Integrand size = 26, antiderivative size = 59 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {\tan (e+f x)}{a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan ^5(e+f x)}{5 a^3 c^3 f} \]
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Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2815, 3852} \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {\tan ^5(e+f x)}{5 a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan (e+f x)}{a^3 c^3 f} \]
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Rule 2815
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(e+f x) \, dx}{a^3 c^3} \\ & = -\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{a^3 c^3 f} \\ & = \frac {\tan (e+f x)}{a^3 c^3 f}+\frac {2 \tan ^3(e+f x)}{3 a^3 c^3 f}+\frac {\tan ^5(e+f x)}{5 a^3 c^3 f} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {\tan (e+f x)+\frac {2}{3} \tan ^3(e+f x)+\frac {1}{5} \tan ^5(e+f x)}{a^3 c^3 f} \]
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Time = 1.63 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68
method | result | size |
default | \(-\frac {\left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )}{a^{3} c^{3} f}\) | \(40\) |
risch | \(\frac {16 i \left (10 \,{\mathrm e}^{4 i \left (f x +e \right )}+5 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{15 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,a^{3} c^{3}}\) | \(65\) |
parallelrisch | \(\frac {\frac {8 \sin \left (3 f x +3 e \right )}{3}+\frac {16 \sin \left (f x +e \right )}{3}+\frac {8 \sin \left (5 f x +5 e \right )}{15}}{a^{3} c^{3} f \left (\cos \left (5 f x +5 e \right )+5 \cos \left (3 f x +3 e \right )+10 \cos \left (f x +e \right )\right )}\) | \(72\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a c f}+\frac {8 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {116 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {8 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(143\) |
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Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {{\left (8 \, \cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sin \left (f x + e\right )}{15 \, a^{3} c^{3} f \cos \left (f x + e\right )^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (53) = 106\).
Time = 3.75 (sec) , antiderivative size = 687, normalized size of antiderivative = 11.64 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\begin {cases} - \frac {30 \tan ^{9}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} + \frac {40 \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} - \frac {116 \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} + \frac {40 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} f} - \frac {30 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} c^{3} f \tan ^{10}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 75 a^{3} c^{3} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} c^{3} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 150 a^{3} c^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} c^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 15 a^{3} c^{3} 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 )^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )}{15 \, a^{3} c^{3} f} \]
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Time = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=\frac {3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )}{15 \, a^{3} c^{3} f} \]
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Time = 7.96 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^3} \, dx=-\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+58\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+15\right )}{15\,a^3\,c^3\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^5} \]
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