Integrand size = 33, antiderivative size = 216 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(23 A+6 C) x}{2 a^3}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d} \]
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Time = 0.80 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4170, 4105, 3872, 2713, 2715, 8} \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {(23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {x (23 A+6 C)}{2 a^3}-\frac {(13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4105
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) (-a (8 A+3 C)+5 a A \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos ^3(c+d x) \left (-9 a^2 (7 A+2 C)+4 a^2 (13 A+3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \cos ^3(c+d x) \left (-12 a^3 (34 A+9 C)+15 a^3 (23 A+6 C) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A+6 C) \int \cos ^2(c+d x) \, dx}{a^3}+\frac {(4 (34 A+9 C)) \int \cos ^3(c+d x) \, dx}{5 a^3} \\ & = -\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(23 A+6 C) \int 1 \, dx}{2 a^3}-\frac {(4 (34 A+9 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d} \\ & = -\frac {(23 A+6 C) x}{2 a^3}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(216)=432\).
Time = 5.06 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (600 (23 A+6 C) d x \cos \left (\frac {d x}{2}\right )+600 (23 A+6 C) d x \cos \left (c+\frac {d x}{2}\right )+6900 A d x \cos \left (c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (c+\frac {3 d x}{2}\right )+6900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+1380 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+1380 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-20410 A \sin \left (\frac {d x}{2}\right )-7020 C \sin \left (\frac {d x}{2}\right )+11110 A \sin \left (c+\frac {d x}{2}\right )+4500 C \sin \left (c+\frac {d x}{2}\right )-15380 A \sin \left (c+\frac {3 d x}{2}\right )-4860 C \sin \left (c+\frac {3 d x}{2}\right )+380 A \sin \left (2 c+\frac {3 d x}{2}\right )+900 C \sin \left (2 c+\frac {3 d x}{2}\right )-4777 A \sin \left (2 c+\frac {5 d x}{2}\right )-1452 C \sin \left (2 c+\frac {5 d x}{2}\right )-1625 A \sin \left (3 c+\frac {5 d x}{2}\right )-300 C \sin \left (3 c+\frac {5 d x}{2}\right )-230 A \sin \left (3 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )-230 A \sin \left (4 c+\frac {7 d x}{2}\right )-60 C \sin \left (4 c+\frac {7 d x}{2}\right )+20 A \sin \left (4 c+\frac {9 d x}{2}\right )+20 A \sin \left (5 c+\frac {9 d x}{2}\right )-5 A \sin \left (5 c+\frac {11 d x}{2}\right )-5 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{3840 a^3 d} \]
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Time = 0.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55
method | result | size |
parallelrisch | \(\frac {-15 \left (\left (-\frac {3416 A}{15}-\frac {312 C}{5}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {43 A}{3}-4 C \right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )-\frac {A \cos \left (5 d x +5 c \right )}{3}+\left (-\frac {10916 A}{15}-\frac {972 C}{5}\right ) \cos \left (d x +c \right )-\frac {7783 A}{15}-\frac {696 C}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-11040 x d \left (A +\frac {6 C}{23}\right )}{960 a^{3} d}\) | \(119\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}-C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}-C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
risch | \(-\frac {23 A x}{2 a^{3}}-\frac {3 x C}{a^{3}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {3 i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {27 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{3} d}+\frac {27 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}-\frac {3 i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {2 i \left (225 A \,{\mathrm e}^{4 i \left (d x +c \right )}+90 C \,{\mathrm e}^{4 i \left (d x +c \right )}+810 A \,{\mathrm e}^{3 i \left (d x +c \right )}+300 C \,{\mathrm e}^{3 i \left (d x +c \right )}+1160 A \,{\mathrm e}^{2 i \left (d x +c \right )}+420 C \,{\mathrm e}^{2 i \left (d x +c \right )}+760 A \,{\mathrm e}^{i \left (d x +c \right )}+270 C \,{\mathrm e}^{i \left (d x +c \right )}+197 A +72 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(293\) |
norman | \(\frac {\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (23 A +6 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{20 a d}-\frac {\left (11 A +6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{15 a d}-\frac {2 \left (19 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}-\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2 a}-\frac {\left (93 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (127 A +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 a d}+\frac {\left (199 A +59 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {\left (207 A +52 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\) | \(296\) |
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Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (23 \, A + 6 \, C\right )} d x - {\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \, A \cos \left (d x + c\right )^{4} + 5 \, {\left (19 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (869 \, A + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (143 \, A + 38 \, C\right )} \cos \left (d x + c\right ) + 544 \, A + 144 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {A {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )} {\left (23 \, A + 6 \, C\right )}}{a^{3}} - \frac {20 \, {\left (51 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 15.01 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\left (17\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {76\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {x\,\left (23\,A+6\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {6\,A+2\,C}{12\,a^3}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}+\frac {6\,A+2\,C}{a^3}+\frac {15\,A-C}{4\,a^3}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \]
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