Integrand size = 26, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {45}{128} a^3 c^5 x+\frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f} \]
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Time = 0.12 (sec) , antiderivative size = 145, 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, 2757, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {3 a^3 c^5 \sin (e+f x) \cos ^5(e+f x)}{16 f}+\frac {15 a^3 c^5 \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac {45 a^3 c^5 \sin (e+f x) \cos (e+f x)}{128 f}+\frac {45}{128} a^3 c^5 x \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx \\ & = \frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {1}{8} \left (9 a^3 c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {1}{8} \left (9 a^3 c^5\right ) \int \cos ^6(e+f x) \, dx \\ & = \frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {1}{16} \left (15 a^3 c^5\right ) \int \cos ^4(e+f x) \, dx \\ & = \frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {1}{64} \left (45 a^3 c^5\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f}+\frac {1}{128} \left (45 a^3 c^5\right ) \int 1 \, dx \\ & = \frac {45}{128} a^3 c^5 x+\frac {9 a^3 c^5 \cos ^7(e+f x)}{56 f}+\frac {45 a^3 c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {15 a^3 c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {3 a^3 c^5 \cos ^5(e+f x) \sin (e+f x)}{16 f}+\frac {a^3 \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{8 f} \\ \end{align*}
Time = 8.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {a^3 c^5 (2520 e+2520 f x+1120 \cos (e+f x)+672 \cos (3 (e+f x))+224 \cos (5 (e+f x))+32 \cos (7 (e+f x))+1792 \sin (2 (e+f x))+280 \sin (4 (e+f x))-7 \sin (8 (e+f x)))}{7168 f} \]
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Time = 3.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(-\frac {c^{5} a^{3} \left (-2520 f x +7 \sin \left (8 f x +8 e \right )-32 \cos \left (7 f x +7 e \right )-224 \cos \left (5 f x +5 e \right )-280 \sin \left (4 f x +4 e \right )-672 \cos \left (3 f x +3 e \right )-1792 \sin \left (2 f x +2 e \right )-1120 \cos \left (f x +e \right )-2048\right )}{7168 f}\) | \(92\) |
risch | \(\frac {45 a^{3} c^{5} x}{128}+\frac {5 c^{5} a^{3} \cos \left (f x +e \right )}{32 f}-\frac {c^{5} a^{3} \sin \left (8 f x +8 e \right )}{1024 f}+\frac {c^{5} a^{3} \cos \left (7 f x +7 e \right )}{224 f}+\frac {c^{5} a^{3} \cos \left (5 f x +5 e \right )}{32 f}+\frac {5 c^{5} a^{3} \sin \left (4 f x +4 e \right )}{128 f}+\frac {3 c^{5} a^{3} \cos \left (3 f x +3 e \right )}{32 f}+\frac {c^{5} a^{3} \sin \left (2 f x +2 e \right )}{4 f}\) | \(148\) |
derivativedivides | \(\frac {-c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 a^{3} c^{5} \cos \left (f x +e \right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(276\) |
default | \(\frac {-c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}+2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 a^{3} c^{5} \cos \left (f x +e \right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(276\) |
parts | \(a^{3} c^{5} x +\frac {2 c^{5} a^{3} \cos \left (f x +e \right )}{f}-\frac {2 c^{5} a^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 c^{5} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{f}+\frac {6 c^{5} a^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {2 c^{5} a^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}-\frac {2 c^{5} a^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7 f}-\frac {c^{5} a^{3} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )}{f}\) | \(289\) |
norman | \(\frac {\frac {4 c^{5} a^{3}}{7 f}+\frac {83 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}+\frac {45 a^{3} c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{3} c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {315 a^{3} c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {1575 a^{3} c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64}+\frac {315 a^{3} c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {315 a^{3} c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32}+\frac {45 a^{3} c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {45 a^{3} c^{5} x \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{128}+\frac {4 c^{5} a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 c^{5} a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 f}+\frac {12 c^{5} a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {12 c^{5} a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {20 c^{5} a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {20 c^{5} a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 c^{5} a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {295 c^{5} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {3 c^{5} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {815 c^{5} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {815 c^{5} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {3 c^{5} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {295 c^{5} a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {83 c^{5} a^{3} \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {45 a^{3} c^{5} x}{128}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{8}}\) | \(526\) |
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {256 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} + 315 \, a^{3} c^{5} f x - 7 \, {\left (16 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} - 24 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 30 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 45 \, a^{3} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{896 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (139) = 278\).
