Integrand size = 33, antiderivative size = 213 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{8} a \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right ) x-\frac {a \left (4 A d \left (c^2+3 c d+d^2\right )-B \left (c^3-4 c^2 d-8 c d^2-4 d^3\right )\right ) \cos (e+f x)}{6 d f}-\frac {a \left (3 (4 A+3 B) d^2-2 c (B c-4 (A+B) d)\right ) \cos (e+f x) \sin (e+f x)}{24 f}+\frac {a (B c-4 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f} \]
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Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a \left (-8 c d (A+B)-3 d^2 (4 A+3 B)+2 B c^2\right ) \sin (e+f x) \cos (e+f x)}{24 f}+\frac {1}{8} a x \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right )-\frac {a \left (4 A d \left (c^2+3 c d+d^2\right )-B \left (c^3-4 c^2 d-8 c d^2-4 d^3\right )\right ) \cos (e+f x)}{6 d f}+\frac {a (B c-4 d (A+B)) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f} \]
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Rule 2813
Rule 2832
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \int (c+d \sin (e+f x))^2 \left (a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}+\frac {\int (c+d \sin (e+f x))^2 (a (4 A+3 B) d-a (B c-4 (A+B) d) \sin (e+f x)) \, dx}{4 d} \\ & = \frac {a (B c-4 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f}+\frac {\int (c+d \sin (e+f x)) \left (a d (12 A c+7 B c+8 A d+8 B d)-a \left (2 B c^2-8 (A+B) c d-3 (4 A+3 B) d^2\right ) \sin (e+f x)\right ) \, dx}{12 d} \\ & = \frac {1}{8} a \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right ) x-\frac {a \left (4 A d \left (c^2+3 c d+d^2\right )-B \left (c^3-4 c^2 d-8 c d^2-4 d^3\right )\right ) \cos (e+f x)}{6 d f}+\frac {a \left (2 B c^2-8 (A+B) c d-3 (4 A+3 B) d^2\right ) \cos (e+f x) \sin (e+f x)}{24 f}+\frac {a (B c-4 (A+B) d) \cos (e+f x) (c+d \sin (e+f x))^2}{12 d f}-\frac {a B \cos (e+f x) (c+d \sin (e+f x))^3}{4 d f} \\ \end{align*}
Time = 1.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a (1+\sin (e+f x)) \left (-24 \left (B \left (4 c^2+6 c d+3 d^2\right )+A \left (4 c^2+8 c d+3 d^2\right )\right ) \cos (e+f x)+8 d (A d+B (2 c+d)) \cos (3 (e+f x))+3 \left (4 \left (4 A \left (2 c^2+2 c d+d^2\right )+B \left (4 c^2+8 c d+3 d^2\right )\right ) f x-8 \left (B (c+d)^2+A d (2 c+d)\right ) \sin (2 (e+f x))+B d^2 \sin (4 (e+f x))\right )\right )}{96 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
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Time = 1.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.81
method | result | size |
parts | \(-\frac {\left (A a \,d^{2}+2 B a c d +d^{2} B a \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {\left (a \,c^{2} A +2 A a c d +B a \,c^{2}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (2 A a c d +A a \,d^{2}+B a \,c^{2}+2 B a c d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+A a \,c^{2} x +\frac {d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(172\) |
parallelrisch | \(\frac {\left (\left (\left (-3 B -3 A \right ) d^{2}-6 d c \left (A +B \right )-3 B \,c^{2}\right ) \sin \left (2 f x +2 e \right )+\left (2 B c +\left (A +B \right ) d \right ) d \cos \left (3 f x +3 e \right )+\frac {3 d^{2} B \sin \left (4 f x +4 e \right )}{8}+\left (\left (-9 A -9 B \right ) d^{2}-24 c \left (A +\frac {3 B}{4}\right ) d -12 c^{2} \left (A +B \right )\right ) \cos \left (f x +e \right )+\left (-8 B +6 f x A +\frac {9}{2} f x B -8 A \right ) d^{2}+12 c \left (f x A +f x B -2 A -\frac {4}{3} B \right ) d +12 c^{2} \left (f x A +\frac {1}{2} f x B -A -B \right )\right ) a}{12 f}\) | \(176\) |
derivativedivides | \(\frac {-a \,c^{2} A \cos \left (f x +e \right )+2 A a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B a \,c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 B a c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a \,c^{2} A \left (f x +e \right )-2 A a c d \cos \left (f x +e \right )+A a \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} \cos \left (f x +e \right )+2 B a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{2} B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(274\) |
