Integrand size = 38, antiderivative size = 92 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f} \]
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Time = 0.30 (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} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}} \]
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Rule 2817
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}+\frac {\int (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx}{2 a} \\ & = \frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f} \\ \end{align*}
Time = 1.45 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {a \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (-12 \cos (2 (e+f x))-3 \cos (4 (e+f x))+8 (9 \sin (e+f x)+\sin (3 (e+f x))))}{96 f} \]
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Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a \left (-3 \left (\cos ^{3}\left (f x +e \right )\right )+4 \cos \left (f x +e \right ) \sin \left (f x +e \right )+8 \tan \left (f x +e \right )+3 \sec \left (f x +e \right )\right )}{12 f}\) | \(73\) |
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {{\left (3 \, a \cos \left (f x + e\right )^{4} - 4 \, {\left (a \cos \left (f x + e\right )^{2} + 2 \, a\right )} \sin \left (f x + e\right ) - 3 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{12 \, f \cos \left (f x + e\right )} \]
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\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \cos ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.53 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {4 \, {\left (3 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 8 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 6 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}\right )} \sqrt {a} \sqrt {c}}{3 \, f} \]
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Time = 10.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (12\,\cos \left (e+f\,x\right )+15\,\cos \left (3\,e+3\,f\,x\right )+3\,\cos \left (5\,e+5\,f\,x\right )-80\,\sin \left (2\,e+2\,f\,x\right )-8\,\sin \left (4\,e+4\,f\,x\right )\right )}{96\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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