Problem number 1436

6.1 Problem number 1436

$\int \frac{(d \sin (e + f x))^{5 / 2}}{(g \cos (e + f x))^{3 / 2} (a + b \sin (e + f x))} d x$

Optimal antiderivative $\frac{a^{2} d^{\frac{5}{2}} \arctan (- 1 + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{d \sin (f x + e)}}) \sqrt{2}}{2 b (a^{2} - b^{2}) f g^{\frac{3}{2}}} - \frac{b d^{\frac{5}{2}} \arctan (- 1 + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{d \sin (f x + e)}}) \sqrt{2}}{2 (a^{2} - b^{2}) f g^{\frac{3}{2}}} + \frac{a^{2} d^{\frac{5}{2}} \arctan (1 + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{d \sin (f x + e)}}) \sqrt{2}}{2 b (a^{2} - b^{2}) f g^{\frac{3}{2}}} - \frac{b d^{\frac{5}{2}} \arctan (1 + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{d \sin (f x + e)}}) \sqrt{2}}{2 (a^{2} - b^{2}) f g^{\frac{3}{2}}} + \frac{a^{2} d^{\frac{5}{2}} \ln (\sqrt{g} + \cot (f x + e) \sqrt{g} - \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{d \sin (f x + e)}}) \sqrt{2}}{4 b (a^{2} - b^{2}) f g^{\frac{3}{2}}} - \frac{b d^{\frac{5}{2}} \ln (\sqrt{g} + \cot (f x + e) \sqrt{g} - \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{d \sin (f x + e)}}) \sqrt{2}}{4 (a^{2} - b^{2}) f g^{\frac{3}{2}}} - \frac{a^{2} d^{\frac{5}{2}} \ln (\sqrt{g} + \cot (f x + e) \sqrt{g} + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{d \sin (f x + e)}}) \sqrt{2}}{4 b (a^{2} - b^{2}) f g^{\frac{3}{2}}} + \frac{b d^{\frac{5}{2}} \ln (\sqrt{g} + \cot (f x + e) \sqrt{g} + \frac{\sqrt{2} \sqrt{d} \sqrt{g \cos (f x + e)}}{\sqrt{d \sin (f x + e)}}) \sqrt{2}}{4 (a^{2} - b^{2}) f g^{\frac{3}{2}}} + \frac{2 a d {(d \sin (f x + e))}^{\frac{3}{2}}}{(a^{2} - b^{2}) f g \sqrt{g \cos (f x + e)}} - \frac{2 a^{3} d^{3} EllipticPi (\frac{\sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{1 + \sin (f x + e)}}, - \frac{\sqrt{- a + b}}{\sqrt{a + b}}, I) \sqrt{2} (\sqrt{\sin} (f x + e))}{b {(- a + b)}^{\frac{3}{2}} {(a + b)}^{\frac{3}{2}} f g^{\frac{3}{2}} \sqrt{d \sin (f x + e)}} + \frac{2 a^{3} d^{3} EllipticPi (\frac{\sqrt{g \cos (f x + e)}}{\sqrt{g} \sqrt{1 + \sin (f x + e)}}, \frac{\sqrt{- a + b}}{\sqrt{a + b}}, I) \sqrt{2} (\sqrt{\sin} (f x + e))}{b {(- a + b)}^{\frac{3}{2}} {(a + b)}^{\frac{3}{2}} f g^{\frac{3}{2}} \sqrt{d \sin (f x + e)}} - \frac{2 b d^{2} \sqrt{d \sin (f x + e)}}{(a^{2} - b^{2}) f g \sqrt{g \cos (f x + e)}} + \frac{2 a d^{2} \sqrt{\frac{1}{2} + \frac{\sin (2 f x + 2 e)}{2}} EllipticE (\cos (e + \frac{π}{4} + f x), \sqrt{2}) \sqrt{g \cos (f x + e)} \sqrt{d \sin (f x + e)}}{\sin (e + \frac{π}{4} + f x) (a^{2} - b^{2}) f g^{2} \sqrt{\sin (2 f x + 2 e)}}$

command

Integrate[(d*Sin[e + f*x])^(5/2)/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]

