# Maple integration test file "Wester Problems.txt"

lst:=[

# Michael Wester

# Gradshteyn and Ryzhik 2.244(8)
[(-5+3*x)^2/(-1+2*x)^(7/2),x,2,(-49/20)/(-1+2*x)^(5/2)+7/2/(-1+2*x)^(3/2)+(-9/4)/sqrt(-1+2*x)],
# => 1/[2 m sqrt (10)] log ([-5 + e^(m x) sqrt (10)]/[-5 - e^(m x) sqrt (10)])

#      [Gradshteyn and Ryzhik 2.314]
[1/((-5)/exp(m*x)+2*exp(m*x)),x,2,-arctanh(exp(m*x)*sqrt(2/5))/(m*sqrt(10))],
# This example involves several symbolic parameters
#   => 1/sqrt(b^2 - a^2) log ([sqrt (b^2 - a^2) tan (x/2) + a + b]/
#                            [sqrt (b^2 - a^2) tan (x/2) - a - b])   (a^2 < b^2)

#      [Gradshteyn and Ryzhik 2.553(3)]
# 
# {1/(a + b*Cos[x]), x, 0, Assumptions -> a^2 < b^2,
# 1/Sqrt[b^2 - a^2]*Log[(Sqrt[b^2 - a^2]*Tan[x/2] + a + b)/
#                       (Sqrt[b^2 - a^2]*Tan[x/2] - a - b)]}

# 
[1/(a+b*cos(x)),x,2,2*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],
# The integral of 1/(a + 3 cos x + 4 sin x) can have 4 different forms

#   depending on the value of a !   [Gradshteyn and Ryzhik 2.558(4)]
[1/(3+3*cos(x)+4*sin(x)),x,2,1/4*log(3+4*tan(1/2*x))],
[1/(4+3*cos(x)+4*sin(x)),x,2,-1/3*log(4+3*cot(1/4*Pi+1/2*x))],
[1/(5+3*cos(x)+4*sin(x)),x,1,(-1)/(2+tan(1/2*x)),1/4*(-4+5*sin(x))/(4*cos(x)-3*sin(x))],

# => (a = 6) 2/sqrt(11) arctan ([3 tan (x/2) + 4]/sqrt(11))
[1/(6+3*cos(x)+4*sin(x)),x,3,x/sqrt(11)+2*arctan((4*cos(x)-3*sin(x))/(6+3*cos(x)+4*sin(x)+sqrt(11)))/sqrt(11)],
# => x log|x^2 - a^2| - 2 x + a log|(x + a)/(x - a)|

#      [Gradshteyn and Ryzhik 2.736(1)]

# {Log[Abs[x^2 - a^2]], x, 0, x*Log[Abs[x^2 - a^2]] - 2*x + a*Log[(x + a)/(x - a)]}
[1/2*log((-a^2+x^2)^2),x,4,-2*x+2*a*arctanh(x/a)+1/2*x*log((-a^2+x^2)^2)]]:
