Integrand size = 26, antiderivative size = 135 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {35 c^4 x}{2 a^2}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )} \]
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Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759, 2758, 2761, 8} \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a \sin (e+f x)+a)^5}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac {35 c^4 x}{2 a^2}+\frac {14 a^4 c^4 \cos ^5(e+f x)}{3 f \left (a^2 \sin (e+f x)+a^2\right )^3} \]
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Rule 8
Rule 2758
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))^6} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}-\frac {1}{3} \left (7 a^2 c^4\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {1}{3} \left (35 c^4\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx \\ & = -\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {\left (35 c^4\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a} \\ & = \frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )}+\frac {\left (35 c^4\right ) \int 1 \, dx}{2 a^2} \\ & = \frac {35 c^4 x}{2 a^2}+\frac {35 c^4 \cos (e+f x)}{2 a^2 f}-\frac {2 a^3 c^4 \cos ^7(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {14 a c^4 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^3}+\frac {35 c^4 \cos ^3(e+f x)}{6 f \left (a^2+a^2 \sin (e+f x)\right )} \\ \end{align*}
Time = 11.89 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.80 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^4 \left (128 \sin \left (\frac {1}{2} (e+f x)\right )-64 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-640 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+210 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+72 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (a+a \sin (e+f x))^2} \]
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Time = 0.86 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 c^{4} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(125\) |
default | \(\frac {2 c^{4} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+6 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+6}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {35 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {32}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(125\) |
risch | \(\frac {35 c^{4} x}{2 a^{2}}+\frac {i c^{4} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {3 c^{4} {\mathrm e}^{i \left (f x +e \right )}}{a^{2} f}+\frac {3 c^{4} {\mathrm e}^{-i \left (f x +e \right )}}{a^{2} f}-\frac {i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {96 i c^{4} {\mathrm e}^{i \left (f x +e \right )}+64 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {160 c^{4}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(150\) |
parallelrisch | \(\frac {c^{4} \left (1260 f x \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+1260 f x \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-420 f x \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+420 f x \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+1443 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+2493 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-63 \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+3 \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+191 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+63 \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+1503 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}{24 f \,a^{2} \left (3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) | \(198\) |
norman | \(\frac {\frac {111 c^{4} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {131 c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a}+\frac {500 c^{4} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {35 c^{4} x}{2 a}+\frac {164 c^{4}}{3 f a}+\frac {105 c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {455 c^{4} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {315 c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {385 c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {315 c^{4} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {455 c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {245 c^{4} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {105 c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {35 c^{4} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {33 c^{4} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {460 c^{4} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {524 c^{4} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {718 c^{4} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {632 c^{4} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}+\frac {1454 c^{4} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}+\frac {815 c^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(490\) |
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Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.56 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=-\frac {3 \, c^{4} \cos \left (f x + e\right )^{4} - 30 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x - 32 \, c^{4} - {\left (105 \, c^{4} f x - 193 \, c^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (105 \, c^{4} f x + 194 \, c^{4}\right )} \cos \left (f x + e\right ) + {\left (3 \, c^{4} \cos \left (f x + e\right )^{3} + 210 \, c^{4} f x + 33 \, c^{4} \cos \left (f x + e\right )^{2} + 32 \, c^{4} + {\left (105 \, c^{4} f x + 226 \, c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2312 vs. \(2 (128) = 256\).
Time = 7.28 (sec) , antiderivative size = 2312, normalized size of antiderivative = 17.13 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (125) = 250\).
Time = 0.30 (sec) , antiderivative size = 903, normalized size of antiderivative = 6.69 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {105 \, {\left (f x + e\right )} c^{4}}{a^{2}} + \frac {6 \, {\left (c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} + \frac {64 \, {\left (3 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, c^{4}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \]
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Time = 10.67 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.16 \[ \int \frac {(c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {35\,c^4\,x}{2\,a^2}-\frac {\frac {35\,c^4\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+786\right )}{6}\right )-\frac {c^4\,\left (105\,e+105\,f\,x+328\right )}{6}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {105\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (315\,e+315\,f\,x+198\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+666\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {175\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (525\,e+525\,f\,x+974\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+868\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {245\,c^4\,\left (e+f\,x\right )}{2}-\frac {c^4\,\left (735\,e+735\,f\,x+1428\right )}{6}\right )}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2} \]
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