Integrand size = 30, antiderivative size = 133 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {\cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{2 f (c-c \sin (e+f x))^{13/2}}-\frac {3 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{5 c f (c-c \sin (e+f x))^{11/2}}+\frac {9 \cos (e+f x)}{20 c^2 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{9/2}} \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2818, 2817} \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {a^3 \cos (e+f x)}{60 c^2 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}} \]
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Rule 2817
Rule 2818
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{3 c} \\ & = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{15 c^2} \\ & = \frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 c f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{60 c^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}} \\ \end{align*}
Time = 8.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {3 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2} (29-15 \cos (2 (e+f x))+36 \sin (e+f x))}{40 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (-1+\sin (e+f x))^6 \sqrt {c-c \sin (e+f x)}} \]
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Time = 3.40 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.32
method | result | size |
default | \(-\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{2} \left (7 \left (\cos ^{5}\left (f x +e \right )\right )+42 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-126 \left (\cos ^{3}\left (f x +e \right )\right )-224 \sin \left (f x +e \right ) \cos \left (f x +e \right )+321 \cos \left (f x +e \right )+242 \tan \left (f x +e \right )-202 \sec \left (f x +e \right )\right )}{60 f \left (\left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-5 \left (\cos ^{4}\left (f x +e \right )\right )-12 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+20 \left (\cos ^{2}\left (f x +e \right )\right )+16 \sin \left (f x +e \right )-16\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{6}}\) | \(175\) |
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Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.22 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {{\left (15 \, a^{2} \cos \left (f x + e\right )^{2} - 18 \, a^{2} \sin \left (f x + e\right ) - 22 \, a^{2}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {{\left (15 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 24 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 10 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{960 \, c^{\frac {13}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12}} \]
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Time = 13.23 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.16 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {464\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,c^7\,f}+\frac {192\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}-\frac {16\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \]
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