Integrand size = 29, antiderivative size = 151 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=a^4 (4 A+B) x+\frac {a^4 (13 A+12 B) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d} \]
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Time = 0.42 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4103, 4081, 3855} \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^4 (13 A+12 B) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{3 d}+a^4 x (4 A+B)+\frac {(A+2 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}+\frac {a B \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d} \]
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Rule 3855
Rule 4081
Rule 4103
Rubi steps \begin{align*} \text {integral}& = \frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos (c+d x) (a+a \sec (c+d x))^3 (a (3 A-B)+3 a (A+2 B) \sec (c+d x)) \, dx \\ & = \frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {1}{6} \int \cos (c+d x) (a+a \sec (c+d x))^2 \left (a^2 (3 A-8 B)+2 a^2 (9 A+11 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) (a+a \sec (c+d x)) \left (-15 a^3 (A+2 B)+3 a^3 (13 A+12 B) \sec (c+d x)\right ) \, dx \\ & = -\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d}-\frac {1}{6} \int \left (-6 a^4 (4 A+B)-3 a^4 (13 A+12 B) \sec (c+d x)\right ) \, dx \\ & = a^4 (4 A+B) x-\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d}+\frac {1}{2} \left (a^4 (13 A+12 B)\right ) \int \sec (c+d x) \, dx \\ & = a^4 (4 A+B) x+\frac {a^4 (13 A+12 B) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac {a B (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {(A+2 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {(9 A+11 B) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1202\) vs. \(2(151)=302\).
Time = 12.29 (sec) , antiderivative size = 1202, normalized size of antiderivative = 7.96 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {(4 A+B) x \cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x))}{16 (B+A \cos (c+d x))}+\frac {(-13 A-12 B) \cos ^5(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x))}{32 d (B+A \cos (c+d x))}+\frac {(13 A+12 B) \cos ^5(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x))}{32 d (B+A \cos (c+d x))}+\frac {A \cos (d x) \cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \sin (c)}{16 d (B+A \cos (c+d x))}+\frac {A \cos (c) \cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \sin (d x)}{16 d (B+A \cos (c+d x))}+\frac {B \cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left (\frac {d x}{2}\right )}{96 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \left (3 A \cos \left (\frac {c}{2}\right )+13 B \cos \left (\frac {c}{2}\right )-3 A \sin \left (\frac {c}{2}\right )-11 B \sin \left (\frac {c}{2}\right )\right )}{192 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {\cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \left (3 A \sin \left (\frac {d x}{2}\right )+5 B \sin \left (\frac {d x}{2}\right )\right )}{12 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {B \cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left (\frac {d x}{2}\right )}{96 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \left (-3 A \cos \left (\frac {c}{2}\right )-13 B \cos \left (\frac {c}{2}\right )-3 A \sin \left (\frac {c}{2}\right )-11 B \sin \left (\frac {c}{2}\right )\right )}{192 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {\cos ^5(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \left (3 A \sin \left (\frac {d x}{2}\right )+5 B \sin \left (\frac {d x}{2}\right )\right )}{12 d (B+A \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
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Time = 3.15 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.28
method | result | size |
parallelrisch | \(\frac {2 \left (-\frac {39 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {12 B}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4}+\frac {39 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {12 B}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4}+2 d \left (A +\frac {B}{4}\right ) x \cos \left (3 d x +3 c \right )+\left (A +2 B \right ) \sin \left (2 d x +2 c \right )+2 \left (A +\frac {5 B}{3}\right ) \sin \left (3 d x +3 c \right )+\frac {A \sin \left (4 d x +4 c \right )}{4}+6 d \left (A +\frac {B}{4}\right ) x \cos \left (d x +c \right )+2 \sin \left (d x +c \right ) \left (A +2 B \right )\right ) a^{4}}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(193\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (d x +c \right )}{d}\) | \(199\) |
default | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (d x +c \right )}{d}\) | \(199\) |
risch | \(4 a^{4} A x +a^{4} x B -\frac {i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{4} \left (3 A \,{\mathrm e}^{5 i \left (d x +c \right )}+12 B \,{\mathrm e}^{5 i \left (d x +c \right )}-24 A \,{\mathrm e}^{4 i \left (d x +c \right )}-36 B \,{\mathrm e}^{4 i \left (d x +c \right )}-48 A \,{\mathrm e}^{2 i \left (d x +c \right )}-84 B \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )} A -12 B \,{\mathrm e}^{i \left (d x +c \right )}-24 A -40 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}\) | \(266\) |
norman | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) x +\left (-12 a^{4} A -3 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-12 a^{4} A -3 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (4 a^{4} A +B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (8 a^{4} A +2 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (8 a^{4} A +2 B \,a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a^{4} \left (11 A +18 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{4} \left (A +2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {2 a^{4} \left (5 A -4 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {4 a^{4} \left (6 A +19 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {4 a^{4} \left (18 A +19 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {a^{4} \left (13 A +12 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (13 A +12 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(356\) |
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Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {12 \, {\left (4 \, A + B\right )} a^{4} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (13 \, A + 12 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (13 \, A + 12 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (3 \, A + 5 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 3 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right ) + 2 \, B a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=a^{4} \left (\int A \cos {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.56 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {48 \, {\left (d x + c\right )} A a^{4} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 12 \, {\left (d x + c\right )} B a^{4} - 3 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} \sin \left (d x + c\right ) + 48 \, A a^{4} \tan \left (d x + c\right ) + 72 \, B a^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.50 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\frac {12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 6 \, {\left (4 \, A a^{4} + B a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (13 \, A a^{4} + 12 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (13 \, A a^{4} + 12 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 76 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 13.54 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.68 \[ \int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {13\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {12\,B\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {20\,B\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {2\,B\,a^4\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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