Optimal. Leaf size=487 \[ \frac {b \left (-5 a^2 d+6 a b c-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d) (a+b \sin (e+f x))}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {3 \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (-5 a^2 d+6 a b c-b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (-3 a^4 d^2+12 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+12 a b^3 c d-b^4 \left (4 c^2-d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A] time = 1.60, antiderivative size = 487, 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 = {2796, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {\left (-2 a^2 b^2 \left (4 c^2+5 d^2\right )+12 a^3 b c d-3 a^4 d^2+12 a b^3 c d-b^4 \left (4 c^2-d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt {c+d \sin (e+f x)}}+\frac {b \left (-5 a^2 d+6 a b c-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d) (a+b \sin (e+f x))}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {3 \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 b f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (-5 a^2 d+6 a b c-b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2796
Rule 2805
Rule 2807
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^3} \, dx &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {\frac {1}{2} (-4 a c+b d)+(b c-2 a d) \sin (e+f x)+\frac {1}{2} b d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {b \left (6 a b c-5 a^2 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d) f (a+b \sin (e+f x))}+\frac {\int \frac {\frac {1}{4} \left (-8 a^3 c d-10 a b^2 c d+b^3 \left (4 c^2-d^2\right )+a^2 b \left (8 c^2+7 d^2\right )\right )+\frac {1}{2} d \left (5 a^2 b c+b^3 c-4 a^3 d-2 a b^2 d\right ) \sin (e+f x)+\frac {1}{4} b d \left (6 a b c-5 a^2 d-b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)}\\ &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {b \left (6 a b c-5 a^2 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d) f (a+b \sin (e+f x))}-\frac {\int \frac {\frac {1}{4} b d \left (3 a^3 c d+9 a b^2 c d-b^3 \left (4 c^2-d^2\right )-a^2 b \left (2 c^2+7 d^2\right )\right )+\frac {3}{4} b d (b c-a d) \left (2 a b c-a^2 d-b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )^2 d (b c-a d)}+\frac {\left (6 a b c-5 a^2 d-b^2 d\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{8 \left (a^2-b^2\right )^2 (b c-a d)}\\ &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {b \left (6 a b c-5 a^2 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d) f (a+b \sin (e+f x))}-\frac {\left (3 \left (2 a b c-a^2 d-b^2 d\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{8 b \left (a^2-b^2\right )^2}-\frac {\left (12 a^3 b c d+12 a b^3 c d-3 a^4 d^2-b^4 \left (4 c^2-d^2\right )-2 a^2 b^2 \left (4 c^2+5 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{8 b \left (a^2-b^2\right )^2 (b c-a d)}+\frac {\left (\left (6 a b c-5 a^2 d-b^2 d\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) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {b \left (6 a b c-5 a^2 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d) f (a+b \sin (e+f x))}+\frac {\left (6 a b c-5 a^2 d-b^2 d\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) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (3 \left (2 a b c-a^2 d-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 b \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (12 a^3 b c d+12 a b^3 c d-3 a^4 d^2-b^4 \left (4 c^2-d^2\right )-2 a^2 b^2 \left (4 c^2+5 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 \left (a^2-b^2\right )^2 (b c-a d) \sqrt {c+d \sin (e+f x)}}\\ &=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {b \left (6 a b c-5 a^2 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d) f (a+b \sin (e+f x))}+\frac {\left (6 a b c-5 a^2 d-b^2 d\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) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 \left (2 a b c-a^2 d-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 b \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (12 a^3 b c d+12 a b^3 c d-3 a^4 d^2-b^4 \left (4 c^2-d^2\right )-2 a^2 b^2 \left (4 c^2+5 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 (a+b)^3 (b c-a d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 7.74, size = 1038, normalized size = 2.13 \[ \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {b \cos (e+f x)}{2 \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-d \cos (e+f x) b^3+6 a c \cos (e+f x) b^2-5 a^2 d \cos (e+f x) b}{4 \left (a^2-b^2\right )^2 (a d-b c) (a+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (16 c d a^3-16 b c^2 a^2-9 b d^2 a^2+14 b^2 c d a-8 b^3 c^2+3 b^3 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^2 a^3-20 b c d a^2+8 b^2 d^2 a-4 b^3 c 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 (-d^2 b^3+6 a c d b^2-5 a^2 d^2 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 (a d-b c) 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 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 9.44, size = 1525, normalized size = 3.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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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