Integrand size = 27, antiderivative size = 144 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}+\frac {4 \cos (e+f x)}{315 f \left (a^6+a^6 \sin (e+f x)\right )} \]
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Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2936, 2829, 2729, 2727} \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4 \cos (e+f x)}{315 f \left (a^6 \sin (e+f x)+a^6\right )}+\frac {4 \cos (e+f x)}{315 f \left (a^3 \sin (e+f x)+a^3\right )^2}+\frac {2 \cos (e+f x)}{105 f \left (a^2 \sin (e+f x)+a^2\right )^3}-\frac {19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5} \]
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Rule 2727
Rule 2729
Rule 2829
Rule 2936
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {\int \frac {-10 a+9 a \sin (e+f x)}{(a+a \sin (e+f x))^4} \, dx}{9 a^3} \\ & = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}-\frac {2 \int \frac {1}{(a+a \sin (e+f x))^3} \, dx}{21 a^3} \\ & = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}-\frac {4 \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{105 a^4} \\ & = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}-\frac {4 \int \frac {1}{a+a \sin (e+f x)} \, dx}{315 a^5} \\ & = \frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}+\frac {4 \cos (e+f x)}{315 f \left (a^6+a^6 \sin (e+f x)\right )} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {378 \cos \left (e+\frac {f x}{2}\right )+210 \cos \left (e+\frac {3 f x}{2}\right )-108 \cos \left (3 e+\frac {5 f x}{2}\right )+225 \cos \left (3 e+\frac {7 f x}{2}\right )+3 \cos \left (5 e+\frac {9 f x}{2}\right )+3150 \sin \left (\frac {f x}{2}\right )+2562 \sin \left (2 e+\frac {3 f x}{2}\right )-900 \sin \left (2 e+\frac {5 f x}{2}\right )-27 \sin \left (4 e+\frac {7 f x}{2}\right )+25 \sin \left (4 e+\frac {9 f x}{2}\right )}{13860 a^6 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
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Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {8 \left (105 \,{\mathrm e}^{6 i \left (f x +e \right )}+21 i {\mathrm e}^{3 i \left (f x +e \right )}-126 \,{\mathrm e}^{4 i \left (f x +e \right )}+9 i {\mathrm e}^{i \left (f x +e \right )}+36 \,{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{315 f \,a^{6} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{9}}\) | \(82\) |
parallelrisch | \(\frac {-\frac {22}{315}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {18 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}-6 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {58 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {14 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}}{f \,a^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}\) | \(113\) |
derivativedivides | \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{6}}\) | \(130\) |
default | \(\frac {-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{6}}\) | \(130\) |
norman | \(\frac {-\frac {22}{315 a f}-\frac {2 \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6 \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {18 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {242 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{315 f a}-\frac {174 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {274 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 f a}-\frac {646 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {862 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {1202 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 f a}-\frac {1762 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {8342 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {9442 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 f a}-\frac {17494 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {23858 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}-\frac {28786 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{11}}\) | \(325\) |
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Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4 \, \cos \left (f x + e\right )^{5} - 16 \, \cos \left (f x + e\right )^{4} - 50 \, \cos \left (f x + e\right )^{3} - 65 \, \cos \left (f x + e\right )^{2} - {\left (4 \, \cos \left (f x + e\right )^{4} + 20 \, \cos \left (f x + e\right )^{3} - 30 \, \cos \left (f x + e\right )^{2} + 35 \, \cos \left (f x + e\right ) + 70\right )} \sin \left (f x + e\right ) + 35 \, \cos \left (f x + e\right ) + 70}{315 \, {\left (a^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{6} f \cos \left (f x + e\right )^{4} - 8 \, a^{6} f \cos \left (f x + e\right )^{3} - 20 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f + {\left (a^{6} f \cos \left (f x + e\right )^{4} - 4 \, a^{6} f \cos \left (f x + e\right )^{3} - 12 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (131) = 262\).
Time = 43.59 (sec) , antiderivative size = 1501, normalized size of antiderivative = 10.42 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (134) = 268\).
Time = 0.23 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (\frac {99 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {81 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {609 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {945 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {315 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 11\right )}}{315 \, {\left (a^{6} + \frac {9 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{6} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{6} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \]
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Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 945 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 609 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 81 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 99 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11\right )}}{315 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \]
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Time = 9.48 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+99\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+81\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+609\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+315\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{315\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \]
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