Integrand size = 22, antiderivative size = 235 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {429 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{40 d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d} \]
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Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3577, 3579, 3567, 3855} \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {429 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{40 d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d} \]
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Rule 3567
Rule 3577
Rule 3579
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\left (13 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^6 \, dx \\ & = -\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {1}{6} \left (143 a^3\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^5 \, dx \\ & = -\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1}{10} \left (429 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1}{40} \left (3003 a^5\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1}{8} \left (1001 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx \\ & = -\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1}{16} \left (3003 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx \\ & = -\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d}-\frac {1}{16} \left (3003 a^8\right ) \int \sec (c+d x) \, dx \\ & = -\frac {3003 a^8 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {3003 i a^8 \sec (c+d x)}{16 d}-\frac {429 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{40 d}-\frac {13 i a^3 \sec (c+d x) (a+i a \tan (c+d x))^5}{6 d}-\frac {2 i a \cos (c+d x) (a+i a \tan (c+d x))^7}{d}-\frac {143 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^4}{30 d}-\frac {1001 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{40 d}-\frac {1001 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{16 d} \\ \end{align*}
Time = 2.76 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {a^8 \cos ^2(c+d x) (\cos (8 c)-i \sin (8 c)) \left (-658944 i \cos (c+d x)+720720 \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+5 (-73216 i \cos (3 (c+d x))-19968 i \cos (5 (c+d x))-1536 i \cos (7 (c+d x))+12870 \sin (c+d x)+22165 \sin (3 (c+d x))+10959 \sin (5 (c+d x))+1536 \sin (7 (c+d x)))\right ) (-i+\tan (c+d x))^8}{3840 d (\cos (d x)+i \sin (d x))^8} \]
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Time = 90.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {128 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{8} \left (62475 \,{\mathrm e}^{11 i \left (d x +c \right )}+246505 \,{\mathrm e}^{9 i \left (d x +c \right )}+416094 \,{\mathrm e}^{7 i \left (d x +c \right )}+364194 \,{\mathrm e}^{5 i \left (d x +c \right )}+163095 \,{\mathrm e}^{3 i \left (d x +c \right )}+29685 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {3003 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {3003 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\) | \(151\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-8 i a^{8} \cos \left (d x +c \right )+a^{8} \sin \left (d x +c \right )}{d}\) | \(522\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{9}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{16 \cos \left (d x +c \right )^{2}}+\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{16}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {35 \sin \left (d x +c \right )}{16}-\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )-28 a^{8} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-8 i a^{8} \cos \left (d x +c \right )+a^{8} \sin \left (d x +c \right )}{d}\) | \(522\) |
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none
Time = 0.25 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.61 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {-30720 i \, a^{8} e^{\left (13 i \, d x + 13 i \, c\right )} - 309270 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} - 953810 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 1446588 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} - 1189188 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 510510 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} - 90090 i \, a^{8} e^{\left (i \, d x + i \, c\right )} - 45045 \, {\left (a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) + 45045 \, {\left (a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{240 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.36 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {3003 a^{8} \left (\frac {\log {\left (e^{i d x} - i e^{- i c} \right )}}{16} - \frac {\log {\left (e^{i d x} + i e^{- i c} \right )}}{16}\right )}{d} + \frac {- 62475 i a^{8} e^{11 i c} e^{11 i d x} - 246505 i a^{8} e^{9 i c} e^{9 i d x} - 416094 i a^{8} e^{7 i c} e^{7 i d x} - 364194 i a^{8} e^{5 i c} e^{5 i d x} - 163095 i a^{8} e^{3 i c} e^{3 i d x} - 29685 i a^{8} e^{i c} e^{i d x}}{120 d e^{12 i c} e^{12 i d x} + 720 d e^{10 i c} e^{10 i d x} + 1800 d e^{8 i c} e^{8 i d x} + 2400 d e^{6 i c} e^{6 i d x} + 1800 d e^{4 i c} e^{4 i d x} + 720 d e^{2 i c} e^{2 i d x} + 120 d} + \begin {cases} - \frac {128 i a^{8} e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\128 a^{8} x e^{i c} & \text {otherwise} \end {cases} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (195) = 390\).
Time = 0.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.69 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=-\frac {5 \, a^{8} {\left (\frac {2 \, {\left (87 \, \sin \left (d x + c\right )^{5} - 136 \, \sin \left (d x + c\right )^{3} + 57 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 96 \, \sin \left (d x + c\right )\right )} + 840 \, a^{8} {\left (\frac {2 \, {\left (9 \, \sin \left (d x + c\right )^{3} - 7 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 16 \, \sin \left (d x + c\right )\right )} + 8400 \, a^{8} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )} + 26880 i \, a^{8} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 8960 i \, a^{8} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} + 768 i \, a^{8} {\left (\frac {15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1}{\cos \left (d x + c\right )^{5}} + 5 \, \cos \left (d x + c\right )\right )} + 6720 \, a^{8} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} + 3840 i \, a^{8} \cos \left (d x + c\right ) - 480 \, a^{8} \sin \left (d x + c\right )}{480 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (195) = 390\).
Time = 1.44 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.93 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\text {Too large to display} \]
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Time = 9.23 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.70 \[ \int \cos (c+d x) (a+i a \tan (c+d x))^8 \, dx=\frac {\frac {3019\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{8}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,2891{}\mathrm {i}}{8}-\frac {52795\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{24}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,45115{}\mathrm {i}}{24}+\frac {22415\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,43757{}\mathrm {i}}{12}-\frac {97811\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{12}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,12977{}\mathrm {i}}{4}+\frac {167237\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{24}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,160729{}\mathrm {i}}{120}-\frac {127113\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{40}-\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,25499{}\mathrm {i}}{120}+\frac {8848\,a^8}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,1{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,6{}\mathrm {i}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,15{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,15{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,6{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}-\frac {3003\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]
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