Integrand size = 26, antiderivative size = 83 \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\frac {2^{\frac {1}{2}+m} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2}-m,-\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec ^5(e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (3+3 \sin (e+f x))^{2+m}}{45 c^3 f} \]
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Time = 0.11 (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} \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\frac {2^{m+\frac {1}{2}} \sec ^5(e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{2}-m,-\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{5 a^2 c^3 f} \]
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Rule 71
Rule 72
Rule 2768
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) (a+a \sin (e+f x))^{3+m} \, dx}{a^3 c^3} \\ & = \frac {\left (\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 {\left (2^{-\frac {1}{2}+m} \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} \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} \\ \end{align*}
\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{3}}d x\]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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