Integrand size = 35, antiderivative size = 226 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A-4 A n+B (3+4 n)) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) (1+2 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.45 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3057, 3066, 2866, 2865, 2864, 129, 440, 2855, 69, 67} \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {(-4 A n+A+B (4 n+3)) \cos (e+f x) \sin ^{-n}(e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{4 a f \sqrt {a \sin (e+f x)+a}}-\frac {(2 n+1) (A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right )}{2 a f \sqrt {a \sin (e+f x)+a}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{2 d f (a \sin (e+f x)+a)^{3/2}} \]
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Rule 67
Rule 69
Rule 129
Rule 440
Rule 2855
Rule 2864
Rule 2865
Rule 2866
Rule 3057
Rule 3066
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {(d \sin (e+f x))^n \left (a d (A+B-A n+B n)+\frac {1}{2} a (A-B) d (1+2 n) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2 d} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {((A-B) (1+2 n)) \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx}{4 a^2}+\frac {\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^3 d} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \sqrt {1+\sin (e+f x)}\right ) \int \frac {(d \sin (e+f x))^n}{\sqrt {1+\sin (e+f x)}} \, dx}{2 a^3 d \sqrt {a+a \sin (e+f x)}}+\frac {((A-B) (1+2 n) \cos (e+f x)) \text {Subst}\left (\int \frac {(d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{4 f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt {1+\sin (e+f x)}\right ) \int \frac {\sin ^n(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx}{2 a^3 d \sqrt {a+a \sin (e+f x)}}+\frac {\left ((A-B) (1+2 n) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {x^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{4 f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) (1+2 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {(1-x)^n}{(2-x) \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{2 a^3 d f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) (1+2 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^n}{2-x^2} \, dx,x,\sqrt {1-\sin (e+f x)}\right )}{a^3 d f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A+3 B-4 A n+4 B n) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) (1+2 n) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(523\) vs. \(2(226)=452\).
Time = 27.19 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.31 \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\sec (e+f x) (d \sin (e+f x))^n \left (a B (1+\sin (e+f x)) \left (a \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} (-\sin (e+f x))^{-n} (1+\sin (e+f x))-\frac {4 \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (-2 a (1+2 n) \operatorname {AppellF1}\left (\frac {1}{2}-n,-\frac {1}{2},-n,\frac {3}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right )+a (-1+2 n) \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (1+\sin (e+f x))\right )}{-1+4 n^2}\right )+A \left (a^2 \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} (-\sin (e+f x))^{-n} (1+\sin (e+f x))^2-\frac {4 a \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}} (1+\sin (e+f x)) \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (2 a (1+2 n) \operatorname {AppellF1}\left (\frac {1}{2}-n,-\frac {1}{2},-n,\frac {3}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right )+a (-1+2 n) \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (1+\sin (e+f x))\right )}{-1+4 n^2}\right )\right )}{8 a^3 f \sqrt {a (1+\sin (e+f x))}} \]
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\[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )\right )}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n} \left (A + B \sin {\left (e + f x \right )}\right )}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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