Integrand size = 30, antiderivative size = 89 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c^2 \cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f} \]
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Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f} \]
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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} \sqrt {c-c \sin (e+f x)}}{5 f}+\frac {1}{5} (2 c) \int (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{10 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{5 f} \\ \end{align*}
Time = 3.52 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.63 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {27 \sqrt {3} c (-1+\sin (e+f x)) (1+\sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} (-20 \cos (2 (e+f x))-5 \cos (4 (e+f x))+70 \sin (e+f x)+5 \sin (3 (e+f x))-\sin (5 (e+f x)))}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
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Time = 2.82 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03
method | result | size |
default | \(-\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, c \,a^{3} \left (2 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+5 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \tan \left (f x +e \right )-5 \sec \left (f x +e \right )\right )}{10 f}\) | \(92\) |
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {{\left (5 \, a^{3} c \cos \left (f x + e\right )^{4} - 5 \, a^{3} c + 2 \, {\left (a^{3} c \cos \left (f x + e\right )^{4} - 2 \, a^{3} c \cos \left (f x + e\right )^{2} - 4 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{10 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.17 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 \, {\left (4 \, a^{3} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 5 \, a^{3} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{5 \, f} \]
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Time = 2.66 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.22 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {a^3\,c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (20\,\cos \left (e+f\,x\right )+25\,\cos \left (3\,e+3\,f\,x\right )+5\,\cos \left (5\,e+5\,f\,x\right )-75\,\sin \left (2\,e+2\,f\,x\right )-4\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )\right )}{80\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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