Integrand size = 26, antiderivative size = 83 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=-\frac {81\ 2^{\frac {1}{2}+m} c^3 \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{\frac {1}{2}-m} (3+3 \sin (e+f x))^{-4+m}}{7 f} \]
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Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2768, 72, 71} \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=-\frac {a^4 c^3 2^{m+\frac {1}{2}} \cos ^7(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1}{2}-m,\frac {9}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{7 f} \]
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Rule 71
Rule 72
Rule 2768
Rule 2815
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} \, dx \\ & = \frac {\left (a^5 c^3 \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 {\left (2^{-\frac {1}{2}+m} a^5 c^3 \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 \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} \\ \end{align*}
Result contains complex when optimal does not.
Time = 15.71 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.93 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\frac {(1+i) 3^m (1+\sin (e+f x))^m (c-c \sin (e+f x))^3 \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (7+2 m) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (-115-58 m-12 m^2+\left (5+6 m+4 m^2\right ) \cos (2 (e+f x))+16 m (4+m) \sin (e+f x)\right ) \left (-1+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^4+15 i \operatorname {Hypergeometric2F1}\left (4+m,7+2 m,2 (4+m),\frac {(1-i) \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}{-i+\tan \left (\frac {1}{2} (e+f x)\right )}\right ) (-i \cos (e+f x)-\sin (e+f x))^m \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )^6\right )}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{3}d x\]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=- c^{3} \left (\int 3 \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin {\left (e + f x \right )}\, dx + \int \left (- 3 \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (a \sin {\left (e + f x \right )} + a\right )^{m} \sin ^{3}{\left (e + f x \right )}\, dx + \int \left (- \left (a \sin {\left (e + f x \right )} + a\right )^{m}\right )\, dx\right ) \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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