet), and if it isn't then a new ticket can be created, possibly with an upstream report (Maxima).

EDIT: i've added ticket #24008. thanks for reporting :)

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Asked: 2017-10-10 22:15:46 +0200

Seen: 232 times

Last updated: Oct 11 '17