Integrand size = 26, antiderivative size = 144 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f} \]
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Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 3852} \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3} \]
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Rule 2751
Rule 2815
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a^2 c^2} \\ & = \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{3 a^2 c^3} \\ & = \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {10 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)} \, dx}{21 a^2 c^4} \\ & = \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \int \sec ^4(e+f x) \, dx}{21 a^2 c^5} \\ & = \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {8 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{21 a^2 c^5 f} \\ & = \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f} \\ \end{align*}
Time = 2.06 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (2397276 \cos (e+f x)+221184 \cos (2 (e+f x))+133182 \cos (3 (e+f x))+98304 \cos (4 (e+f x))-399546 \cos (5 (e+f x))-8192 \cos (6 (e+f x))+294912 \sin (e+f x)-1797957 \sin (2 (e+f x))+16384 \sin (3 (e+f x))-799092 \sin (4 (e+f x))-49152 \sin (5 (e+f x))+66591 \sin (6 (e+f x)))}{1032192 a^2 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^2} \]
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Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {32 \left (i+36 \,{\mathrm e}^{5 i \left (f x +e \right )}-12 i {\mathrm e}^{2 i \left (f x +e \right )}-27 i {\mathrm e}^{4 i \left (f x +e \right )}-6 \,{\mathrm e}^{i \left (f x +e \right )}+2 \,{\mathrm e}^{3 i \left (f x +e \right )}\right )}{63 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{5} a^{2}}\) | \(100\) |
parallelrisch | \(\frac {-\frac {38}{63}+6 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {68 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {26 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {100 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-\frac {28 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+12 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {470 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21}-\frac {26 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}}{f \,c^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(181\) |
derivativedivides | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{5} f}\) | \(193\) |
default | \(\frac {-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{5} f}\) | \(193\) |
norman | \(\frac {-\frac {28 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {38}{63 a c f}+\frac {6 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {12 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {26 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {470 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a c f}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 a c f}-\frac {68 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {26 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a c f}+\frac {100 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(286\) |
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Time = 0.34 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {16 \, \cos \left (f x + e\right )^{6} - 72 \, \cos \left (f x + e\right )^{4} + 30 \, \cos \left (f x + e\right )^{2} + 2 \, {\left (24 \, \cos \left (f x + e\right )^{4} - 20 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) + 7}{63 \, {\left (3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} - {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (131) = 262\).
Time = 20.31 (sec) , antiderivative size = 3186, normalized size of antiderivative = 22.12 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (137) = 274\).
Time = 0.21 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.60 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (\frac {51 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {39 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {450 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {306 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {294 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {378 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {273 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {189 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {63 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - 19\right )}}{63 \, {\left (a^{2} c^{5} - \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{2} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )} f} \]
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Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {21 \, {\left (21 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19\right )}}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {3591 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 19656 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 56196 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 95760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 107730 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 79464 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 38484 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10944 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1615}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2016 \, f} \]
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Time = 9.01 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {2\,\left (63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}-189\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+273\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-378\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+294\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+306\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-450\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+235\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+39\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-51\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+19\right )}{63\,a^2\,c^5\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^9\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
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