Integrand size = 26, antiderivative size = 161 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )} \]
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Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759, 2758, 2761, 8} \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {63 c^5 x}{2 a^3}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^3} \]
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Rule 8
Rule 2758
Rule 2759
Rule 2761
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^5 c^5\right ) \int \frac {\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^8} \, dx \\ & = -\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}-\frac {1}{5} \left (9 a^3 c^5\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx \\ & = -\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}+\frac {1}{5} \left (21 a c^5\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx \\ & = -\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {\left (21 c^5\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{a} \\ & = -\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a^2} \\ & = -\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int 1 \, dx}{2 a^3} \\ & = -\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )} \\ \end{align*}
Time = 16.79 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.88 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-896 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+448 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2304 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-160 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sin (2 (e+f x))\right )}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^3} \]
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Time = 1.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
default | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
risch | \(-\frac {63 c^{5} x}{2 a^{3}}-\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {4 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{a^{3} f}-\frac {4 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{a^{3} f}+\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} f}-\frac {32 \left (-105 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 i c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+25 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}-65 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+18 c^{5}\right )}{5 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(179\) |
parallelrisch | \(\frac {\left (\left (-2520 f x -4892\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (1260 f x +682\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (252 f x +\frac {5038}{5}\right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (-2520 f x -3044\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-1260 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+252 f x \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )-3286 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-214 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+27 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-\sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )+27 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right ) c^{5}}{8 f \,a^{3} \left (10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(236\) |
norman | \(\frac {-\frac {945 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}-\frac {431 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a}-\frac {63 c^{5} x}{2 a}-\frac {9765 c^{5} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {4095 c^{5} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2516 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6071 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {7915 c^{5} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {7736 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {5477 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {309 c^{5} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {496 c^{5}}{5 f a}-\frac {4095 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {945 c^{5} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {956 c^{5} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {1179 c^{5} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {4412 c^{5} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2036 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {65 c^{5} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {6363 c^{5} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {7640 c^{5} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {19741 c^{5} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {9765 c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(658\) |
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Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.77 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {5 \, c^{5} \cos \left (f x + e\right )^{5} + 70 \, c^{5} \cos \left (f x + e\right )^{4} - 1260 \, c^{5} f x - 64 \, c^{5} + 7 \, {\left (45 \, c^{5} f x + 113 \, c^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (945 \, c^{5} f x - 502 \, c^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (315 \, c^{5} f x + 646 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (5 \, c^{5} \cos \left (f x + e\right )^{4} - 65 \, c^{5} \cos \left (f x + e\right )^{3} + 1260 \, c^{5} f x - 64 \, c^{5} - 3 \, {\left (105 \, c^{5} f x - 242 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (315 \, c^{5} f x + 614 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{10 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3643 vs. \(2 (153) = 306\).
Time = 23.34 (sec) , antiderivative size = 3643, normalized size of antiderivative = 22.63 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1496 vs. \(2 (149) = 298\).
Time = 0.32 (sec) , antiderivative size = 1496, normalized size of antiderivative = 9.29 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Time = 0.36 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {10 \, {\left (c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 16 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 16 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} + \frac {64 \, {\left (10 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 85 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 55 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, c^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{10 \, f} \]
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Time = 11.55 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.26 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {\frac {63\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+4310\right )}{10}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+992\right )}{10}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+650\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+3090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+8814\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+7610\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+12230\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+11090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+14702\right )}{10}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2}-\frac {63\,c^5\,x}{2\,a^3} \]
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