default | \(\frac {-a \,c^{2} A \cos \left (f x +e \right )+2 A a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A a \,d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B a \,c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 B a c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+d^{2} B a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+a \,c^{2} A \left (f x +e \right )-2 A a c d \cos \left (f x +e \right )+A a \,d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a \,c^{2} \cos \left (f x +e \right )+2 B a c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {d^{2} B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\) | \(274\) |
risch | \(A a \,c^{2} x +A a c d x +\frac {A a \,d^{2} x}{2}+\frac {B a \,c^{2} x}{2}+B a c d x +\frac {3 B a \,d^{2} x}{8}-\frac {a \cos \left (f x +e \right ) A \,c^{2}}{f}-\frac {2 a \cos \left (f x +e \right ) A c d}{f}-\frac {3 a \cos \left (f x +e \right ) A \,d^{2}}{4 f}-\frac {a \cos \left (f x +e \right ) B \,c^{2}}{f}-\frac {3 a \cos \left (f x +e \right ) c d B}{2 f}-\frac {3 a \cos \left (f x +e \right ) d^{2} B}{4 f}+\frac {d^{2} B a \sin \left (4 f x +4 e \right )}{32 f}+\frac {d^{2} a \cos \left (3 f x +3 e \right ) A}{12 f}+\frac {d a \cos \left (3 f x +3 e \right ) B c}{6 f}+\frac {d^{2} a \cos \left (3 f x +3 e \right ) B}{12 f}-\frac {\sin \left (2 f x +2 e \right ) A a c d}{2 f}-\frac {\sin \left (2 f x +2 e \right ) A a \,d^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a \,c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a c d}{2 f}-\frac {\sin \left (2 f x +2 e \right ) d^{2} B a}{4 f}\) | \(307\) |
norman | \(\frac {\left (a \,c^{2} A +A a c d +\frac {1}{2} A a \,d^{2}+\frac {1}{2} B a \,c^{2}+B a c d +\frac {3}{8} d^{2} B a \right ) x +\left (a \,c^{2} A +A a c d +\frac {1}{2} A a \,d^{2}+\frac {1}{2} B a \,c^{2}+B a c d +\frac {3}{8} d^{2} B a \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (4 a \,c^{2} A +4 A a c d +2 A a \,d^{2}+2 B a \,c^{2}+4 B a c d +\frac {3}{2} d^{2} B a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (4 a \,c^{2} A +4 A a c d +2 A a \,d^{2}+2 B a \,c^{2}+4 B a c d +\frac {3}{2} d^{2} B a \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (6 a \,c^{2} A +6 A a c d +3 A a \,d^{2}+3 B a \,c^{2}+6 B a c d +\frac {9}{4} d^{2} B a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {6 a \,c^{2} A +12 A a c d +4 A a \,d^{2}+6 B a \,c^{2}+8 B a c d +4 d^{2} B a}{3 f}-\frac {2 \left (a \,c^{2} A +2 A a c d +B a \,c^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (3 a \,c^{2} A +6 A a c d +2 A a \,d^{2}+3 B a \,c^{2}+4 B a c d +2 d^{2} B a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (9 a \,c^{2} A +18 A a c d +8 A a \,d^{2}+9 B a \,c^{2}+16 B a c d +8 d^{2} B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +3 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +3 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +11 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a \left (8 A c d +4 A \,d^{2}+4 B \,c^{2}+8 c d B +11 d^{2} B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(648\) |
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Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.75 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {8 \, {\left (2 \, B a c d + {\left (A + B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, {\left (2 \, A + B\right )} a c^{2} + 8 \, {\left (A + B\right )} a c d + {\left (4 \, A + 3 \, B\right )} a d^{2}\right )} f x - 24 \, {\left ({\left (A + B\right )} a c^{2} + 2 \, {\left (A + B\right )} a c d + {\left (A + B\right )} a d^{2}\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, B a d^{2} \cos \left (f x + e\right )^{3} - {\left (4 \, B a c^{2} + 8 \, {\left (A + B\right )} a c d + {\left (4 \, A + 5 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (201) = 402\).
