Integrand size = 38, antiderivative size = 190 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {8 c^3 \cos (e+f x) \log (1+\sin (e+f x))}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.46 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2920, 2819, 2816, 2746, 31} \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {8 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a \sin (e+f x)+a}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(c-c \sin (e+f x))^{7/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a c} \\ & = \frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a} \\ & = \frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(4 c) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a} \\ & = \frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (8 c^2\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a} \\ & = \frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (8 c^3 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (8 c^3 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {8 c^3 \cos (e+f x) \log (1+\sin (e+f x))}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Time = 8.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)} \left (-12 \cos (2 (e+f x))+192 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-87 \sin (e+f x)+\sin (3 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {\left (\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\left (\cos ^{4}\left (f x +e \right )\right )-5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-6 \left (\cos ^{3}\left (f x +e \right )\right )-22 \cos \left (f x +e \right ) \sin \left (f x +e \right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+48 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )+17 \left (\cos ^{2}\left (f x +e \right )\right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \cos \left (f x +e \right )+48 \cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-16 \sin \left (f x +e \right )+6 \cos \left (f x +e \right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+48 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-16\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}{3 f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a}\) | \(274\) |
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\[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {a} c^{\frac {5}{2}} {\left (\frac {6 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\sqrt {2} {\left (2 \, a^{4} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 3 \, a^{4} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 6 \, a^{4} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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