Integrand size = 27, antiderivative size = 454 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (9-b^2\right ) f (3+b \sin (e+f x))^2}+\frac {b \left (18 b c-45 d-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (9-b^2\right )^2 (b c-3 d) f (3+b \sin (e+f x))}+\frac {\left (18 b c-45 d-b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 \left (9-b^2\right )^2 (b c-3 d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {3 \left (6 b c-9 d-b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b \left (9-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (324 b c d+36 b^3 c d-243 d^2-b^4 \left (4 c^2-d^2\right )-18 b^2 \left (4 c^2+5 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (3-b)^2 b (3+b)^3 (b c-3 d) f \sqrt {c+d \sin (e+f x)}} \]
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Time = 1.00 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2875, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\frac {b \left (-5 a^2 d+6 a b c-b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (b c-a d) (a+b \sin (e+f x))}+\frac {b \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {3 \left (a^2 (-d)+2 a b c-b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b f \left (a^2-b^2\right )^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (-5 a^2 d+6 a b c-b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{4 f \left (a^2-b^2\right )^2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (-3 a^4 d^2+12 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+12 a b^3 c d-b^4 \left (4 c^2-d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{4 b f (a-b)^2 (a+b)^3 (b c-a d) \sqrt {c+d \sin (e+f x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2875
Rule 2884
Rule 2886
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \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 ) \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}}}{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 ) \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 (a+b)^3 (b c-a d) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.68 (sec) , antiderivative size = 988, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {b \cos (e+f x)}{2 \left (-9+b^2\right ) (3+b \sin (e+f x))^2}+\frac {18 b^2 c \cos (e+f x)-45 b d \cos (e+f x)-b^3 d \cos (e+f x)}{4 \left (-9+b^2\right )^2 (b c-3 d) (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (144 b c^2+8 b^3 c^2-432 c d-42 b^2 c d+81 b d^2-3 b^3 d^2\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 (180 b c d+4 b^3 c d-432 d^2-24 b^2 d^2\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 (-18 b^2 c d+45 b d^2+b^3 d^2\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}}}}{16 (-3+b)^2 (3+b)^2 (b c-3 d) f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1524\) vs. \(2(560)=1120\).
Time = 15.26 (sec) , antiderivative size = 1525, normalized size of antiderivative = 3.36
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Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+b \sin (e+f x))^3} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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