Integrand size = 34, antiderivative size = 142 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}-\frac {a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^2}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^4-c^4 \sin (e+f x)\right )} \]
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Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3046, 2936, 2829, 2729, 2727} \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^4-c^4 \sin (e+f x)\right )}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^2}-\frac {a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}+\frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4} \]
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Rule 2727
Rule 2729
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))^5} \, dx \\ & = \frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}+\frac {a \int \frac {-A c-8 B c-7 B c \sin (e+f x)}{(c-c \sin (e+f x))^3} \, dx}{7 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}-\frac {a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}-\frac {(a (2 A-5 B)) \int \frac {1}{(c-c \sin (e+f x))^2} \, dx}{35 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}-\frac {a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^2}-\frac {(a (2 A-5 B)) \int \frac {1}{c-c \sin (e+f x)} \, dx}{105 c^3} \\ & = \frac {2 a (A+B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}-\frac {a (A+15 B) \cos (e+f x)}{35 c f (c-c \sin (e+f x))^3}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^2}-\frac {a (2 A-5 B) \cos (e+f x)}{105 f \left (c^4-c^4 \sin (e+f x)\right )} \\ \end{align*}
Time = 6.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.23 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a \left (35 (4 A-B) \cos \left (e+\frac {f x}{2}\right )-42 A \cos \left (e+\frac {3 f x}{2}\right )+2 A \cos \left (3 e+\frac {7 f x}{2}\right )-5 B \cos \left (3 e+\frac {7 f x}{2}\right )+70 A \sin \left (\frac {f x}{2}\right )+140 B \sin \left (\frac {f x}{2}\right )+105 B \sin \left (2 e+\frac {3 f x}{2}\right )+14 A \sin \left (2 e+\frac {5 f x}{2}\right )-35 B \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{420 c^4 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
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Time = 0.91 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(-\frac {2 \left (A \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (B -2 A \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (13 A -B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {2 \left (-5 A +2 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {13 A \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {\left (-\frac {8 A}{5}+B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3}+\frac {23 A}{105}-\frac {B}{21}\right ) a}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(130\) |
risch | \(-\frac {2 i a \left (140 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-35 i B \,{\mathrm e}^{4 i \left (f x +e \right )}+105 B \,{\mathrm e}^{5 i \left (f x +e \right )}-42 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-70 A \,{\mathrm e}^{3 i \left (f x +e \right )}-140 B \,{\mathrm e}^{3 i \left (f x +e \right )}+2 i A -14 A \,{\mathrm e}^{i \left (f x +e \right )}-5 i B +35 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{105 f \,c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7}}\) | \(133\) |
derivativedivides | \(\frac {2 a \left (-\frac {68 A +60 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {48 A +48 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {16 A +16 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {56 A +40 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {28 A +14 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\) | \(159\) |
default | \(\frac {2 a \left (-\frac {68 A +60 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {48 A +48 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {16 A +16 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {56 A +40 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {28 A +14 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\) | \(159\) |
norman | \(\frac {\frac {\left (4 a A -2 B a \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {46 a A -10 B a}{105 c f}-\frac {2 a A \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {\left (16 a A -10 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 c f}+\frac {2 \left (22 a A -10 B a \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {\left (38 a A -2 B a \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {\left (44 a A -20 B a \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}+\frac {2 \left (46 a A -20 B a \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {2 \left (184 a A -10 B a \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {\left (638 a A -20 B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {2 \left (1024 a A -40 B a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(321\) |
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Time = 0.25 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.77 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {{\left (2 \, A - 5 \, B\right )} a \cos \left (f x + e\right )^{4} + 4 \, {\left (2 \, A - 5 \, B\right )} a \cos \left (f x + e\right )^{3} - 3 \, {\left (3 \, A + 10 \, B\right )} a \cos \left (f x + e\right )^{2} + 15 \, {\left (A + B\right )} a \cos \left (f x + e\right ) + 30 \, {\left (A + B\right )} a - {\left ({\left (2 \, A - 5 \, B\right )} a \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, A - 5 \, B\right )} a \cos \left (f x + e\right )^{2} - 15 \, {\left (A + B\right )} a \cos \left (f x + e\right ) - 30 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1831 vs. \(2 (124) = 248\).
Time = 8.62 (sec) , antiderivative size = 1831, normalized size of antiderivative = 12.89 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1080 vs. \(2 (138) = 276\).
Time = 0.24 (sec) , antiderivative size = 1080, normalized size of antiderivative = 7.61 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Time = 0.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.24 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (105 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 210 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 105 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 455 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 350 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 140 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 273 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 56 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 35 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 23 \, A a - 5 \, B a\right )}}{105 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \]
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Time = 12.49 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.61 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {15\,B\,a}{4}-\frac {171\,A\,a}{2}+\frac {353\,A\,a\,\cos \left (e+f\,x\right )}{8}+\frac {5\,B\,a\,\cos \left (e+f\,x\right )}{4}+\frac {595\,A\,a\,\sin \left (e+f\,x\right )}{8}-35\,B\,a\,\sin \left (e+f\,x\right )+\frac {43\,A\,a\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {25\,A\,a\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {5\,B\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {5\,B\,a\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {77\,A\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {21\,A\,a\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {35\,B\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{105\,c^4\,f\,\left (\frac {35\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{8}-\frac {21\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{8}-\frac {7\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{8}+\frac {\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{8}\right )} \]
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