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an integral with arccos

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.

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

an integral with arccos

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.

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

an integral with arccosarccos 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.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

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