Integrand size = 26, antiderivative size = 76 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f} \]
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Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 3852} \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )} \]
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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)} \, dx}{a^2 c^2} \\ & = \frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \int \sec ^4(e+f x) \, dx}{5 a^2 c^3} \\ & = \frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^2 c^3 f} \\ & = \frac {\sec ^3(e+f x)}{5 a^2 f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{5 a^2 c^3 f}+\frac {4 \tan ^3(e+f x)}{15 a^2 c^3 f} \\ \end{align*}
Time = 1.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {1050 \cos (e+f x)+256 \cos (2 (e+f x))+350 \cos (3 (e+f x))+128 \cos (4 (e+f x))+768 \sin (e+f x)-350 \sin (2 (e+f x))+256 \sin (3 (e+f x))-175 \sin (4 (e+f x))}{1920 a^2 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {\frac {32 \,{\mathrm e}^{3 i \left (f x +e \right )}}{5}-\frac {32 i {\mathrm e}^{2 i \left (f x +e \right )}}{15}+\frac {32 \,{\mathrm e}^{i \left (f x +e \right )}}{15}-\frac {16 i}{15}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{3} a^{2}}\) | \(77\) |
parallelrisch | \(\frac {-\frac {2}{5}+\frac {2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {10 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+2 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {26 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}+\frac {14 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}}{f \,c^{3} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(129\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {5}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {5}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{3} f}\) | \(133\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {5}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {5}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {3}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{2} c^{3} f}\) | \(133\) |
norman | \(\frac {-\frac {10 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2}{5 a c f}+\frac {2 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5 a c f}+\frac {14 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}+\frac {2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {26 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(198\) |
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Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {8 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sin \left (f x + e\right ) - 1}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} f \cos \left (f x + e\right )^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (66) = 132\).
Time = 4.83 (sec) , antiderivative size = 1418, normalized size of antiderivative = 18.66 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (71) = 142\).
Time = 0.22 (sec) , antiderivative size = 335, normalized size of antiderivative = 4.41 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {13 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {15 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 3\right )}}{15 \, {\left (a^{2} c^{3} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {6 \, a^{2} c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{2} c^{3} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a^{2} c^{3} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \]
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Time = 0.37 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {\frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13\right )}}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {165 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 480 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 650 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 400 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 113}{a^{2} c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{120 \, f} \]
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Time = 7.83 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^3} \, dx=-\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+25\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+13\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+9\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+3\right )}{15\,a^2\,c^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^5\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \]
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