Integrand size = 30, antiderivative size = 134 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\cos (e+f x)}{4 f (3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{12 f (3+3 \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}+\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{36 f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2822, 2820, 3855} \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \text {arctanh}(\sin (e+f x))}{4 a^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}} \]
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Rule 2820
Rule 2822
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}+\frac {\int \frac {1}{(a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}} \, dx}{2 a} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{4 a f (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{4 a^2} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{4 a f (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) \int \sec (e+f x) \, dx}{4 a^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x)}{4 a f (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}+\frac {\text {arctanh}(\sin (e+f x)) \cos (e+f x)}{4 a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-10-3 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )+\cos (2 (e+f x)) \left (2+\log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )+3 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-2 \left (5+2 \log \left (1-\tan \left (\frac {1}{2} (e+f x)\right )\right )-2 \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin (e+f x)\right )}{72 \sqrt {3} f (1+\sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(421\) vs. \(2(122)=244\).
Time = 3.54 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.15
method | result | size |
default | \(\frac {\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-\left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+\left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+2 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-2 \left (\cos ^{3}\left (f x +e \right )\right )+\left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-\left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+3 \sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos ^{2}\left (f x +e \right )+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \cos \left (f x +e \right )-2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \cos \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )-1}{4 f \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+2 \sin \left (f x +e \right )+\cos \left (f x +e \right )+2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{2}}\) | \(422\) |
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Time = 0.32 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.81 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\left [\frac {{\left (\cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt {a c} \log \left (-\frac {a c \cos \left (f x + e\right )^{3} - 2 \, a c \cos \left (f x + e\right ) - 2 \, \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3}}\right ) + 2 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\sin \left (f x + e\right ) + 2\right )}}{8 \, {\left (a^{3} c f \cos \left (f x + e\right )^{3} - 2 \, a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} c f \cos \left (f x + e\right )\right )}}, -\frac {{\left (\cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) - \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\sin \left (f x + e\right ) + 2\right )}}{4 \, {\left (a^{3} c f \cos \left (f x + e\right )^{3} - 2 \, a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} c f \cos \left (f x + e\right )\right )}}\right ] \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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Time = 0.55 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {2 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{\frac {5}{2}} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, \log \left ({\left | \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right )}{a^{\frac {5}{2}} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1}{a^{\frac {5}{2}} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{16 \, f} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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