Integrand size = 36, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {2^{\frac {1}{2}+m} a^4 c^3 (B (3-m)-A (4+m)) \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-4+m}}{7 f (4+m)}-\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)} \]
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Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2939, 2768, 72, 71} \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {a^4 c^3 2^{m+\frac {1}{2}} (B (3-m)-A (m+4)) \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{7 f (m+4)}-\frac {a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f (m+4)} \]
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Rule 71
Rule 72
Rule 2768
Rule 2939
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} (A+B \sin (e+f x)) \, dx \\ & = -\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\left (a^3 c^3 \left (A-\frac {B (3-m)}{4+m}\right )\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} \, dx \\ & = -\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\frac {\left (a^5 c^3 \left (A-\frac {B (3-m)}{4+m}\right ) \cos ^7(e+f x)\right ) \text {Subst}\left (\int (a-a x)^{5/2} (a+a x)^{-\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2} (a+a \sin (e+f x))^{7/2}} \\ & = -\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)}+\frac {\left (2^{-\frac {1}{2}+m} a^5 c^3 \left (A-\frac {B (3-m)}{4+m}\right ) \cos ^7(e+f x) (a+a \sin (e+f x))^{-4+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m} (a-a x)^{5/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{7/2}} \\ & = -\frac {2^{\frac {1}{2}+m} a^4 c^3 \left (A-\frac {B (3-m)}{4+m}\right ) \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-4+m}}{7 f}-\frac {a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f (4+m)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 26.95 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.17 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=-\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^3 (\cos (e+f x)+i (1+\sin (e+f x))) \left (-\frac {10 (4 A-3 B) \operatorname {Hypergeometric2F1}(1,1+m,1-m,i \cos (e+f x)-\sin (e+f x))}{m}+\frac {2 (15 A-13 B) \operatorname {Hypergeometric2F1}(1,2+m,2-m,i \cos (e+f x)-\sin (e+f x)) (-i \cos (e+f x)+\sin (e+f x))}{-1+m}+\frac {2 (15 A-13 B) \operatorname {Hypergeometric2F1}(1,m,-m,i \cos (e+f x)-\sin (e+f x)) (i \cos (e+f x)+\sin (e+f x))}{1+m}+\frac {4 (3 A-4 B) \operatorname {Hypergeometric2F1}(1,-1+m,-1-m,i \cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))-i \sin (2 (e+f x)))}{2+m}+\frac {4 (3 A-4 B) \operatorname {Hypergeometric2F1}(1,3+m,3-m,i \cos (e+f x)-\sin (e+f x)) (\cos (2 (e+f x))+i \sin (2 (e+f x)))}{-2+m}-\frac {2 i (A-3 B) \operatorname {Hypergeometric2F1}(1,-2+m,-2-m,i \cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))-i \sin (3 (e+f x)))}{3+m}+\frac {2 i (A-3 B) \operatorname {Hypergeometric2F1}(1,4+m,4-m,i \cos (e+f x)-\sin (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))}{-3+m}+\frac {B \operatorname {Hypergeometric2F1}(1,-3+m,-3-m,i \cos (e+f x)-\sin (e+f x)) (\cos (4 (e+f x))-i \sin (4 (e+f x)))}{4+m}+\frac {B \operatorname {Hypergeometric2F1}(1,5+m,5-m,i \cos (e+f x)-\sin (e+f x)) (\cos (4 (e+f x))+i \sin (4 (e+f x)))}{-4+m}\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{3}d x\]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\int { -{\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=- c^{3} \left (\int \left (- A \left (a \sin {\left (e + f x \right )} + a\right )^{m}\right )\, dx + \int 3 A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- 3 A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int A \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{3}{\left (e + f x \right )}\, dx + \int \left (- B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\right )\, dx + \int 3 B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\, dx + \int \left (- 3 B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int B \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\int { -{\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\int { -{\left (B \sin \left (f x + e\right ) + A\right )} {\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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