Integrand size = 38, antiderivative size = 45 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 a f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2920, 2817} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 a f \sqrt {c-c \sin (e+f x)}} \]
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Rule 2817
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 a f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(45)=90\).
Time = 7.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.64 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{5/2} (-28 \cos (2 (e+f x))+\cos (4 (e+f x))+56 \sin (e+f x)-8 \sin (3 (e+f x)))}{32 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {c-c \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(39)=78\).
Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.36
method | result | size |
default | \(-\frac {\left (\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\left (\cos ^{4}\left (f x +e \right )\right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 \left (\cos ^{3}\left (f x +e \right )\right )-7 \cos \left (f x +e \right ) \sin \left (f x +e \right )+4 \left (\cos ^{2}\left (f x +e \right )\right )-\sin \left (f x +e \right )-8 \cos \left (f x +e \right )+1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{2} \left (1+\cos \left (f x +e \right )\right )}{4 f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(151\) |
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (39) = 78\).
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.18 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {{\left (a^{2} \cos \left (f x + e\right )^{4} - 8 \, a^{2} \cos \left (f x + e\right )^{2} + 7 \, a^{2} - 4 \, {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{4 \, c f \cos \left (f x + e\right )} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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none
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {4 \, a^{\frac {5}{2}} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sqrt {c} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
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Time = 2.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^2\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (28\,\cos \left (e+f\,x\right )+27\,\cos \left (3\,e+3\,f\,x\right )-\cos \left (5\,e+5\,f\,x\right )-48\,\sin \left (2\,e+2\,f\,x\right )+8\,\sin \left (4\,e+4\,f\,x\right )\right )}{32\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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