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integral 1/(1/(1-x^2))^(3/2) dx
Expanding the integrand 1/(1/(1-x^2))^(3/2) gives -2 sqrt(1/(1-x^2)) x^2+sqrt(1/(1-x^2))+sqrt(1/(1-x^2)) x^4:
= integral (-2 sqrt(1/(1-x^2)) x^2+sqrt(1/(1-x^2))+sqrt(1/(1-x^2)) x^4) dx
Integrate the sum term by term and factor out constants:
= integral sqrt(1/(1-x^2)) dx-2 integral x^2 sqrt(1/(1-x^2)) dx+ integral x^4 sqrt(1/(1-x^2)) dx
For the integrand x^2 sqrt(1/(1-x^2)), simplify powers:
= integral sqrt(1/(1-x^2)) dx-2 integral x^2/sqrt(1-x^2) dx+ integral x^4 sqrt(1/(1-x^2)) dx
For the integrand, x^2/sqrt(1-x^2) substitute x = sin(u) and dx = cos(u) du. Then sqrt(1-x^2) = sqrt(1-sin^2(u)) = cos(u) and u = sin^(-1)(x):
= -2 integral sin^2(u) du+ integral sqrt(1/(1-x^2)) dx+ integral x^4 sqrt(1/(1-x^2)) dx
For the integrand x^4 sqrt(1/(1-x^2)), simplify powers:
= -2 integral sin^2(u) du+ integral sqrt(1/(1-x^2)) dx+ integral x^4/sqrt(1-x^2) dx
For the integrand, x^4/sqrt(1-x^2) substitute x = sin(s) and dx = cos(s) ds. Then sqrt(1-x^2) = sqrt(1-sin^2(s)) = cos(s) and s = sin^(-1)(x):
= integral sin^4(s) ds-2 integral sin^2(u) du+ integral sqrt(1/(1-x^2)) dx
For the integrand sqrt(1/(1-x^2)), simplify powers:
= integral sin^4(s) ds-2 integral sin^2(u) du+ integral 1/sqrt(1-x^2) dx
The integral of 1/sqrt(1-x^2) is sin^(-1)(x):
= integral sin^4(s) ds-2 integral sin^2(u) du+sin^(-1)(x)
Write sin^2(u) as 1/2-1/2 cos(2 u):
= integral sin^4(s) ds-2 integral (1/2-1/2 cos(2 u)) du+sin^(-1)(x)
Integrate the sum term by term and factor out constants:
= integral sin^4(s) ds-2 integral 1/2 du+ integral cos(2 u) du+sin^(-1)(x)
The integral of 1/2 is u/2:
= integral sin^4(s) ds-u+ integral cos(2 u) du+sin^(-1)(x)
For the integrand cos(2 u), substitute p = 2 u and dp = 2 du:
= 1/2 integral cos(p) dp+ integral sin^4(s) ds-u+sin^(-1)(x)
The integral of cos(p) is sin(p):
= (sin(p))/2+ integral sin^4(s) ds-u+sin^(-1)(x)
Use the reduction formula, integral sin^m(s) ds = -(cos(s) sin^(m-1)(s))/m + (m-1)/m integral sin^(-2+m)(s) ds, where m = 4:
= (sin(p))/2-1/4 sin^3(s) cos(s)+3/4 integral sin^2(s) ds-u+sin^(-1)(x)
Write sin^2(s) as 1/2-1/2 cos(2 s):
= (sin(p))/2-1/4 sin^3(s) cos(s)+3/4 integral (1/2-1/2 cos(2 s)) ds-u+sin^(-1)(x)
Integrate the sum term by term and factor out constants:
= (sin(p))/2-1/4 sin^3(s) cos(s)+3/4 integral 1/2 ds-3/8 integral cos(2 s) ds-u+sin^(-1)(x)
For the integrand cos(2 s), substitute w = 2 s and dw = 2 ds:
= (sin(p))/2-1/4 sin^3(s) cos(s)+3/4 integral 1/2 ds-u-3/16 integral cos(w) dw+sin^(-1)(x)
The integral of 1/2 is s/2:
= (sin(p))/2+(3 s)/8-1/4 sin^3(s) cos(s)-u-3/16 integral cos(w) dw+sin^(-1)(x)
The integral of cos(w) is sin(w):
= (sin(p))/2+(3 s)/8-1/4 sin^3(s) cos(s)-u-(3 sin(w))/16+sin^(-1)(x)+constant
Substitute back for w = 2 s:
= (sin(p))/2+(3 s)/8-1/4 sin^3(s) cos(s)-3/8 sin(s) cos(s)-u+sin^(-1)(x)+constant
Substitute back for p = 2 u:
= (3 s)/8-1/4 sin^3(s) cos(s)-3/8 sin(s) cos(s)-u+sin(u) cos(u)+sin^(-1)(x)+constant
Substitute back for s = sin^(-1)(x):
= -u+sin(u) cos(u)-3/8 sqrt(1-x^2) x-1/4 sqrt(1-x^2) x^3+11/8 sin^(-1)(x)+constant
Substitute back for u = sin^(-1)(x):
= 5/8 sqrt(1-x^2) x-1/4 sqrt(1-x^2) x^3+3/8 sin^(-1)(x)+constant
Factor the answer a different way:
= 1/8 (x sqrt(1-x^2) (5-2 x^2)+3 sin^(-1)(x))+constant
Which is equivalent for restricted x values to:
= 1/8 sqrt(1/(1-x^2)) (2 x^5-7 x^3+3 sqrt(1-x^2) sin^(-1)(x)+5 x)+constant