Integrand size = 26, antiderivative size = 34 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]
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Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2815, 2750} \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]
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Rule 2750
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).
Time = 2.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 \left (35 \cos \left (\frac {1}{2} (e+f x)\right )-21 \cos \left (\frac {3}{2} (e+f x)\right )-7 \cos \left (\frac {5}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{28 c^4 f (-1+\sin (e+f x))^4} \]
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Time = 0.97 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.88
method | result | size |
parallelrisch | \(-\frac {2 a^{3} \left (\frac {1}{7}+\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )+5 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(64\) |
risch | \(-\frac {2 \left (7 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-35 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+21 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-a^{3}\right )}{7 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(71\) |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\) | \(118\) |
default | \(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\) | \(118\) |
norman | \(\frac {-\frac {2 a^{3}}{7 c f}-\frac {48 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {202 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {352 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {42 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {16 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(177\) |
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (33) = 66\).
Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.53 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} - {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )}{7 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (29) = 58\).
Time = 14.26 (sec) , antiderivative size = 619, normalized size of antiderivative = 18.21 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\begin {cases} - \frac {14 a^{3} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {70 a^{3} \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {42 a^{3} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {2 a^{3}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )^{3}}{\left (- c \sin {\left (e \right )} + c\right )^{4}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (33) = 66\).
Time = 0.22 (sec) , antiderivative size = 1045, normalized size of antiderivative = 30.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (33) = 66\).
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.15 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (7 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}\right )}}{7 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \]
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Time = 6.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {2\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+21\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{7\,c^4\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \]
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