Integrand size = 48, antiderivative size = 149 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=a^3 (b B-a C) x+\frac {b \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^2 \left (9 a b B-a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {b^3 (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \]
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Time = 0.35 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4126, 4003, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=a^3 x (b B-a C)+\frac {b^2 \left (a^2 (-C)+9 a b B+2 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {b \left (-4 a^3 C+6 a^2 b B+2 a b^2 C+b^3 B\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^3 (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac {b^2 C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4003
Rule 4126
Rule 4133
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b \sec (c+d x))^3 \left (b^2 (b B-a C)+b^3 C \sec (c+d x)\right ) \, dx}{b^2} \\ & = \frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {\int (a+b \sec (c+d x)) \left (3 a^2 b^2 (b B-a C)+b^3 \left (6 a b B-3 a^2 C+2 b^2 C\right ) \sec (c+d x)+b^4 (3 b B+2 a C) \sec ^2(c+d x)\right ) \, dx}{3 b^2} \\ & = \frac {b^3 (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {\int \left (6 a^3 b^2 (b B-a C)+3 b^3 \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right ) \sec (c+d x)+2 b^4 \left (9 a b B-a^2 C+2 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{6 b^2} \\ & = a^3 (b B-a C) x+\frac {b^3 (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \left (b^2 \left (9 a b B-a^2 C+2 b^2 C\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (b \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right )\right ) \int \sec (c+d x) \, dx \\ & = a^3 (b B-a C) x+\frac {b \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^3 (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {\left (b^2 \left (9 a b B-a^2 C+2 b^2 C\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = a^3 (b B-a C) x+\frac {b \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b^2 \left (9 a b B-a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {b^3 (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b^2 C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=\frac {6 a^3 (b B-a C) d x+3 b \left (6 a^2 b B+b^3 B-4 a^3 C+2 a b^2 C\right ) \text {arctanh}(\sin (c+d x))+3 b^3 (b B+2 a C+2 (3 a B+b C) \cos (c+d x)) \sec (c+d x) \tan (c+d x)+2 b^4 C \tan ^3(c+d x)}{6 d} \]
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Time = 0.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.95
method | result | size |
parts | \(a^{3} \left (B b -C a \right ) x +\frac {\left (B \,b^{4}+2 C a \,b^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 B \,a^{2} b^{2}-2 a^{3} b C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {C \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {3 B a \,b^{3} \tan \left (d x +c \right )}{d}\) | \(141\) |
derivativedivides | \(\frac {B \,a^{3} b \left (d x +c \right )-a^{4} C \left (d x +c \right )+3 B \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-2 a^{3} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B a \,b^{3} \tan \left (d x +c \right )+2 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(183\) |
default | \(\frac {B \,a^{3} b \left (d x +c \right )-a^{4} C \left (d x +c \right )+3 B \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-2 a^{3} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B a \,b^{3} \tan \left (d x +c \right )+2 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(183\) |
parallelrisch | \(\frac {-27 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B \,a^{2} b +\frac {1}{6} B \,b^{3}-\frac {2}{3} a^{3} C +\frac {1}{3} C a \,b^{2}\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+27 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (B \,a^{2} b +\frac {1}{6} B \,b^{3}-\frac {2}{3} a^{3} C +\frac {1}{3} C a \,b^{2}\right ) b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 a^{3} d x \left (B b -C a \right ) \cos \left (3 d x +3 c \right )+3 \left (B \,b^{4}+2 C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (9 B a \,b^{3}+2 C \,b^{4}\right ) \sin \left (3 d x +3 c \right )+9 a^{3} d x \left (B b -C a \right ) \cos \left (d x +c \right )+9 \sin \left (d x +c \right ) b^{3} \left (a B +\frac {2 C b}{3}\right )}{3 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(257\) |
norman | \(\frac {\left (-B \,a^{3} b +a^{4} C \right ) x +\left (-3 B \,a^{3} b +3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (B \,a^{3} b -a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (3 B \,a^{3} b -3 a^{4} C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {4 b^{3} \left (9 a B +C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {b^{3} \left (6 a B -B b -2 C a +2 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {b^{3} \left (6 a B +B b +2 C a +2 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}-\frac {b \left (6 B \,a^{2} b +B \,b^{3}-4 a^{3} C +2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (6 B \,a^{2} b +B \,b^{3}-4 a^{3} C +2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(298\) |
risch | \(B \,a^{3} b x -a^{4} x C -\frac {i b^{3} \left (3 B b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 C a \,{\mathrm e}^{5 i \left (d x +c \right )}-18 a B \,{\mathrm e}^{4 i \left (d x +c \right )}-36 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-12 C b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B b \,{\mathrm e}^{i \left (d x +c \right )}-6 C a \,{\mathrm e}^{i \left (d x +c \right )}-18 a B -4 C b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{2} b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{4}}{2 d}-\frac {2 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{2} b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{4}}{2 d}+\frac {2 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(323\) |
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Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.34 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {12 \, {\left (C a^{4} - B a^{3} b\right )} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, C a^{3} b - 6 \, B a^{2} b^{2} - 2 \, C a b^{3} - B b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, C a^{3} b - 6 \, B a^{2} b^{2} - 2 \, C a b^{3} - B b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, C b^{4} + 2 \, {\left (9 \, B a b^{3} + 2 \, C b^{4}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=- \int C a^{4}\, dx - \int \left (- B a^{3} b\right )\, dx - \int \left (- B b^{4} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int \left (- C b^{4} \sec ^{4}{\left (c + d x \right )}\right )\, dx - \int \left (- 3 B a b^{3} \sec ^{2}{\left (c + d x \right )}\right )\, dx - \int \left (- 3 B a^{2} b^{2} \sec {\left (c + d x \right )}\right )\, dx - \int \left (- 2 C a b^{3} \sec ^{3}{\left (c + d x \right )}\right )\, dx - \int 2 C a^{3} b \sec {\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.37 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {12 \, {\left (d x + c\right )} C a^{4} - 12 \, {\left (d x + c\right )} B a^{3} b - 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{4} + 6 \, C a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3 \, B b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 36 \, B a^{2} b^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 36 \, B a b^{3} \tan \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (142) = 284\).
Time = 0.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.02 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=-\frac {6 \, {\left (C a^{4} - B a^{3} b\right )} {\left (d x + c\right )} + 3 \, {\left (4 \, C a^{3} b - 6 \, B a^{2} b^{2} - 2 \, C a b^{3} - B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, C a^{3} b - 6 \, B a^{2} b^{2} - 2 \, C a b^{3} - B b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (18 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 19.61 (sec) , antiderivative size = 576, normalized size of antiderivative = 3.87 \[ \int (a+b \sec (c+d x))^2 \left (a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {B\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {C\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {C\,b^4\,\sin \left (c+d\,x\right )}{2}+\frac {3\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{4}-\frac {3\,C\,a^4\,\cos \left (c+d\,x\right )\,\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 )}{2}-\frac {B\,b^4\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4}+\frac {3\,B\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {C\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{2}-\frac {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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {B\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{4}+\frac {B\,a^3\,b\,\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 (3\,c+3\,d\,x\right )}{2}-\frac {B\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,9{}\mathrm {i}}{2}-\frac {C\,a\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{2}+C\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}-\frac {B\,a^2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{2}+\frac {3\,B\,a^3\,b\,\cos \left (c+d\,x\right )\,\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 )}{2}-\frac {C\,a\,b^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{2}+C\,a^3\,b\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
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