Integrand size = 28, antiderivative size = 176 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{432 \sqrt {2} c^{3/2} f}+\frac {7 \cos (e+f x)}{432 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{324 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{810 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{135 c^3 f} \]
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Time = 0.24 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2815, 2754, 2766, 2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} a^3 c^{3/2} f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2766
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{a^3 c^3} \\ & = -\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \sec ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx}{10 a^3 c^2} \\ & = -\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{12 a^3 c} \\ & = -\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{8 a^3} \\ & = \frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{32 a^3 c} \\ & = \frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac {7 \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{16 a^3 c f} \\ & = \frac {7 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} a^3 c^{3/2} f}+\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.51 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\frac {1}{51840}+\frac {i}{51840}\right ) \cos (e+f x) \left (840 \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+(1-i) (206+350 \cos (2 (e+f x))-231 \sin (e+f x)+105 \sin (3 (e+f x)))\right )}{c f (-1+\sin (e+f x)) (1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \]
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Time = 1.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {105 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -105 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +42 c^{\frac {7}{2}} \sin \left (f x +e \right )-350 \left (\sin ^{2}\left (f x +e \right )\right ) c^{\frac {7}{2}}-210 \left (\sin ^{3}\left (f x +e \right )\right ) c^{\frac {7}{2}}+278 c^{\frac {7}{2}}}{480 c^{\frac {9}{2}} a^{3} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(170\) |
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Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + \cos \left (f x + e\right )^{3}\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (175 \, \cos \left (f x + e\right )^{2} + 21 \, {\left (5 \, \cos \left (f x + e\right )^{2} - 4\right )} \sin \left (f x + e\right ) - 36\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{960 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (164) = 328\).
Time = 0.39 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.41 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\frac {15 \, \sqrt {2} {\left (\frac {14 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{a^{3} c^{\frac {3}{2}} {\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 )} - \frac {210 \, \sqrt {2} \log \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}\right )}{a^{3} c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {15 \, \sqrt {2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{a^{3} c^{\frac {3}{2}} {\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 )} - \frac {32 \, \sqrt {2} {\left (29 \, \sqrt {c} + \frac {100 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {170 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {120 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}\right )}}{a^{3} c^{2} {\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} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{1920 \, f} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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