Integrand size = 30, antiderivative size = 134 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^3 \cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{15 f \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {c \cos (e+f x) (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f} \]
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Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^3 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 f \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f} \]
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Rule 2817
Rule 2819
Rubi steps \begin{align*} \text {integral}& = \frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}+\frac {1}{3} (2 c) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx \\ & = \frac {2 c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f}+\frac {1}{15} \left (4 c^2\right ) \int (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {c^3 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{15 f \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 f} \\ \end{align*}
Time = 4.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.19 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {9 \sqrt {3} c^2 (-1+\sin (e+f x))^2 (1+\sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} (-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x))+600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x)))}{320 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
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Time = 4.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (-5 \left (\cos ^{5}\left (f x +e \right )\right )+6 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+8 \sin \left (f x +e \right ) \cos \left (f x +e \right )+16 \tan \left (f x +e \right )+5 \sec \left (f x +e \right )\right )}{30 f}\) | \(94\) |
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {{\left (5 \, a^{3} c^{2} \cos \left (f x + e\right )^{6} - 5 \, a^{3} c^{2} - 2 \, {\left (3 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{3} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Time = 0.41 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.52 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {16 \, {\left (10 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 36 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} + 45 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 20 \, a^{3} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}\right )} \sqrt {a} \sqrt {c}}{15 \, f} \]
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Time = 10.95 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.93 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a^3\,c^2\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (75\,\cos \left (e+f\,x\right )+105\,\cos \left (3\,e+3\,f\,x\right )+35\,\cos \left (5\,e+5\,f\,x\right )+5\,\cos \left (7\,e+7\,f\,x\right )-700\,\sin \left (2\,e+2\,f\,x\right )-112\,\sin \left (4\,e+4\,f\,x\right )-12\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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