Integrand size = 25, antiderivative size = 629 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d e^2 \sqrt {e \sin (c+d x)}} \]
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Time = 2.14 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2773, 2943, 2945, 2946, 2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac {a \left (8 a^2+69 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{12 d e^2 \left (a^2-b^2\right )^3 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {11 a b}{4 d e \left (a^2-b^2\right )^2 (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}-\frac {b}{2 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 d e \left (a^2-b^2\right )^3 (e \sin (c+d x))^{3/2}} \]
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2773
Rule 2781
Rule 2884
Rule 2886
Rule 2943
Rule 2945
Rule 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {\int \frac {-2 a+\frac {7}{2} b \cos (c+d x)}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx}{2 \left (a^2-b^2\right )} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+7 b^2\right )-\frac {55}{4} a b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (-4 a^4+60 a^2 b^2+21 b^4\right )-\frac {1}{8} a b \left (8 a^2+69 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^3 e^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\left (7 b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{8 \left (a^2-b^2\right )^3 e^2}+\frac {\left (a \left (8 a^2+69 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d e}+\frac {\left (a \left (8 a^2+69 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2 \sqrt {e \sin (c+d x)}} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d e}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt {e \sin (c+d x)}}-\frac {\left (7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt {e \sin (c+d x)}} \\ & = -\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (-a^2+b^2\right )^{7/2} d e^2}+\frac {\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 \left (-a^2+b^2\right )^{7/2} d e^2} \\ & = \frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}+\frac {7 b^{5/2} \left (9 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {11 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \cos (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \sin (c+d x))^{3/2}}+\frac {a \left (8 a^2+69 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 a b^2 \left (9 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt {-a^2+b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.26 (sec) , antiderivative size = 1308, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\frac {\left (\frac {b^3}{2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {15 a b^3}{4 \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {2 \left (-3 a^2 b-b^3+a^3 \cos (c+d x)+3 a b^2 \cos (c+d x)\right ) \csc ^2(c+d x)}{3 \left (a^2-b^2\right )^3}\right ) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac {\sin ^{\frac {5}{2}}(c+d x) \left (\frac {2 \left (8 a^3 b+69 a b^3\right ) \cos ^2(c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {a \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)} \sqrt {1-\sin ^2(c+d x)}}{\left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},2,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (8 a^4-120 a^2 b^2-42 b^4\right ) \cos (c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )-\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\left (-a^2+b^2\right )^{3/4}}+\frac {5 a \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^2(c+d x)} \left (5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )-2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{24 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2680\) vs. \(2(649)=1298\).
Time = 11.22 (sec) , antiderivative size = 2681, normalized size of antiderivative = 4.26
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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