Integrand size = 31, antiderivative size = 140 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{21 d}+\frac {5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac {10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac {a^3 (2 A-B) \tan ^5(c+d x)}{21 d} \]
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Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2934, 2755, 3852} \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (2 A-B) \tan ^5(c+d x)}{21 d}+\frac {10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac {5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{21 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^3}{9 d} \]
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Rule 2755
Rule 2934
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {1}{3} (a (2 A-B)) \int \sec ^8(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{21 d}+\frac {1}{21} \left (5 a^3 (2 A-B)\right ) \int \sec ^6(c+d x) \, dx \\ & = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{21 d}-\frac {\left (5 a^3 (2 A-B)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{21 d} \\ & = \frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^3}{9 d}+\frac {2 (2 A-B) \sec ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{21 d}+\frac {5 a^3 (2 A-B) \tan (c+d x)}{21 d}+\frac {10 a^3 (2 A-B) \tan ^3(c+d x)}{63 d}+\frac {a^3 (2 A-B) \tan ^5(c+d x)}{21 d} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 (-42 B+27 (-2 A+B) \cos (2 (c+d x))+12 (-2 A+B) \cos (4 (c+d x))+2 A \cos (6 (c+d x))-B \cos (6 (c+d x))-72 A \sin (c+d x)+36 B \sin (c+d x)-4 A \sin (3 (c+d x))+2 B \sin (3 (c+d x))+12 A \sin (5 (c+d x))-6 B \sin (5 (c+d x)))}{252 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^9 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {16 i a^{3} \left (72 i A \,{\mathrm e}^{5 i \left (d x +c \right )}-36 i B \,{\mathrm e}^{5 i \left (d x +c \right )}+42 B \,{\mathrm e}^{6 i \left (d x +c \right )}+4 i A \,{\mathrm e}^{3 i \left (d x +c \right )}+54 A \,{\mathrm e}^{4 i \left (d x +c \right )}-2 i B \,{\mathrm e}^{3 i \left (d x +c \right )}-27 B \,{\mathrm e}^{4 i \left (d x +c \right )}-12 i A \,{\mathrm e}^{i \left (d x +c \right )}+24 A \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i B \,{\mathrm e}^{i \left (d x +c \right )}-12 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2 A +B \right )}{63 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{9} d}\) | \(181\) |
parallelrisch | \(-\frac {2 \left (A \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13 A}{3}-2 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (A +3 B \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 \left (7 A -B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (17 A +2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (-25 A +9 B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (235 A -128 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63}+\frac {\left (13 A +25 B \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}+\frac {\left (-17 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{21}+\frac {19 A}{63}+\frac {B}{63}\right ) a^{3}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{9}}\) | \(240\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+\frac {A \,a^{3}}{3 \cos \left (d x +c \right )^{9}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(535\) |
default | \(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+\frac {A \,a^{3}}{3 \cos \left (d x +c \right )^{9}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-A \,a^{3} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {B \,a^{3}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(535\) |
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Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.34 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {8 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{6} - 36 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{2} + 7 \, {\left (A - 2 \, B\right )} a^{3} + {\left (24 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, A - B\right )} a^{3} \cos \left (d x + c\right )^{2} - 7 \, {\left (2 \, A - B\right )} a^{3}\right )} \sin \left (d x + c\right )}{63 \, {\left (3 \, d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3} - {\left (d \cos \left (d x + c\right )^{5} - 4 \, d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (130) = 260\).
Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.93 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} A a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a^{3} + {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} B a^{3} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} A a^{3}}{\cos \left (d x + c\right )^{9}} - \frac {15 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} B a^{3}}{\cos \left (d x + c\right )^{9}} + \frac {105 \, A a^{3}}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a^{3}}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.81 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {\frac {21 \, {\left (21 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19 \, A a^{3} - 13 \, B a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} + \frac {3591 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 315 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 19656 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 756 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 56196 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4200 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 95760 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 11340 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 107730 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 14994 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 79464 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13356 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 38484 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6768 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10944 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2196 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1615 \, A a^{3} - 209 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{9}}}{2016 \, d} \]
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Time = 13.84 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.30 \[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {63\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {171\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {145\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {49\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {21\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {75\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {21\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {41\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {7\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}-\frac {617\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {329\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {145\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {113\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {115\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {19\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {109\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {35\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {43\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {59\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {47\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{2016\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^9} \]
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