Integrand size = 21, antiderivative size = 182 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {85 a^3 x}{16}+\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.34 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2951, 2717, 2715, 8, 2713, 3852, 3853, 3855} \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {85 a^3 x}{16} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \sin ^3(c+d x) \tan ^3(c+d x) \, dx \\ & = -\frac {\int \left (8 a^9+6 a^9 \cos (c+d x)-6 a^9 \cos ^2(c+d x)-8 a^9 \cos ^3(c+d x)+3 a^9 \cos ^5(c+d x)+a^9 \cos ^6(c+d x)-3 a^9 \sec ^2(c+d x)-a^9 \sec ^3(c+d x)\right ) \, dx}{a^6} \\ & = -8 a^3 x-a^3 \int \cos ^6(c+d x) \, dx+a^3 \int \sec ^3(c+d x) \, dx-\left (3 a^3\right ) \int \cos ^5(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx-\left (6 a^3\right ) \int \cos (c+d x) \, dx+\left (6 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (8 a^3\right ) \int \cos ^3(c+d x) \, dx \\ & = -8 a^3 x-\frac {6 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx-\frac {1}{6} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (8 a^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -5 a^3 x+\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{8} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -5 a^3 x+\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{16} \left (5 a^3\right ) \int 1 \, dx \\ & = -\frac {85 a^3 x}{16}+\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {a^3 \sec ^2(c+d x) \left (10200 c+10200 d x-1920 \text {arctanh}(\sin (c+d x)) \cos ^2(c+d x)+10200 (c+d x) \cos (2 (c+d x))-460 \sin (c+d x)-8145 \sin (2 (c+d x))+1156 \sin (3 (c+d x))-1120 \sin (4 (c+d x))-268 \sin (5 (c+d x))+55 \sin (6 (c+d x))+36 \sin (7 (c+d x))+5 \sin (8 (c+d x))\right )}{3840 d} \]
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Time = 2.52 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(\frac {a^{3} \left (-10200 d x \cos \left (2 d x +2 c \right )-960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )+960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )-10200 d x +460 \sin \left (d x +c \right )+8145 \sin \left (2 d x +2 c \right )+1120 \sin \left (4 d x +4 c \right )-55 \sin \left (6 d x +6 c \right )-5 \sin \left (8 d x +8 c \right )-36 \sin \left (7 d x +7 c \right )+268 \sin \left (5 d x +5 c \right )-1156 \sin \left (3 d x +3 c \right )-960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{1920 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(199\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(232\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(232\) |
parts | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) | \(240\) |
risch | \(-\frac {85 a^{3} x}{16}+\frac {17 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}+\frac {81 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {81 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {15 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {17 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}-\frac {15 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{64 d}\) | \(264\) |
norman | \(\frac {-\frac {85 a^{3} x}{16}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {425 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{16}+\frac {77 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {277 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 d}+\frac {997 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 d}+\frac {3933 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{40 d}+\frac {6169 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {4319 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 d}-\frac {1039 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d}-\frac {93 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(366\) |
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Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {1275 \, a^{3} d x \cos \left (d x + c\right )^{2} - 60 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 60 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (40 \, a^{3} \cos \left (d x + c\right )^{7} + 144 \, a^{3} \cos \left (d x + c\right )^{6} + 50 \, a^{3} \cos \left (d x + c\right )^{5} - 448 \, a^{3} \cos \left (d x + c\right )^{4} - 645 \, a^{3} \cos \left (d x + c\right )^{3} + 544 \, a^{3} \cos \left (d x + c\right )^{2} - 720 \, a^{3} \cos \left (d x + c\right ) - 120 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.32 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {96 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 80 \, {\left (4 \, \sin \left (d x + c\right )^{3} - \frac {6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} a^{3} + 360 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{3}}{960 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {1275 \, {\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {240 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (795 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4025 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7614 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5634 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 14.53 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.43 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {85\,a^3\,x}{16}+\frac {-\frac {93\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{8}-\frac {1039\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{24}-\frac {4319\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{120}+\frac {6169\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{120}+\frac {3933\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{40}+\frac {997\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {277\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {77\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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