Integrand size = 27, antiderivative size = 58 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a+a \sin (e+f x))^5} \]
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Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2938, 2750} \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {\cos ^5(e+f x)}{7 f (a \sin (e+f x)+a)^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a \sin (e+f x)+a)^5} \]
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Rule 2750
Rule 2938
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}+\frac {6 \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^5} \, dx}{7 a} \\ & = \frac {\cos ^5(e+f x)}{7 f (a+a \sin (e+f x))^6}-\frac {6 \cos ^5(e+f x)}{35 a f (a+a \sin (e+f x))^5} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(58)=116\).
Time = 1.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {4585 \cos \left (e+\frac {f x}{2}\right )-2982 \cos \left (e+\frac {3 f x}{2}\right )-1148 \cos \left (3 e+\frac {5 f x}{2}\right )+197 \cos \left (3 e+\frac {7 f x}{2}\right )+2275 \sin \left (\frac {f x}{2}\right )+1134 \sin \left (2 e+\frac {3 f x}{2}\right )-224 \sin \left (2 e+\frac {5 f x}{2}\right )+\sin \left (4 e+\frac {7 f x}{2}\right )}{4620 a^6 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
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Time = 0.53 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.50
method | result | size |
parallelrisch | \(\frac {-\frac {2}{35}-2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {4 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}}{f \,a^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}\) | \(87\) |
risch | \(-\frac {2 \left (35 i {\mathrm e}^{5 i \left (f x +e \right )}+35 \,{\mathrm e}^{6 i \left (f x +e \right )}-70 i {\mathrm e}^{3 i \left (f x +e \right )}-140 \,{\mathrm e}^{4 i \left (f x +e \right )}+7 i {\mathrm e}^{i \left (f x +e \right )}+91 \,{\mathrm e}^{2 i \left (f x +e \right )}-6\right )}{35 f \,a^{6} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{7}}\) | \(94\) |
derivativedivides | \(\frac {\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {224}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{6}}\) | \(100\) |
default | \(\frac {\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {224}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{f \,a^{6}}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.36 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\frac {6 \, \cos \left (f x + e\right )^{4} - 11 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )^{2} + {\left (6 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} - 10 \, \cos \left (f x + e\right ) - 20\right )} \sin \left (f x + e\right ) + 10 \, \cos \left (f x + e\right ) + 20}{35 \, {\left (a^{6} f \cos \left (f x + e\right )^{4} - 3 \, a^{6} f \cos \left (f x + e\right )^{3} - 8 \, a^{6} f \cos \left (f x + e\right )^{2} + 4 \, a^{6} f \cos \left (f x + e\right ) + 8 \, a^{6} f - {\left (a^{6} f \cos \left (f x + e\right )^{3} + 4 \, a^{6} f \cos \left (f x + e\right )^{2} - 4 \, a^{6} f \cos \left (f x + e\right ) - 8 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (49) = 98\).
Time = 93.37 (sec) , antiderivative size = 901, normalized size of antiderivative = 15.53 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (54) = 108\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.64 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (\frac {7 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {14 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {70 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {35 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + 1\right )}}{35 \, {\left (a^{6} + \frac {7 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {21 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} f} \]
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Time = 0.49 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 70 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 14 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{35 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{7}} \]
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Time = 9.86 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.71 \[ \int \frac {\cos ^4(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx=-\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-14\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+70\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-35\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+35\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}{35\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \]
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