Integrand size = 27, antiderivative size = 499 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\frac {d \left (18 b c d-45 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (9-b^2\right ) f}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (9-b^2\right ) f (3+b \sin (e+f x))}+\frac {\left (261 b c d^2-405 d^3+b^3 \left (3 c^3-20 c d^2\right )-3 b^2 \left (9 c^2 d-12 d^3\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^3 \left (9-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (648 b c d^3-1215 d^4-36 b^3 c d \left (c^2+3 d^2\right )+18 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^4 \left (9-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-3 d)^3 \left (6 b c+45 d-7 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3-b) b^4 (3+b)^2 f \sqrt {c+d \sin (e+f x)}} \]
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Time = 1.31 (sec) , antiderivative size = 534, 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 = {2871, 3128, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\frac {d \left (-5 a^2 d^2+6 a b c d-\left (b^2 \left (3 c^2-2 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 f \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac {\left (5 a^2 d+2 a b c-7 b^2 d\right ) (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b^4 f (a-b) (a+b)^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (-15 a^3 d^3+29 a^2 b c d^2-a b^2 \left (9 c^2 d-12 d^3\right )+b^3 \left (3 c^3-20 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 b^3 f \left (a^2-b^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (-15 a^4 d^4+24 a^3 b c d^3+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )-12 a b^3 c d \left (c^2+3 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 b^4 f \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2871
Rule 2884
Rule 2886
Rule 3081
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))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} \left (7 b^2 c^2 d+3 a^2 d^3-2 a b c \left (c^2+4 d^2\right )\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+\frac {1}{2} d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {2 \int \frac {\frac {1}{4} \left (21 b^3 c^3 d+15 a^2 b c d^3-5 a^3 d^4-a b^2 \left (6 c^4+27 c^2 d^2-2 d^4\right )\right )+\frac {1}{2} d \left (5 a^3 c d^2+b^3 d \left (18 c^2+d^2\right )-a b^2 c \left (3 c^2+14 d^2\right )-a^2 b \left (9 c^2 d-2 d^3\right )\right ) \sin (e+f x)-\frac {1}{4} d \left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b^2 \left (a^2-b^2\right )} \\ & = \frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {2 \int \frac {-\frac {1}{4} d \left (21 b^4 c^3 d-15 a^4 c d^3-9 a^2 b^2 c d \left (c^2-3 d^2\right )+a^3 b \left (29 c^2 d^2-5 d^4\right )-a b^3 \left (3 c^4+47 c^2 d^2-2 d^4\right )\right )-\frac {1}{4} d \left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \sin (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 ) d}+\frac {\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{6 b^3 \left (a^2-b^2\right )} \\ & = \frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left ((b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^4 \left (a^2-b^2\right )}-\frac {\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{6 b^4 \left (a^2-b^2\right )}+\frac {\left (\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\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}{6 b^3 \left (a^2-b^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\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)}}{3 b^3 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\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}{2 b^4 \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 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}{6 b^4 \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}} \\ & = \frac {d \left (6 a b c d-5 a^2 d^2-b^2 \left (3 c^2-2 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (29 a^2 b c d^2-15 a^3 d^3+b^3 \left (3 c^3-20 c d^2\right )-a b^2 \left (9 c^2 d-12 d^3\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)}}{3 b^3 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (24 a^3 b c d^3-15 a^4 d^4-12 a b^3 c d \left (c^2+3 d^2\right )+2 a^2 b^2 d^2 \left (c^2+8 d^2\right )+b^4 \left (3 c^4+16 c^2 d^2+2 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}}}{3 b^4 \left (a^2-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^3 \left (2 a b c+5 a^2 d-7 b^2 d\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}}}{(a-b) b^4 (a+b)^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 9.34 (sec) , antiderivative size = 1058, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^3 \cos (e+f x)}{3 b^2}+\frac {-b^3 c^3 \cos (e+f x)+9 b^2 c^2 d \cos (e+f x)-27 b c d^2 \cos (e+f x)+27 d^3 \cos (e+f x)}{b^2 \left (-9+b^2\right ) (3+b \sin (e+f x))}\right )}{f}+\frac {-\frac {2 \left (-36 b^2 c^4+39 b^3 c^3 d-135 b^2 c^2 d^2+9 b c d^3+20 b^3 c d^3+135 d^4-24 b^2 d^4\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 (-36 b^2 c^3 d-324 b c^2 d^2+72 b^3 c^2 d^2+540 c d^3-168 b^2 c d^3+72 b d^4+4 b^3 d^4\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 (3 b^3 c^3 d-27 b^2 c^2 d^2+261 b c d^3-20 b^3 c d^3-405 d^4+36 b^2 d^4\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}}}}{12 (-3+b) b^2 (3+b) f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1885\) vs. \(2(612)=1224\).
Time = 17.97 (sec) , antiderivative size = 1886, normalized size of antiderivative = 3.78
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^{7/2}}{(3+b \sin (e+f x))^2} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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