Integrand size = 33, antiderivative size = 162 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^3 A x+\frac {a^3 (28 A+20 B+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \]
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Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4139, 4002, 3999, 3852, 8, 3855} \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (28 A+20 B+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac {(12 A+20 B+15 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{24 d}+a^3 A x+\frac {(4 B+3 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{12 a d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rule 4139
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\int (a+a \sec (c+d x))^3 (4 a A+a (4 B+3 C) \sec (c+d x)) \, dx}{4 a} \\ & = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {\int (a+a \sec (c+d x))^2 \left (12 a^2 A+a^2 (12 A+20 B+15 C) \sec (c+d x)\right ) \, dx}{12 a} \\ & = \frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {\int (a+a \sec (c+d x)) \left (24 a^3 A+15 a^3 (4 A+4 B+3 C) \sec (c+d x)\right ) \, dx}{24 a} \\ & = a^3 A x+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {1}{8} \left (5 a^3 (4 A+4 B+3 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (a^3 (28 A+20 B+15 C)\right ) \int \sec (c+d x) \, dx \\ & = a^3 A x+\frac {a^3 (28 A+20 B+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac {\left (5 a^3 (4 A+4 B+3 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d} \\ & = a^3 A x+\frac {a^3 (28 A+20 B+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac {C (a+a \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {(4 B+3 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{12 a d}+\frac {(12 A+20 B+15 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \\ \end{align*}
Time = 5.50 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (24 A d x+(84 A+60 B+45 C) \text {arctanh}(\sin (c+d x))+3 \left (24 A+32 (B+C)+(4 A+12 B+15 C) \sec (c+d x)+2 C \sec ^3(c+d x)\right ) \tan (c+d x)+8 (B+3 C) \tan ^3(c+d x)\right )}{24 d} \]
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Time = 0.64 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.27
method | result | size |
parts | \(a^{3} A x +\frac {\left (3 a^{3} A +B \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {\left (B \,a^{3}+3 a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{3} A +3 B \,a^{3}+3 a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}+a^{3} C \right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(206\) |
parallelrisch | \(\frac {a^{3} \left (-14 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {5 B}{7}+\frac {15 C}{28}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+14 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {5 B}{7}+\frac {15 C}{28}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 d x A \cos \left (2 d x +2 c \right )+d x A \cos \left (4 d x +4 c \right )+\left (6 A +\frac {26 B}{3}+10 C \right ) \sin \left (2 d x +2 c \right )+\left (A +3 B +\frac {15 C}{4}\right ) \sin \left (3 d x +3 c \right )+\left (3 C +3 A +\frac {11 B}{3}\right ) \sin \left (4 d x +4 c \right )+\left (3 B +\frac {23 C}{4}+A \right ) \sin \left (d x +c \right )+3 d x A \right )}{d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(226\) |
norman | \(\frac {a^{3} A x +a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 a^{3} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {5 a^{3} \left (4 A +4 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a^{3} \left (28 A +49 C +44 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (204 A +220 B +165 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {a^{3} \left (228 A +292 B +219 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a^{3} \left (28 A +20 B +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{3} \left (28 A +20 B +15 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(270\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )}{d}\) | \(297\) |
default | \(\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 a^{3} A \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{3} A \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} C \tan \left (d x +c \right )}{d}\) | \(297\) |
risch | \(a^{3} A x -\frac {i a^{3} \left (12 A \,{\mathrm e}^{7 i \left (d x +c \right )}+36 B \,{\mathrm e}^{7 i \left (d x +c \right )}+45 C \,{\mathrm e}^{7 i \left (d x +c \right )}-72 A \,{\mathrm e}^{6 i \left (d x +c \right )}-72 B \,{\mathrm e}^{6 i \left (d x +c \right )}-24 C \,{\mathrm e}^{6 i \left (d x +c \right )}+12 A \,{\mathrm e}^{5 i \left (d x +c \right )}+36 B \,{\mathrm e}^{5 i \left (d x +c \right )}+69 C \,{\mathrm e}^{5 i \left (d x +c \right )}-216 A \,{\mathrm e}^{4 i \left (d x +c \right )}-264 B \,{\mathrm e}^{4 i \left (d x +c \right )}-216 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{3 i \left (d x +c \right )}-36 B \,{\mathrm e}^{3 i \left (d x +c \right )}-69 C \,{\mathrm e}^{3 i \left (d x +c \right )}-216 A \,{\mathrm e}^{2 i \left (d x +c \right )}-280 B \,{\mathrm e}^{2 i \left (d x +c \right )}-264 C \,{\mathrm e}^{2 i \left (d x +c \right )}-12 A \,{\mathrm e}^{i \left (d x +c \right )}-36 B \,{\mathrm e}^{i \left (d x +c \right )}-45 C \,{\mathrm e}^{i \left (d x +c \right )}-72 A -88 B -72 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(424\) |
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Time = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, A a^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (28 \, A + 20 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (28 \, A + 20 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (9 \, A + 11 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 12 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 8 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 6 \, C a^{3}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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\[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A\, dx + \int 3 A \sec {\left (c + d x \right )}\, dx + \int 3 A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 3 B \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{3}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{4}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (152) = 304\).
Time = 0.24 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.17 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {48 \, {\left (d x + c\right )} A a^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 144 \, A a^{3} \tan \left (d x + c\right ) + 144 \, B a^{3} \tan \left (d x + c\right ) + 48 \, C a^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.86 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {24 \, {\left (d x + c\right )} A a^{3} + 3 \, {\left (28 \, A a^{3} + 20 \, B a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (28 \, A a^{3} + 20 \, B a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (60 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 204 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 220 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 228 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 292 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 219 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 84 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 132 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 147 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
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Time = 18.63 (sec) , antiderivative size = 611, normalized size of antiderivative = 3.77 \[ \int (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {9\,A\,a^3\,\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 )+\frac {63\,A\,a^3\,\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 )}{2}+\frac {45\,B\,a^3\,\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 )}{2}+\frac {135\,C\,a^3\,\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 )}{8}+9\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )+\frac {3\,A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {9\,A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{2}+13\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )+\frac {9\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{2}+\frac {11\,B\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{2}+15\,C\,a^3\,\sin \left (2\,c+2\,d\,x\right )+\frac {45\,C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{8}+\frac {9\,C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{2}+\frac {3\,A\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {9\,B\,a^3\,\sin \left (c+d\,x\right )}{2}+\frac {69\,C\,a^3\,\sin \left (c+d\,x\right )}{8}+12\,A\,a^3\,\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 )\,\cos \left (2\,c+2\,d\,x\right )+3\,A\,a^3\,\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 )\,\cos \left (4\,c+4\,d\,x\right )+42\,A\,a^3\,\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 )\,\cos \left (2\,c+2\,d\,x\right )+\frac {21\,A\,a^3\,\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 )\,\cos \left (4\,c+4\,d\,x\right )}{2}+30\,B\,a^3\,\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 )\,\cos \left (2\,c+2\,d\,x\right )+\frac {15\,B\,a^3\,\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 )\,\cos \left (4\,c+4\,d\,x\right )}{2}+\frac {45\,C\,a^3\,\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 )\,\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {45\,C\,a^3\,\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 )\,\cos \left (4\,c+4\,d\,x\right )}{8}}{12\,d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {\cos \left (4\,c+4\,d\,x\right )}{8}+\frac {3}{8}\right )} \]
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