Integrand size = 28, antiderivative size = 177 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\frac {35 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{576 \sqrt {2} c^{5/2} f}+\frac {35 \cos (e+f x)}{576 c f (c-c \sin (e+f x))^{3/2}}+\frac {7 \sec (e+f x)}{216 c f (c-c \sin (e+f x))^{3/2}}-\frac {35 \sec (e+f x)}{432 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {\sec ^3(e+f x)}{27 c^2 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.25 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2815, 2766, 2760, 2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\frac {35 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{64 \sqrt {2} a^2 c^{5/2} f}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {35 \sec (e+f x)}{48 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2760
Rule 2766
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{a^2 c^2} \\ & = -\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {7 \int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{6 a^2 c} \\ & = \frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {35 \int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{48 a^2 c^2} \\ & = \frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}-\frac {35 \sec (e+f x)}{48 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {35 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{32 a^2 c} \\ & = \frac {35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}-\frac {35 \sec (e+f x)}{48 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}+\frac {35 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{128 a^2 c^2} \\ & = \frac {35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}-\frac {35 \sec (e+f x)}{48 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {35 \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 )}{64 a^2 c^2 f} \\ & = \frac {35 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{64 \sqrt {2} a^2 c^{5/2} f}+\frac {35 \cos (e+f x)}{64 a^2 c f (c-c \sin (e+f x))^{3/2}}+\frac {7 \sec (e+f x)}{24 a^2 c f (c-c \sin (e+f x))^{3/2}}-\frac {35 \sec (e+f x)}{48 a^2 c^2 f \sqrt {c-c \sin (e+f x)}}-\frac {\sec ^3(e+f x)}{3 a^2 c^2 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=-\frac {\left (\frac {1}{13824}+\frac {i}{13824}\right ) \sec ^3(e+f x) \left (840 \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+(1-i) (102+70 \cos (2 (e+f x))-329 \sin (e+f x)-105 \sin (3 (e+f x)))\right )}{c^2 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.80 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32
method | result | size |
default | \(-\frac {105 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {2}\, c^{2}+70 \left (\sin ^{2}\left (f x +e \right )\right ) c^{\frac {7}{2}}-210 \left (\sin ^{3}\left (f x +e \right )\right ) c^{\frac {7}{2}}-210 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) \sqrt {2}\, c^{2}+322 c^{\frac {7}{2}} \sin \left (f x +e \right )+105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}}-86 c^{\frac {7}{2}}}{384 c^{\frac {11}{2}} a^{2} \left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(233\) |
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Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - \cos \left (f x + e\right )^{3}\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (35 \, \cos \left (f x + e\right )^{2} - 7 \, {\left (15 \, \cos \left (f x + e\right )^{2} + 8\right )} \sin \left (f x + e\right ) + 8\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{768 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} f \cos \left (f x + e\right )^{3}\right )}} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {1}{c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a^{2}} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (165) = 330\).
Time = 0.36 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.58 \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\frac {\frac {420 \, \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^{2} c^{\frac {5}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (\sqrt {c} - \frac {24 \, \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 {210 \, \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}}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}{a^{2} c^{3} {\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 )} + \frac {256 \, \sqrt {2} {\left (5 \, \sqrt {c} + \frac {9 \, \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 {6 \, \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}}\right )}}{a^{2} c^{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} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, {\left (\frac {24 \, \sqrt {2} a^{2} c^{\frac {7}{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 )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {\sqrt {2} a^{2} c^{\frac {7}{2}} {\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^{4} c^{6}}}{3072 \, f} \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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