Problem number 176

5.1 Problem number 176

$\int \frac{\cos^{2} (a + b x)}{\sin^{\frac{7}{2}} (2 a + 2 b x)} d x$

Optimal antiderivative $\frac{3 \sqrt{\frac{1}{2} + \frac{\sin (2 b x + 2 a)}{2}} EllipticE (\cos (a + \frac{π}{4} + b x), \sqrt{2})}{10 \sin (a + \frac{π}{4} + b x) b} - \frac{\cos^{2} (b x + a)}{5 b \sin {(2 b x + 2 a)}^{\frac{5}{2}}} - \frac{3 \cos (2 b x + 2 a)}{10 b \sqrt{\sin (2 b x + 2 a)}}$

command

int(cos(b*x+a)^2/sin(2*b*x+2*a)^(7/2),x)

Maple 2022.1 output

$\int \frac{\cos^{2} (x b + a)}{\sin {(2 x b + 2 a)}^{\frac{7}{2}}} d x$

Maple 2021.1 output

$\frac{\sqrt{2} (- \frac{8 \sqrt{2}}{5 \sin {(2 b x + 2 a)}^{\frac{5}{2}}} + \frac{4 \sqrt{2} (6 \sqrt{1 + \sin (2 b x + 2 a)} \sqrt{- 2 \sin (2 b x + 2 a) + 2} \sqrt{- \sin (2 b x + 2 a)} (\sin^{2} (2 b x + 2 a)) EllipticE (\sqrt{1 + \sin (2 b x + 2 a)}, \frac{\sqrt{2}}{2}) - 3 \sqrt{1 + \sin (2 b x + 2 a)} \sqrt{- 2 \sin (2 b x + 2 a) + 2} \sqrt{- \sin (2 b x + 2 a)} (\sin^{2} (2 b x + 2 a)) EllipticF (\sqrt{1 + \sin (2 b x + 2 a)}, \frac{\sqrt{2}}{2}) + 6 (\sin^{4} (2 b x + 2 a)) - 4 (\sin^{2} (2 b x + 2 a)) - 2)}{5 \sin {(2 b x + 2 a)}^{\frac{5}{2}} \cos (2 b x + 2 a)})}{32 b}$

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