Integrand size = 34, antiderivative size = 104 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3}-\frac {a (A+11 B) c \cos (e+f x)}{15 f \left (c^2-c^2 \sin (e+f x)\right )^2}-\frac {a (A-4 B) \cos (e+f x)}{15 f \left (c^3-c^3 \sin (e+f x)\right )} \]
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Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3046, 2936, 2829, 2727} \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=-\frac {a (A-4 B) \cos (e+f x)}{15 f \left (c^3-c^3 \sin (e+f x)\right )}-\frac {a c (A+11 B) \cos (e+f x)}{15 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3} \]
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Rule 2727
Rule 2829
Rule 2936
Rule 3046
Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx \\ & = \frac {2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3}+\frac {a \int \frac {-A c-6 B c-5 B c \sin (e+f x)}{(c-c \sin (e+f x))^2} \, dx}{5 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3}-\frac {a (A+11 B) \cos (e+f x)}{15 c f (c-c \sin (e+f x))^2}-\frac {(a (A-4 B)) \int \frac {1}{c-c \sin (e+f x)} \, dx}{15 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{5 f (c-c \sin (e+f x))^3}-\frac {a (A+11 B) \cos (e+f x)}{15 c f (c-c \sin (e+f x))^2}-\frac {a (A-4 B) \cos (e+f x)}{15 f \left (c^3-c^3 \sin (e+f x)\right )} \\ \end{align*}
Time = 6.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.41 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {a \left (15 (A-B) \cos \left (e+\frac {f x}{2}\right )-5 (A-B) \cos \left (e+\frac {3 f x}{2}\right )+5 A \sin \left (\frac {f x}{2}\right )+25 B \sin \left (\frac {f x}{2}\right )+15 B \sin \left (2 e+\frac {3 f x}{2}\right )+A \sin \left (2 e+\frac {5 f x}{2}\right )-4 B \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{30 c^3 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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Time = 0.75 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(-\frac {2 \left (A \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-A +B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (5 A +B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {\left (-A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3}+\frac {4 A}{15}-\frac {B}{15}\right ) a}{f \,c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(94\) |
derivativedivides | \(\frac {2 a \left (-\frac {14 A +10 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {8 A +8 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 B +6 A}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{3}}\) | \(115\) |
default | \(\frac {2 a \left (-\frac {14 A +10 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {8 A +8 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 B +6 A}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{3}}\) | \(115\) |
risch | \(\frac {-\frac {10 B a \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}-2 i B a \,{\mathrm e}^{3 i \left (f x +e \right )}+\frac {2 i B a \,{\mathrm e}^{i \left (f x +e \right )}}{3}+2 B a \,{\mathrm e}^{4 i \left (f x +e \right )}-\frac {2 i A a \,{\mathrm e}^{i \left (f x +e \right )}}{3}-\frac {2 A a \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}-\frac {2 a A}{15}+\frac {8 B a}{15}+2 i A a \,{\mathrm e}^{3 i \left (f x +e \right )}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} f \,c^{3}}\) | \(127\) |
norman | \(\frac {-\frac {8 a A -2 B a}{15 c f}-\frac {2 a A \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {10 \left (a A -B a \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {2 \left (a A -B a \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {\left (2 a A -2 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 c f}-\frac {2 \left (11 a A +B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {2 \left (11 a A +B a \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {2 \left (7 a A -7 B a \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {2 \left (23 a A +3 B a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(262\) |
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Time = 0.27 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.76 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=-\frac {{\left (A - 4 \, B\right )} a \cos \left (f x + e\right )^{3} - {\left (2 \, A + 7 \, B\right )} a \cos \left (f x + e\right )^{2} + 3 \, {\left (A + B\right )} a \cos \left (f x + e\right ) + 6 \, {\left (A + B\right )} a + {\left ({\left (A - 4 \, B\right )} a \cos \left (f x + e\right )^{2} + 3 \, {\left (A + B\right )} a \cos \left (f x + e\right ) + 6 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )}{15 \, {\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f - {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (92) = 184\).
Time = 4.34 (sec) , antiderivative size = 1035, normalized size of antiderivative = 9.95 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (101) = 202\).
Time = 0.23 (sec) , antiderivative size = 737, normalized size of antiderivative = 7.09 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.26 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (15 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 15 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 25 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, A a - B a\right )}}{15 \, c^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}} \]
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Time = 12.46 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {11\,A\,a\,\cos \left (e+f\,x\right )}{2}-\frac {B\,a}{4}-\frac {41\,A\,a}{4}+\frac {B\,a\,\cos \left (e+f\,x\right )}{2}+5\,A\,a\,\sin \left (e+f\,x\right )-5\,B\,a\,\sin \left (e+f\,x\right )+\frac {3\,A\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {3\,B\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {5\,A\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {5\,B\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{15\,c^3\,f\,\left (\frac {5\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {5\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}+\frac {\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}\right )} \]
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