Mathematica 13.1 output

$$Aborted$

Mathematica 12.3 output

$\frac{2 \cot (e + f x) \csc (e + f x) (d \sin (e + f x))^{5 / 2} (a \sin (e + f x) - b)}{(a^{2} - b^{2}) f (g \cos (e + f x))^{3 / 2}} - \frac{\cos^{\frac{3}{2}} (e + f x) (d \sin (e + f x))^{5 / 2} (- \frac{2 (3 a^{2} - b^{2}) (a F_{1} (\frac{3}{4}; \frac{1}{4}, 1; \frac{7}{4}; \cos^{2} (e + f x), \frac{b^{2} \cos^{2} (e + f x)}{b^{2} - a^{2}}) - b F_{1} (\frac{3}{4}; - \frac{1}{4}, 1; \frac{7}{4}; \cos^{2} (e + f x), \frac{b^{2} \cos^{2} (e + f x)}{b^{2} - a^{2}})) \cos^{\frac{3}{2}} (e + f x) (a + b \sqrt{1 - \cos^{2} (e + f x)}) \sin^{\frac{3}{2}} (e + f x)}{3 (a^{2} - b^{2}) {(1 - \cos^{2} (e + f x))}^{3 / 4} (a + b \sin (e + f x))} - \frac{\cos (2 (e + f x)) \sqrt{\tan (e + f x)} (\sqrt{\tan^{2} (e + f x) + 1} a + b \tan (e + f x)) (24 b (b^{2} - a^{2}) F_{1} (\frac{7}{4}; \frac{1}{2}, 1; \frac{11}{4}; - \tan^{2} (e + f x), (\frac{b^{2}}{a^{2}} - 1) \tan^{2} (e + f x)) \tan^{\frac{7}{2}} (e + f x) + 56 b (b^{2} - 3 a^{2}) F_{1} (\frac{3}{4}; \frac{1}{2}, 1; \frac{7}{4}; - \tan^{2} (e + f x), (\frac{b^{2}}{a^{2}} - 1) \tan^{2} (e + f x)) \tan^{\frac{3}{2}} (e + f x) + 21 a^{3 / 2} (- \frac{4 \sqrt{2} \tan^{- 1} (1 - \frac{\sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)}}{\sqrt{a}}) a^{2}}{\sqrt[4]{a^{2} - b^{2}}} + \frac{4 \sqrt{2} \tan^{- 1} (\frac{\sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)}}{\sqrt{a}} + 1) a^{2}}{\sqrt[4]{a^{2} - b^{2}}} - \frac{2 \sqrt{2} \log (- a + \sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)} \sqrt{a} - \sqrt{a^{2} - b^{2}} \tan (e + f x)) a^{2}}{\sqrt[4]{a^{2} - b^{2}}} + \frac{2 \sqrt{2} \log (a + \sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)} \sqrt{a} + \sqrt{a^{2} - b^{2}} \tan (e + f x)) a^{2}}{\sqrt[4]{a^{2} - b^{2}}} + 4 \sqrt{2} \tan^{- 1} (1 - \sqrt{2} \sqrt{\tan (e + f x)}) a^{3 / 2} - 4 \sqrt{2} \tan^{- 1} (\sqrt{2} \sqrt{\tan (e + f x)} + 1) a^{3 / 2} + 2 \sqrt{2} \log (\tan (e + f x) - \sqrt{2} \sqrt{\tan (e + f x)} + 1) a^{3 / 2} - 2 \sqrt{2} \log (\tan (e + f x) + \sqrt{2} \sqrt{\tan (e + f x)} + 1) a^{3 / 2} + \frac{8 b \tan^{\frac{3}{2}} (e + f x) \sqrt{a}}{\sqrt{\tan^{2} (e + f x) + 1}} + \frac{2 \sqrt{2} b^{2} \tan^{- 1} (1 - \frac{\sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)}}{\sqrt{a}})}{\sqrt[4]{a^{2} - b^{2}}} - \frac{2 \sqrt{2} b^{2} \tan^{- 1} (\frac{\sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)}}{\sqrt{a}} + 1)}{\sqrt[4]{a^{2} - b^{2}}} + \frac{\sqrt{2} b^{2} \log (- a + \sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)} \sqrt{a} - \sqrt{a^{2} - b^{2}} \tan (e + f x))}{\sqrt[4]{a^{2} - b^{2}}} - \frac{\sqrt{2} b^{2} \log (a + \sqrt{2} \sqrt[4]{a^{2} - b^{2}} \sqrt{\tan (e + f x)} \sqrt{a} + \sqrt{a^{2} - b^{2}} \tan (e + f x))}{\sqrt[4]{a^{2} - b^{2}}}))}{84 a b \cos^{\frac{3}{2}} (e + f x) (a + b \sin (e + f x)) (\tan^{2} (e + f x) - 1) \sqrt{\tan^{2} (e + f x) + 1} \sqrt{\sin (e + f x)}})}{(a - b) (a + b) f (g \cos (e + f x))^{3 / 2} \sin^{\frac{5}{2}} (e + f x)}$

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