Integrand size = 33, antiderivative size = 181 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{8} a^4 (52 A+35 C) x+\frac {4 a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d} \]
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Time = 0.74 (sec) , antiderivative size = 181, 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 = {3123, 3055, 3047, 3102, 2814, 3855} \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {4 a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {(12 A-35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (52 A+35 C)-\frac {(12 A-7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}-\frac {a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d} \]
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Rule 2814
Rule 3047
Rule 3055
Rule 3102
Rule 3123
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^4 (4 a A-a (4 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a} \\ & = -\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^3 \left (16 a^2 A-a^2 (12 A-7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{4 a} \\ & = -\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^2 \left (48 a^3 A-a^3 (12 A-35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a} \\ & = -\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x)) \left (96 a^4 A+15 a^4 (4 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a} \\ & = -\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (96 a^5 A+\left (96 a^5 A+15 a^5 (4 A+7 C)\right ) \cos (c+d x)+15 a^5 (4 A+7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a} \\ & = \frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (96 a^5 A+3 a^5 (52 A+35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a} \\ & = \frac {1}{8} a^4 (52 A+35 C) x+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\left (4 a^4 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} a^4 (52 A+35 C) x+\frac {4 a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d} \\ \end{align*}
Time = 5.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.87 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (12 (52 A+35 C) x-\frac {384 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {384 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {96 (4 A+7 C) \cos (d x) \sin (c)}{d}+\frac {24 (A+7 C) \cos (2 d x) \sin (2 c)}{d}+\frac {32 C \cos (3 d x) \sin (3 c)}{d}+\frac {3 C \cos (4 d x) \sin (4 c)}{d}+\frac {96 (4 A+7 C) \cos (c) \sin (d x)}{d}+\frac {24 (A+7 C) \cos (2 c) \sin (2 d x)}{d}+\frac {32 C \cos (3 c) \sin (3 d x)}{d}+\frac {3 C \cos (4 c) \sin (4 d x)}{d}+\frac {96 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {96 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{1536} \]
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Time = 6.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {\left (-32 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+32 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+\left (16 A +\frac {88 C}{3}\right ) \sin \left (2 d x +2 c \right )+\left (A +\frac {57 C}{8}\right ) \sin \left (3 d x +3 c \right )+\frac {4 \sin \left (4 d x +4 c \right ) C}{3}+\frac {\sin \left (5 d x +5 c \right ) C}{8}+52 \left (A +\frac {35 C}{52}\right ) x d \cos \left (d x +c \right )+9 \left (A +\frac {7 C}{9}\right ) \sin \left (d x +c \right )\right ) a^{4}}{8 d \cos \left (d x +c \right )}\) | \(144\) |
parts | \(\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +6 C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{4} A +4 C \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +C \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(187\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (d x +c \right )+6 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \,a^{4} \sin \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+C \,a^{4} \left (d x +c \right )}{d}\) | \(197\) |
default | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (d x +c \right )+6 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \,a^{4} \sin \left (d x +c \right )+a^{4} A \tan \left (d x +c \right )+C \,a^{4} \left (d x +c \right )}{d}\) | \(197\) |
risch | \(\frac {13 a^{4} x A}{2}+\frac {35 a^{4} C x}{8}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} A}{8 d}-\frac {7 i {\mathrm e}^{2 i \left (d x +c \right )} C \,a^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{4}}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{4}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} A}{8 d}+\frac {7 i {\mathrm e}^{-2 i \left (d x +c \right )} C \,a^{4}}{8 d}+\frac {2 i a^{4} A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {4 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (4 d x +4 c \right ) C \,a^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{4}}{3 d}\) | \(271\) |
norman | \(\frac {\left (-\frac {13}{2} a^{4} A -\frac {35}{8} C \,a^{4}\right ) x +\left (-\frac {117}{2} a^{4} A -\frac {315}{8} C \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {65}{2} a^{4} A -\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {65}{2} a^{4} A -\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {13}{2} a^{4} A +\frac {35}{8} C \,a^{4}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} a^{4} A +\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {65}{2} a^{4} A +\frac {175}{8} C \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {117}{2} a^{4} A +\frac {315}{8} C \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{4} \left (4 A +7 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {a^{4} \left (36 A -25 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{4} \left (44 A +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{4} \left (108 A +245 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{4} \left (132 A +791 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {a^{4} \left (276 A +395 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a^{4} \left (828 A +617 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {4 a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {4 a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(453\) |
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Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.87 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (52 \, A + 35 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 48 \, A a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, A a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 32 \, C a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (3 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.07 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 576 \, {\left (d x + c\right )} A a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 96 \, {\left (d x + c\right )} C a^{4} + 192 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, A a^{4} \sin \left (d x + c\right ) + 384 \, C a^{4} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.35 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {96 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (52 \, A a^{4} + 35 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (84 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 276 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 0.67 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.29 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,C\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {13\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {35\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {4\,C\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {27\,C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d} \]
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