Integrand size = 27, antiderivative size = 752 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {d \left (972 b c d^2-2835 d^3+b^4 d \left (45 c^2-8 d^2\right )-54 b^3 c \left (c^2+5 d^2\right )+9 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (9-b^2\right )^2 f}+\frac {(b c-3 d)^2 \left (18 b c+63 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2}+\frac {\left (14985 b c d^3-25515 d^4-b^5 c d \left (51 c^2-104 d^2\right )-405 b^2 d^2 \left (3 c^2-13 d^2\right )-9 b^3 c d \left (21 c^2+361 d^2\right )+27 b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (9-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (36450 b c d^4-76545 d^5-324 b^3 c d^2 \left (4 c^2+29 d^2\right )+81 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+18 b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-9 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d)^3 \left (540 b c d-132 b^3 c d+2835 d^2+18 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b^5 (3+b)^3 f \sqrt {c+d \sin (e+f x)}} \]
[Out]
Time = 2.08 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2871, 3126, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\left (35 d^2 a^4+20 b c d a^3+2 b^2 \left (4 c^2-43 d^2\right ) a^2-44 b^3 c d a+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} (b c-a d)^3}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt {c+d \sin (e+f x)}}+\frac {\cos (e+f x) (c+d \sin (e+f x))^{5/2} (b c-a d)^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (7 d a^2+6 b c a-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2} (b c-a d)^2}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {d \left (-35 d^3 a^4+36 b c d^2 a^3+b^2 d \left (9 c^2+61 d^2\right ) a^2-18 b^3 c \left (c^2+5 d^2\right ) a+b^4 d \left (45 c^2-8 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}-\frac {\left (-105 d^5 a^6+150 b c d^4 a^5+b^2 d^3 \left (26 c^2+223 d^2\right ) a^4-12 b^3 c d^2 \left (4 c^2+29 d^2\right ) a^3-b^4 d \left (33 c^4+70 d^2 c^2+128 d^4\right ) a^2+6 b^5 c \left (3 c^4+38 d^2 c^2+48 d^4\right ) a-b^6 d \left (57 c^4+136 d^2 c^2+8 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (-105 d^4 a^5+185 b c d^3 a^4-15 b^2 d^2 \left (3 c^2-13 d^2\right ) a^3-b^3 c d \left (21 c^2+361 d^2\right ) a^2+9 b^4 \left (2 c^4+17 d^2 c^2-8 d^4\right ) a-b^5 c d \left (51 c^2-104 d^2\right )\right ) E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2871
Rule 2884
Rule 2886
Rule 3081
Rule 3126
Rule 3128
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} \left (5 d (b c-a d)^2+4 b c \left (2 b c d-a \left (c^2+d^2\right )\right )\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)+\frac {1}{2} d \left (6 a b c d-7 a^2 d^2-b^2 \left (3 c^2-4 d^2\right )\right ) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{4} \left (28 a^3 b c d^3-21 a^4 d^4-2 a b^3 c d \left (27 c^2+47 d^2\right )+b^4 c^2 \left (4 c^2+63 d^2\right )+a^2 b^2 \left (8 c^4+27 c^2 d^2+39 d^4\right )\right )+\frac {1}{2} d \left (7 a^4 c d^2-b^4 c \left (c^2-16 d^2\right )+a^2 b^2 c \left (7 c^2-5 d^2\right )-2 a^3 b d \left (3 c^2-d^2\right )-4 a b^3 d \left (3 c^2+2 d^2\right )\right ) \sin (e+f x)-\frac {1}{4} d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {\frac {1}{8} \left (-99 a^4 b c d^4+35 a^5 d^5+a^3 b^2 d^3 \left (75 c^2-61 d^2\right )+3 b^5 c^3 \left (4 c^2+63 d^2\right )-a b^4 d \left (162 c^4+327 c^2 d^2-8 d^4\right )+3 a^2 b^3 c \left (8 c^4+33 c^2 d^2+69 d^4\right )\right )-\frac {1}{4} d \left (35 a^5 c d^3+a^3 b^2 c d \left (9 c^2-91 d^2\right )+a b^4 c d \left (63 c^2+128 d^2\right )-a^4 b \left (57 c^2 d^2-14 d^4\right )-b^5 \left (3 c^4+120 c^2 d^2+4 d^4\right )-a^2 b^3 \left (15 c^4-69 c^2 d^2+28 d^4\right )\right ) \sin (e+f x)+\frac {1}{8} d \left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2} \\ & = \frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{8} d \left (105 a^6 c d^4+3 a^4 b^2 c d^2 \left (15 c^2-98 d^2\right )-5 a^5 b d^3 \left (37 c^2-7 d^2\right )+3 b^6 c^3 \left (4 c^2+63 d^2\right )+a^3 b^3 d \left (21 c^4+436 c^2 d^2-61 d^4\right )-a b^5 d \left (111 c^4+431 c^2 d^2-8 d^4\right )+3 a^2 b^4 c \left (2 c^4-18 c^2 d^2+93 d^4\right )\right )+\frac {1}{8} d \left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{24 b^4 \left (a^2-b^2\right )^2} \\ & = \frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^5 \left (a^2-b^2\right )^2}-\frac {\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{24 b^5 \left (a^2-b^2\right )^2}+\frac {\left (\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{24 b^4 \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{8 b^5 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{24 b^5 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {d \left (36 a^3 b c d^2-35 a^4 d^3+b^4 d \left (45 c^2-8 d^2\right )-18 a b^3 c \left (c^2+5 d^2\right )+a^2 b^2 d \left (9 c^2+61 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{12 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (6 a b c+7 a^2 d-13 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (185 a^4 b c d^3-105 a^5 d^4-b^5 c d \left (51 c^2-104 d^2\right )-15 a^3 b^2 d^2 \left (3 c^2-13 d^2\right )-a^2 b^3 c d \left (21 c^2+361 d^2\right )+9 a b^4 \left (2 c^4+17 c^2 d^2-8 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{12 b^4 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (150 a^5 b c d^4-105 a^6 d^5-12 a^3 b^3 c d^2 \left (4 c^2+29 d^2\right )+a^4 b^2 d^3 \left (26 c^2+223 d^2\right )-b^6 d \left (57 c^4+136 c^2 d^2+8 d^4\right )+6 a b^5 c \left (3 c^4+38 c^2 d^2+48 d^4\right )-a^2 b^4 d \left (33 c^4+70 c^2 d^2+128 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{12 b^5 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^3 \left (20 a^3 b c d-44 a b^3 c d+35 a^4 d^2+2 a^2 b^2 \left (4 c^2-43 d^2\right )+b^4 \left (4 c^2+63 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^5 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.11 (sec) , antiderivative size = 1403, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^4 \cos (e+f x)}{3 b^3}+\frac {-b^4 c^4 \cos (e+f x)+12 b^3 c^3 d \cos (e+f x)-54 b^2 c^2 d^2 \cos (e+f x)+108 b c d^3 \cos (e+f x)-81 d^4 \cos (e+f x)}{2 b^3 \left (-9+b^2\right ) (3+b \sin (e+f x))^2}+\frac {18 b^4 c^4 \cos (e+f x)-63 b^3 c^3 d \cos (e+f x)-17 b^5 c^3 d \cos (e+f x)-405 b^2 c^2 d^2 \cos (e+f x)+153 b^4 c^2 d^2 \cos (e+f x)+2187 b c d^3 \cos (e+f x)-459 b^3 c d^3 \cos (e+f x)-2673 d^4 \cos (e+f x)+459 b^2 d^4 \cos (e+f x)}{4 b^3 \left (-9+b^2\right )^2 (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (432 b^3 c^5+24 b^5 c^5-918 b^4 c^4 d+1593 b^3 c^3 d^2+327 b^5 c^3 d^2+2835 b^2 c^2 d^3-1503 b^4 c^2 d^3-1053 b c d^4+477 b^3 c d^4+104 b^5 c d^4-8505 d^5+1971 b^2 d^5-168 b^4 d^5\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (540 b^3 c^4 d+12 b^5 c^4 d-972 b^2 c^3 d^2-756 b^4 c^3 d^2+18468 b c^2 d^3-2484 b^3 c^2 d^3+480 b^5 c^2 d^3-34020 c d^4+9828 b^2 c d^4-1536 b^4 c d^4-4536 b d^5+1008 b^3 d^5+16 b^5 d^5\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-54 b^4 c^4 d+189 b^3 c^3 d^2+51 b^5 c^3 d^2+1215 b^2 c^2 d^3-459 b^4 c^2 d^3-14985 b c d^4+3249 b^3 c d^4-104 b^5 c d^4+25515 d^5-5265 b^2 d^5+216 b^4 d^5\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{48 (-3+b)^2 b^3 (3+b)^2 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2774\) vs. \(2(885)=1770\).
Time = 90.07 (sec) , antiderivative size = 2775, normalized size of antiderivative = 3.69
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{9/2}}{(3+b \sin (e+f x))^3} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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