Integrand size = 28, antiderivative size = 100 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 c^2 \cos (e+f x) (3+3 \sin (e+f x))^m}{f \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m)} \]
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Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 c^2 \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3)} \]
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Rule 2817
Rule 2819
Rubi steps \begin{align*} \text {integral}& = \frac {2 c \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m)}+\frac {(4 c) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{3+2 m} \\ & = \frac {8 c^2 \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m)} \\ \end{align*}
Time = 2.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2\ 3^m c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^m \sqrt {c-c \sin (e+f x)} (-5-2 m+(1+2 m) \sin (e+f x))}{f (1+2 m) (3+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.45 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 \, {\left ({\left (2 \, c m + c\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c m + 5 \, c\right )} \cos \left (f x + e\right ) - {\left ({\left (2 \, c m + c\right )} \cos \left (f x + e\right ) - 4 \, c\right )} \sin \left (f x + e\right ) + 4 \, c\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{4 \, f m^{2} + 8 \, f m + {\left (4 \, f m^{2} + 8 \, f m + 3 \, f\right )} \cos \left (f x + e\right ) - {\left (4 \, f m^{2} + 8 \, f m + 3 \, f\right )} \sin \left (f x + e\right ) + 3 \, f} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (96) = 192\).
Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.93 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (4 \, m^{2} + 8 \, m + 3\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \]
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\[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Time = 1.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int (3+3 \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx=-\frac {c\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (10\,\cos \left (e+f\,x\right )-\sin \left (2\,e+2\,f\,x\right )+4\,m\,\cos \left (e+f\,x\right )-2\,m\,\sin \left (2\,e+2\,f\,x\right )\right )}{f\,\left (\sin \left (e+f\,x\right )-1\right )\,\left (4\,m^2+8\,m+3\right )} \]
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