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I have the Solution (partly) The function was wrong, it came up because i wrecked it with a faulty transformation.

However I was able to do the first Step of integration with the "faulty" function.

IN:

var('a b')
ftrick = -sqrt(r^2*a + b)
BItrick = integral(ftrick, (r,r1,r2))


OUT:

Is  a  positive or negative?


IN:

view('AHA!')
assume(a<0)
assume(b>0)
BItrick = integral(ftrick, (r,r1,r2))
view(BItrick)
forget()
BItrick2 = BItrick.substitute(a == -2*cos(phi)^2)
BItrick2 = BItrick2.substitute(b == 25 )
view(BItrick2)


BItrick2 =

1/4*(6*sqrt(-18*cos(phi)^2 + 25)*cos(phi)^2 +
25*sqrt(2)*sqrt(cos(phi)^2)*arcsin(3/5*sqrt(2)*sqrt(cos(phi)^2)))/cos(ph\
i)^2 - 5/4*(2*sqrt(-50*cos(phi)^2 + 25)*cos(phi)^2 +
5*sqrt(2)*sqrt(cos(phi)^2)*arcsin(sqrt(2)*sqrt(cos(phi)^2)))/cos(phi)^2


When I try to go on sage says it doent know wheather cos(phi) ist positive or negative. So i divided the integral.

forget()
assume(cos(phi) >0)
erg1 = integral(BItrick2, (phi,phi1,0.5*pi))
forget()
assume(cos(phi) <0)
erg2 = integral(BItrick2, (phi,0.5*pi,1.5*pi))
forget()
assume(cos(phi) >0)
erg3 = integral(BItrick2, (phi,1.5*pi,2*pi))
erg = erg1+erg2+erg3


Sage does calculate for 2 minutes and then it gives me the input. The system is unable to integrate.

If you have problems with Integrations try using assume() and forget()

Thank you for your Help.