Integrand size = 34, antiderivative size = 86 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=-\frac {2^{\frac {3}{2}+m} a^4 c^3 \cos ^9(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-\frac {1}{2}-m,\frac {11}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^{-4+m}}{9 f} \]
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Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2919, 2768, 72, 71} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=-\frac {a^4 c^3 2^{m+\frac {3}{2}} \cos ^9(e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^{m-4} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-m-\frac {1}{2},\frac {11}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{9 f} \]
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Rule 71
Rule 72
Rule 2768
Rule 2919
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^8(e+f x) (a+a \sin (e+f x))^{-3+m} \, dx \\ & = \frac {\left (a^5 c^3 \cos ^9(e+f x)\right ) \text {Subst}\left (\int (a-a x)^{7/2} (a+a x)^{\frac {1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{9/2} (a+a \sin (e+f x))^{9/2}} \\ & = \frac {\left (2^{\frac {1}{2}+m} a^5 c^3 \cos ^9(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)^{7/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{9/2}} \\ & = -\frac {2^{\frac {3}{2}+m} a^4 c^3 \cos ^9(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-\frac {1}{2}-m,\frac {11}{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}}{9 f} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=-\frac {2^{\frac {3}{2}+m} c^3 \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-\frac {1}{2}-m,\frac {11}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^4 (1+\sin (e+f x))^{-\frac {1}{2}-m} (a (1+\sin (e+f x)))^m}{9 f} \]
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\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{3}d x\]
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\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\text {Timed out} \]
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\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int { -{\left (c \sin \left (f x + e\right ) - c\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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