Optimal. Leaf size=682 \[ \frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {b \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (-35 a^4 d^2+28 a^3 b c d-2 a^2 b^2 \left (4 c^2-19 d^2\right )-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A] time = 2.77, antiderivative size = 682, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {d \left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (a^2 b^2 d \left (13 c^2-29 d^2\right )+8 a^4 d^3-6 a b^3 c \left (c^2-d^2\right )-b^4 d \left (7 c^2-15 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (-2 a^2 b^2 \left (4 c^2-19 d^2\right )+28 a^3 b c d-35 a^4 d^2-4 a b^3 c d-b^4 \left (4 c^2+15 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f (a-b)^2 (a+b)^3 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \cos (e+f x)}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {b \left (-11 a^2 d+6 a b c+5 b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2802
Rule 2805
Rule 2807
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {\frac {1}{2} \left (-4 a b c+4 a^2 d-5 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+\frac {3}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {\frac {1}{4} \left (-16 a^3 b c d+10 a b^3 c d+8 a^4 d^2+a^2 b^2 \left (8 c^2-29 d^2\right )+b^4 \left (4 c^2+15 d^2\right )\right )+\frac {1}{2} b d \left (3 a^2 b c+3 b^3 c-8 a^3 d+2 a b^2 d\right ) \sin (e+f x)-\frac {1}{4} b^2 d \left (6 a b c-11 a^2 d+5 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {\frac {1}{8} \left (-8 a^5 c d^3+2 a b^4 c d \left (3 c^2-7 d^2\right )-8 a^3 b^2 c d \left (3 c^2-5 d^2\right )+24 a^4 b d^2 \left (c^2-d^2\right )+b^5 \left (4 c^4+11 c^2 d^2-15 d^4\right )+a^2 b^3 \left (8 c^4-41 c^2 d^2+33 d^4\right )\right )-\frac {1}{4} d \left (4 a^4 b c d^2+4 a^5 d^3-b^5 c \left (c^2-5 d^2\right )+4 a^3 b^2 d \left (3 c^2-5 d^2\right )-2 a b^4 d \left (3 c^2-5 d^2\right )-a^2 b^3 c \left (5 c^2+3 d^2\right )\right ) \sin (e+f x)-\frac {1}{8} b d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {\frac {1}{8} b^2 d \left (c^2-d^2\right ) \left (11 a^3 b c d+a b^3 c d-24 a^4 d^2-a^2 b^2 \left (2 c^2-33 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right )+\frac {1}{8} b^3 d (b c-a d) \left (6 a b c-11 a^2 d+5 b^2 d\right ) \left (c^2-d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right )^2 d (b c-a d)^3 \left (c^2-d^2\right )}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (b \left (6 a b c-11 a^2 d+5 b^2 d\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^2}-\frac {\left (b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)^3}-\frac {\left (\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\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}{8 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\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)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (b \left (6 a b c-11 a^2 d+5 b^2 d\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}{8 \left (a^2-b^2\right )^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 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 \left (a^2-b^2\right )^2 (b c-a d)^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {d \left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \left (6 a b c-11 a^2 d+5 b^2 d\right ) \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\left (8 a^4 d^3+a^2 b^2 d \left (13 c^2-29 d^2\right )-b^4 d \left (7 c^2-15 d^2\right )-6 a b^3 c \left (c^2-d^2\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)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \left (6 a b c-11 a^2 d+5 b^2 d\right ) F\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}}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {b \left (28 a^3 b c d-4 a b^3 c d-35 a^4 d^2-2 a^2 b^2 \left (4 c^2-19 d^2\right )-b^4 \left (4 c^2+15 d^2\right )\right ) \Pi \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 (a+b)^3 (b c-a d)^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 9.41, size = 1318, normalized size = 1.93 \[ \frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 \cos (e+f x) d^4}{(b c-a d)^3 \left (c^2-d^2\right ) (c+d \sin (e+f x))}-\frac {7 d \cos (e+f x) b^5+6 a c \cos (e+f x) b^4-13 a^2 d \cos (e+f x) b^3}{4 \left (a^2-b^2\right )^2 (a d-b c)^3 (a+b \sin (e+f x))}+\frac {b^3 \cos (e+f x)}{2 \left (a^2-b^2\right ) (a d-b c)^2 (a+b \sin (e+f x))^2}\right )}{f}+\frac {-\frac {2 \left (16 c d^3 a^5+56 b d^4 a^4-48 b c^2 d^2 a^4-80 b^2 c d^3 a^3+48 b^2 c^3 d a^3-16 b^3 c^4 a^2-95 b^3 d^4 a^2+95 b^3 c^2 d^2 a^2+34 b^4 c d^3 a-18 b^4 c^3 d a-8 b^5 c^4+45 b^5 d^4-29 b^5 c^2 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{(a+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (16 d^4 a^5+16 b c d^3 a^4-80 b^2 d^4 a^3+48 b^2 c^2 d^2 a^3-12 b^3 c d^3 a^2-20 b^3 c^3 d a^2+40 b^4 d^4 a-24 b^4 c^2 d^2 a+20 b^5 c d^3-4 b^5 c^3 d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+a d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\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 {\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-15 d^4 b^5+7 c^2 d^2 b^5-6 a c d^3 b^4+6 a c^3 d b^4+29 a^2 d^4 b^3-13 a^2 c^2 d^2 b^3-8 a^4 d^4 b\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\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 {\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 (c-d) (c+d) (a d-b c)^3 f} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 13.15, size = 2099, normalized size = 3.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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