Time = 0.23 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.68 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\begin {cases} A a c^{2} x - \frac {A a c^{2} \cos {\left (e + f x \right )}}{f} + A a c d x \sin ^{2}{\left (e + f x \right )} + A a c d x \cos ^{2}{\left (e + f x \right )} - \frac {A a c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 A a c d \cos {\left (e + f x \right )}}{f} + \frac {A a d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {A a d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {A a d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {A a d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {B a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {B a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {B a c^{2} \cos {\left (e + f x \right )}}{f} + B a c d x \sin ^{2}{\left (e + f x \right )} + B a c d x \cos ^{2}{\left (e + f x \right )} - \frac {2 B a c d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a c d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 B a c d \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 B a d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 B a d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {5 B a d^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {B a d^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a d^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {2 B a d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (c + d \sin {\left (e \right )}\right )^{2} \left (a \sin {\left (e \right )} + a\right ) & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.24 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {96 \, {\left (f x + e\right )} A a c^{2} + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c d + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c d + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c d + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a d^{2} + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a d^{2} + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a d^{2} + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a d^{2} - 96 \, A a c^{2} \cos \left (f x + e\right ) - 96 \, B a c^{2} \cos \left (f x + e\right ) - 192 \, A a c d \cos \left (f x + e\right )}{96 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {B a d^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{8} \, {\left (8 \, A a c^{2} + 4 \, B a c^{2} + 8 \, A a c d + 8 \, B a c d + 4 \, A a d^{2} + 3 \, B a d^{2}\right )} x + \frac {{\left (2 \, B a c d + A a d^{2} + B a d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (4 \, A a c^{2} + 4 \, B a c^{2} + 8 \, A a c d + 6 \, B a c d + 3 \, A a d^{2} + 3 \, B a d^{2}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (B a c^{2} + 2 \, A a c d + 2 \, B a c d + A a d^{2} + B a d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
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Time = 15.70 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.57 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A\,c^2+4\,A\,d^2+4\,B\,c^2+3\,B\,d^2+8\,A\,c\,d+8\,B\,c\,d\right )}{4\,\left (2\,A\,a\,c^2+A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )}\right )\,\left (8\,A\,c^2+4\,A\,d^2+4\,B\,c^2+3\,B\,d^2+8\,A\,c\,d+8\,B\,c\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a\,c^2+2\,B\,a\,c^2+4\,A\,a\,c\,d\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,A\,a\,c^2+4\,A\,a\,d^2+6\,B\,a\,c^2+4\,B\,a\,d^2+12\,A\,a\,c\,d+8\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,A\,a\,c^2+\frac {16\,A\,a\,d^2}{3}+6\,B\,a\,c^2+\frac {16\,B\,a\,d^2}{3}+12\,A\,a\,c\,d+\frac {32\,B\,a\,c\,d}{3}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a\,d^2+B\,a\,c^2+\frac {3\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a\,d^2+B\,a\,c^2+\frac {11\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a\,d^2+B\,a\,c^2+\frac {11\,B\,a\,d^2}{4}+2\,A\,a\,c\,d+2\,B\,a\,c\,d\right )+2\,A\,a\,c^2+\frac {4\,A\,a\,d^2}{3}+2\,B\,a\,c^2+\frac {4\,B\,a\,d^2}{3}+4\,A\,a\,c\,d+\frac {8\,B\,a\,c\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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