Time = 0.75 (sec) , antiderivative size = 740, normalized size of antiderivative = 5.10 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\begin {cases} - \frac {35 a^{3} c^{5} x \sin ^{8}{\left (e + f x \right )}}{128} - \frac {35 a^{3} c^{5} x \sin ^{6}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{32} + \frac {5 a^{3} c^{5} x \sin ^{6}{\left (e + f x \right )}}{8} - \frac {105 a^{3} c^{5} x \sin ^{4}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{64} + \frac {15 a^{3} c^{5} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{8} - \frac {35 a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{6}{\left (e + f x \right )}}{32} + \frac {15 a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{8} - a^{3} c^{5} x \sin ^{2}{\left (e + f x \right )} - \frac {35 a^{3} c^{5} x \cos ^{8}{\left (e + f x \right )}}{128} + \frac {5 a^{3} c^{5} x \cos ^{6}{\left (e + f x \right )}}{8} - a^{3} c^{5} x \cos ^{2}{\left (e + f x \right )} + a^{3} c^{5} x + \frac {93 a^{3} c^{5} \sin ^{7}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{128 f} - \frac {2 a^{3} c^{5} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {511 a^{3} c^{5} \sin ^{5}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{384 f} - \frac {11 a^{3} c^{5} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 a^{3} c^{5} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {6 a^{3} c^{5} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {385 a^{3} c^{5} \sin ^{3}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{384 f} - \frac {5 a^{3} c^{5} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {16 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {8 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {6 a^{3} c^{5} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {35 a^{3} c^{5} \sin {\left (e + f x \right )} \cos ^{7}{\left (e + f x \right )}}{128 f} - \frac {5 a^{3} c^{5} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{8 f} + \frac {a^{3} c^{5} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {32 a^{3} c^{5} \cos ^{7}{\left (e + f x \right )}}{35 f} + \frac {16 a^{3} c^{5} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {4 a^{3} c^{5} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {2 a^{3} c^{5} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{3} \left (- c \sin {\left (e \right )} + c\right )^{5} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (134) = 268\).
Time = 0.21 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.94 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {6144 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} + 43008 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} + 215040 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{5} - 35 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 1120 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} - 53760 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{5} + 107520 \, {\left (f x + e\right )} a^{3} c^{5} + 215040 \, a^{3} c^{5} \cos \left (f x + e\right )}{107520 \, f} \]
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Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {45}{128} \, a^{3} c^{5} x + \frac {a^{3} c^{5} \cos \left (7 \, f x + 7 \, e\right )}{224 \, f} + \frac {a^{3} c^{5} \cos \left (5 \, f x + 5 \, e\right )}{32 \, f} + \frac {3 \, a^{3} c^{5} \cos \left (3 \, f x + 3 \, e\right )}{32 \, f} + \frac {5 \, a^{3} c^{5} \cos \left (f x + e\right )}{32 \, f} - \frac {a^{3} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {5 \, a^{3} c^{5} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {a^{3} c^{5} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 9.00 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.57 \[ \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5 \, dx=\frac {a^3\,c^5\,\left (\frac {315\,e}{2}+581\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+256\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2065\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+5376\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+5376\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+5705\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+8960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-5705\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+8960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+1792\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-2065\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}+1792\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-581\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}+\frac {315\,f\,x}{2}+1260\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+4410\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+8820\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+11025\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+8820\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+4410\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )+1260\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (e+f\,x\right )+\frac {315\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}\,\left (e+f\,x\right )}{2}+256\right )}{448\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^8} \]
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