Optimal. Leaf size=605 \[ \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {\left (5 a^2 d+6 a b c-11 b^2 d\right ) (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {\left (15 a^4 d^2+12 a^3 b c d+2 a^2 b^2 \left (4 c^2-19 d^2\right )-36 a b^3 c d+b^4 \left (4 c^2+35 d^2\right )\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt {c+d \sin (e+f x)}}+\frac {3 \left (5 a^4 d^3+4 a^3 b c d^2+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^4 f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (-15 a^4 d^3+8 a^3 b c d^2+a^2 b^2 d \left (5 c^2+29 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+b^4 d \left (13 c^2-8 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 2.20, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2792, 3047, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac {3 \left (a^2 b^2 d \left (c^2-11 d^2\right )+4 a^3 b c d^2+5 a^4 d^3-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^4 f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (a^2 b^2 d \left (5 c^2+29 d^2\right )+8 a^3 b c d^2-15 a^4 d^3-2 a b^3 c \left (3 c^2+13 d^2\right )+b^4 d \left (13 c^2-8 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^3 f \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 b^2 \left (4 c^2-19 d^2\right )+12 a^3 b c d+15 a^4 d^2-36 a b^3 c d+b^4 \left (4 c^2+35 d^2\right )\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b^4 f (a-b)^2 (a+b)^3 \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {\left (5 a^2 d+6 a b c-11 b^2 d\right ) (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2792
Rule 2805
Rule 2807
Rule 3002
Rule 3047
Rule 3059
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{7/2}}{(a+b \sin (e+f x))^3} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{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}{2} \left (3 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 (2 a b c d-5 a^2 d^2-b^2 \left (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+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {\frac {1}{4} \left (8 a^3 b c d^3-5 a^4 d^4-2 a b^3 c d \left (21 c^2+19 d^2\right )+b^4 c^2 \left (4 c^2+35 d^2\right )+a^2 b^2 \left (8 c^4+19 c^2 d^2+11 d^4\right )\right )+\frac {1}{2} d \left (5 a^4 c d^2+a^2 b^2 c \left (5 c^2-3 d^2\right )-2 a^3 b d \left (c^2-d^2\right )-8 a b^3 d \left (2 c^2+d^2\right )+b^4 c \left (c^2+16 d^2\right )\right ) \sin (e+f x)-\frac {1}{4} d \left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-\frac {1}{4} d (b c-a d) \left (2 a^2 b^2 c^3+4 b^4 c^3+7 a^3 b c^2 d-25 a b^3 c^2 d+15 a^4 c d^2-32 a^2 b^2 c d^2+35 b^4 c d^2+5 a^3 b d^3-11 a b^3 d^3\right )+\frac {3}{4} d (b c-a d) \left (2 a b^3 c^3-a^2 b^2 c^2 d-5 b^4 c^2 d-4 a^3 b c d^2+10 a b^3 c d^2-5 a^4 d^3+11 a^2 b^2 d^3-8 b^4 d^3\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}+\frac {\left ((b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 d^2\right )\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 b^4 \left (a^2-b^2\right )^2}-\frac {\left (\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 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 b^3 \left (a^2-b^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=\frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 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 b^3 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\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}{8 b^4 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left ((b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 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^4 \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {(b c-a d)^2 \left (6 a b c+5 a^2 d-11 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{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))^{3/2}}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (8 a^3 b c d^2-15 a^4 d^3+b^4 d \left (13 c^2-8 d^2\right )-2 a b^3 c \left (3 c^2+13 d^2\right )+a^2 b^2 d \left (5 c^2+29 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 b^3 \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {3 (b c-a d) \left (4 a^3 b c d^2+5 a^4 d^3+a^2 b^2 d \left (c^2-11 d^2\right )-2 a b^3 c \left (c^2+5 d^2\right )+b^4 d \left (5 c^2+8 d^2\right )\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 b^4 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^2 \left (12 a^3 b c d-36 a b^3 c d+15 a^4 d^2+2 a^2 b^2 \left (4 c^2-19 d^2\right )+b^4 \left (4 c^2+35 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 b^4 (a+b)^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 8.35, size = 1323, normalized size = 2.19 \[ \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)}{2 b^2 \left (b^2-a^2\right ) (a+b \sin (e+f x))^2}+\frac {7 d^3 \cos (e+f x) a^4-8 b c d^2 \cos (e+f x) a^3-13 b^2 d^3 \cos (e+f x) a^2-5 b^2 c^2 d \cos (e+f x) a^2+6 b^3 c^3 \cos (e+f x) a+26 b^3 c d^2 \cos (e+f x) a-13 b^4 c^2 d \cos (e+f x)}{4 b^2 \left (b^2-a^2\right )^2 (a+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (8 c^4 b^4+8 d^4 b^4+57 c^2 d^2 b^4-50 a c d^3 b^3-78 a c^3 d b^3+16 a^2 c^4 b^2-7 a^2 d^4 b^2+33 a^2 c^2 d^2 b^2+8 a^3 c d^3 b+5 a^4 d^4\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 (20 c d^3 a^4+8 b d^4 a^3-8 b c^2 d^2 a^3-12 b^2 c d^3 a^2+20 b^2 c^3 d a^2-32 b^3 d^4 a-64 b^3 c^2 d^2 a+64 b^4 c d^3+4 b^4 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 (-8 d^4 b^4+13 c^2 d^2 b^4-26 a c d^3 b^3-6 a c^3 d b^3+29 a^2 d^4 b^2+5 a^2 c^2 d^2 b^2+8 a^3 c d^3 b-15 a^4 d^4\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 b^2 (a+b)^2 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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 11.60, size = 2237, normalized size = 3.70 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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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