Integrand size = 28, antiderivative size = 161 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{405 f}+\frac {1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{81 f}-\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{27 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{81 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{81 c^2 f} \]
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Time = 0.30 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753, 2752} \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}-\frac {4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}-\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac {1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{15/2} \, dx}{a^3 c^3} \\ & = \frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}+\frac {16 \int \sec ^6(e+f x) (c-c \sin (e+f x))^{13/2} \, dx}{3 a^3 c^2} \\ & = \frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}+\frac {64 \int \sec ^6(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{a^3 c} \\ & = -\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}-\frac {512 \int \sec ^6(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{a^3} \\ & = \frac {1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f}-\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f}+\frac {(2048 c) \int \sec ^6(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{3 a^3} \\ & = -\frac {4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac {1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 f}-\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{3 a^3 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 a^3 c^2 f} \\ \end{align*}
Time = 8.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {c^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-5649+2740 \cos (2 (e+f x))+5 \cos (4 (e+f x))-7800 \sin (e+f x)+200 \sin (3 (e+f x)))}{1620 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3} \]
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Time = 179.74 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.57
method | result | size |
default | \(-\frac {2 c^{5} \left (\sin \left (f x +e \right )-1\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-100 \left (\sin ^{3}\left (f x +e \right )\right )-690 \left (\sin ^{2}\left (f x +e \right )\right )-900 \sin \left (f x +e \right )-363\right )}{15 a^{3} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(91\) |
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Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (5 \, c^{4} \cos \left (f x + e\right )^{4} + 680 \, c^{4} \cos \left (f x + e\right )^{2} - 1048 \, c^{4} + 100 \, {\left (c^{4} \cos \left (f x + e\right )^{2} - 10 \, c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (156) = 312\).
Time = 0.30 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.93 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {2 \, {\left (363 \, c^{\frac {9}{2}} + \frac {1800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5301 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {11600 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {21343 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {30200 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {40065 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {40800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {40065 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {30200 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {21343 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {11600 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac {5301 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {1800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + \frac {363 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}}\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {9}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (156) = 312\).
Time = 0.39 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.65 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {8 \, \sqrt {2} \sqrt {c} {\left (\frac {5 \, {\left (11 \, c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {24 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {9 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{3}} - \frac {73 \, c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {320 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {490 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {240 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5}}\right )}}{15 \, f} \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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