Integrand size = 26, antiderivative size = 124 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2} \]
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Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2759, 2761, 8} \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}-\frac {7 c^4 x}{a^3}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a \sin (e+f x)+a)^2} \]
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Rule 8
Rule 2759
Rule 2761
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^4\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^7} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}-\frac {1}{5} \left (7 a^2 c^4\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^5} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}+\frac {1}{3} \left (7 c^4\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^3} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a^2} \\ & = -\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2}-\frac {\left (7 c^4\right ) \int 1 \, dx}{a^3} \\ & = -\frac {7 c^4 x}{a^3}-\frac {7 c^4 \cos (e+f x)}{a^3 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {14 a c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {14 c^4 \cos ^3(e+f x)}{3 a f (a+a \sin (e+f x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(124)=248\).
Time = 12.48 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.18 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (96 \sin \left (\frac {1}{2} (e+f x)\right )-48 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-256 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+464 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-105 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-15 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5\right ) (c-c \sin (e+f x))^4}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^3} \]
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Time = 1.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {2 c^{4} \left (-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(102\) |
default | \(\frac {2 c^{4} \left (-\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-7 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {8}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(102\) |
risch | \(-\frac {7 c^{4} x}{a^{3}}-\frac {c^{4} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}-\frac {c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}-\frac {16 \left (120 i c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+45 \,{\mathrm e}^{4 i \left (f x +e \right )} c^{4}-100 i c^{4} {\mathrm e}^{i \left (f x +e \right )}-170 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+29 c^{4}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(137\) |
parallelrisch | \(-\frac {c^{4} \left (2100 f x \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+2100 f x \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-210 f x \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+1050 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-210 f x \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-1050 f x \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+2465 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+4215 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-15 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+205 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+2685 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-873 \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-15 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-655 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}{30 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(248\) |
norman | \(\frac {-\frac {98 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {35 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}-\frac {286 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {334 c^{4}}{15 f a}-\frac {497 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {588 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {588 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {1366 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {860 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {16036 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {2404 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {1102 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {638 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {66 c^{4} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {16 c^{4} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {210 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {357 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {497 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {98 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {10814 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {357 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {210 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {35 c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {7 c^{4} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {3686 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {11524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f a}-\frac {7 c^{4} x}{a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(574\) |
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Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (118) = 236\).
Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.06 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {15 \, c^{4} \cos \left (f x + e\right )^{4} - 420 \, c^{4} f x - 48 \, c^{4} + {\left (105 \, c^{4} f x + 277 \, c^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (315 \, c^{4} f x - 134 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 74 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (15 \, c^{4} \cos \left (f x + e\right )^{3} - 420 \, c^{4} f x + 48 \, c^{4} + {\left (105 \, c^{4} f x - 262 \, c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (35 \, c^{4} f x + 66 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2314 vs. \(2 (119) = 238\).
Time = 13.82 (sec) , antiderivative size = 2314, normalized size of antiderivative = 18.66 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (118) = 236\).
Time = 0.33 (sec) , antiderivative size = 1096, normalized size of antiderivative = 8.84 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{3}} + \frac {30 \, c^{4}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} + \frac {16 \, {\left (15 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 60 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 130 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 80 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19 \, c^{4}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \]
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Time = 10.57 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.34 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx=\frac {7\,c^4\,\left (e+f\,x\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+1430\right )}{15}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+334\right )}{15}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (35\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (525\,e+525\,f\,x+240\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+990\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (77\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1155\,e+1155\,f\,x+2684\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2470\right )}{15}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (105\,c^4\,\left (e+f\,x\right )-\frac {c^4\,\left (1575\,e+1575\,f\,x+2540\right )}{15}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {7\,c^4\,x}{a^3} \]
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