Integrand size = 38, antiderivative size = 92 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {a \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{15 c f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{6 c f} \]
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2819, 2817} \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{6 c f}-\frac {a \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{15 c f \sqrt {a \sin (e+f x)+a}} \]
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Rule 2817
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2} \, dx}{a c} \\ & = -\frac {\cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{6 c f}+\frac {\int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx}{3 c} \\ & = -\frac {a \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{15 c f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{6 c f} \\ \end{align*}
Time = 2.54 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (405 \cos (2 (e+f x))+90 \cos (4 (e+f x))-5 \cos (6 (e+f x))+1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x)))}{960 f} \]
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Time = 1.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3} \left (5 \left (\cos ^{5}\left (f x +e \right )\right )+18 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-30 \left (\cos ^{3}\left (f x +e \right )\right )-16 \cos \left (f x +e \right ) \sin \left (f x +e \right )-32 \tan \left (f x +e \right )+25 \sec \left (f x +e \right )\right )}{30 f}\) | \(101\) |
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {{\left (5 \, c^{3} \cos \left (f x + e\right )^{6} - 30 \, c^{3} \cos \left (f x + e\right )^{4} + 25 \, c^{3} + 2 \, {\left (9 \, c^{3} \cos \left (f x + e\right )^{4} - 8 \, c^{3} \cos \left (f x + e\right )^{2} - 16 \, c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.11 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {32 \, {\left (5 \, c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 6 \, c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}\right )} \sqrt {a} \sqrt {c}}{15 \, f} \]
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Time = 12.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.32 \[ \int \cos ^2(e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (405\,\cos \left (e+f\,x\right )+495\,\cos \left (3\,e+3\,f\,x\right )+85\,\cos \left (5\,e+5\,f\,x\right )-5\,\cos \left (7\,e+7\,f\,x\right )+1100\,\sin \left (2\,e+2\,f\,x\right )-16\,\sin \left (4\,e+4\,f\,x\right )-36\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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