Integrand size = 31, antiderivative size = 188 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (30 A+23 C) x+\frac {a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}-\frac {a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d} \]
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Time = 0.44 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3125, 3047, 3102, 2830, 2724, 2717, 2715, 8, 2713} \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d}+\frac {a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {1}{16} a^3 x (30 A+23 C)+\frac {(30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2724
Rule 2830
Rule 3047
Rule 3102
Rule 3125
Rubi steps \begin{align*} \text {integral}& = \frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {\int \cos (c+d x) (a+a \cos (c+d x))^3 (2 a (3 A+C)+3 a C \cos (c+d x)) \, dx}{6 a} \\ & = \frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^3 \left (2 a (3 A+C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \, dx}{6 a} \\ & = \frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac {\int (a+a \cos (c+d x))^3 \left (12 a^2 C+a^2 (30 A+7 C) \cos (c+d x)\right ) \, dx}{30 a^2} \\ & = \frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac {1}{40} (30 A+23 C) \int (a+a \cos (c+d x))^3 \, dx \\ & = \frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac {1}{40} (30 A+23 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx \\ & = \frac {1}{40} a^3 (30 A+23 C) x+\frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac {1}{40} \left (a^3 (30 A+23 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{40} a^3 (30 A+23 C) x+\frac {3 a^3 (30 A+23 C) \sin (c+d x)}{40 d}+\frac {3 a^3 (30 A+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}+\frac {1}{80} \left (3 a^3 (30 A+23 C)\right ) \int 1 \, dx-\frac {\left (a^3 (30 A+23 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{40 d} \\ & = \frac {1}{16} a^3 (30 A+23 C) x+\frac {a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {(30 A+7 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{120 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{10 a d}-\frac {a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.65 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (900 c C+1800 A d x+1380 C d x+120 (26 A+21 C) \sin (c+d x)+15 (64 A+63 C) \sin (2 (c+d x))+240 A \sin (3 (c+d x))+380 C \sin (3 (c+d x))+30 A \sin (4 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x)))}{960 d} \]
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Time = 7.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {\left (\left (32 A +\frac {63 C}{2}\right ) \sin \left (2 d x +2 c \right )+\left (8 A +\frac {38 C}{3}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {9 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {6 \sin \left (5 d x +5 c \right ) C}{5}+\frac {\sin \left (6 d x +6 c \right ) C}{6}+\left (104 A +84 C \right ) \sin \left (d x +c \right )+60 \left (A +\frac {23 C}{30}\right ) x d \right ) a^{3}}{32 d}\) | \(106\) |
risch | \(\frac {15 a^{3} A x}{8}+\frac {23 a^{3} C x}{16}+\frac {13 a^{3} A \sin \left (d x +c \right )}{4 d}+\frac {21 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {C \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 \sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{3}}{32 d}+\frac {9 \sin \left (4 d x +4 c \right ) C \,a^{3}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{3}}{4 d}+\frac {19 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3}}{d}+\frac {63 \sin \left (2 d x +2 c \right ) C \,a^{3}}{64 d}\) | \(189\) |
parts | \(\frac {\left (A \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 A \,a^{3}+C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(215\) |
derivativedivides | \(\frac {A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(245\) |
default | \(\frac {A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{3} \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(245\) |
norman | \(\frac {\frac {a^{3} \left (30 A +23 C \right ) x}{16}+\frac {7 a^{3} \left (14 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {33 a^{3} \left (30 A +23 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {17 a^{3} \left (30 A +23 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{3} \left (30 A +23 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{3} \left (30 A +23 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (30 A +23 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{3} \left (342 A +211 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (1250 A +969 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(329\) |
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Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.67 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A + 23 \, C\right )} a^{3} d x + {\left (40 \, C a^{3} \cos \left (d x + c\right )^{5} + 144 \, C a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (15 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \, {\left (45 \, A + 34 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (170) = 340\).
Time = 0.42 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.44 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {5 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {5 C a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 C a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {5 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{3} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.27 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (30 \, A a^{3} + 23 \, C a^{3}\right )} x + \frac {{\left (2 \, A a^{3} + 9 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (12 \, A a^{3} + 19 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (64 \, A a^{3} + 63 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (26 \, A a^{3} + 21 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 2.59 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.68 \[ \int \cos (c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {15\,A\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {85\,A\,a^3}{4}+\frac {391\,C\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {99\,A\,a^3}{2}+\frac {759\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {125\,A\,a^3}{2}+\frac {969\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {171\,A\,a^3}{4}+\frac {211\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {49\,A\,a^3}{4}+\frac {105\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (30\,A+23\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (30\,A+23\,C\right )}{8\,\left (\frac {15\,A\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )}\right )\,\left (30\,A+23\,C\right )}{8\,d} \]
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