Integrand size = 36, antiderivative size = 148 \[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {2^{\frac {1}{2}+m} (A (2-m)-B (3+m)) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2}-m,-\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{2+m}}{5 a^2 c^3 f (2-m)}+\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)} \]
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Time = 0.22 (sec) , antiderivative size = 148, 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} \[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {B \sec ^5(e+f x) (a \sin (e+f x)+a)^{m+3}}{a^3 c^3 f (2-m)}+\frac {2^{m+\frac {1}{2}} (A (2-m)-B (m+3)) \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2}-m,-\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{5 a^2 c^3 f (2-m)} \]
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Rule 71
Rule 72
Rule 2768
Rule 2939
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} (A+B \sin (e+f x)) \, dx}{a^3 c^3} \\ & = \frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (A-\frac {B (3+m)}{2-m}\right ) \int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} \, dx}{a^3 c^3} \\ & = \frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (\left (A-\frac {B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f} \\ & = \frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)}+\frac {\left (2^{-\frac {1}{2}+m} \left (A-\frac {B (3+m)}{2-m}\right ) \sec ^5(e+f x) (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m}}{(a-a x)^{7/2}} \, dx,x,\sin (e+f x)\right )}{a c^3 f} \\ & = \frac {2^{\frac {1}{2}+m} \left (A-\frac {B (3+m)}{2-m}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2}-m,-\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{2+m}}{5 a^2 c^3 f}+\frac {B \sec ^5(e+f x) (a+a \sin (e+f x))^{3+m}}{a^3 c^3 f (2-m)} \\ \end{align*}
\[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )}{\left (c -c \sin \left (f x +e \right )\right )^{3}}d x\]
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\[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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\[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^m (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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