Integrand size = 24, antiderivative size = 205 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {1155 \text {arctanh}(\sin (c+d x))}{8 a^8 d}+\frac {1155 \sec (c+d x) \tan (c+d x)}{8 a^8 d}+\frac {385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )} \]
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Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3581, 3853, 3855} \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {1155 \text {arctanh}(\sin (c+d x))}{8 a^8 d}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {385 \tan (c+d x) \sec ^3(c+d x)}{4 a^8 d}+\frac {1155 \tan (c+d x) \sec (c+d x)}{8 a^8 d}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7} \]
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Rule 3581
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {11 \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{3 a^2} \\ & = \frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}+\frac {33 \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{a^4} \\ & = \frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}+\frac {231 \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6} \\ & = \frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {385 \int \sec ^5(c+d x) \, dx}{a^8} \\ & = \frac {385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {1155 \int \sec ^3(c+d x) \, dx}{4 a^8} \\ & = \frac {1155 \sec (c+d x) \tan (c+d x)}{8 a^8 d}+\frac {385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac {1155 \int \sec (c+d x) \, dx}{8 a^8} \\ & = \frac {1155 \text {arctanh}(\sin (c+d x))}{8 a^8 d}+\frac {1155 \sec (c+d x) \tan (c+d x)}{8 a^8 d}+\frac {385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac {2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac {22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac {66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac {154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1704\) vs. \(2(205)=410\).
Time = 7.39 (sec) , antiderivative size = 1704, normalized size of antiderivative = 8.31 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {1155 \cos (8 c) \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(c+d x) (\cos (d x)+i \sin (d x))^8}{8 d (a+i a \tan (c+d x))^8}+\frac {1155 \cos (8 c) \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(c+d x) (\cos (d x)+i \sin (d x))^8}{8 d (a+i a \tan (c+d x))^8}+\frac {\cos (3 d x) \sec ^8(c+d x) \left (\frac {32}{3} i \cos (5 c)-\frac {32}{3} \sin (5 c)\right ) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\cos (d x) \sec ^8(c+d x) (-160 i \cos (7 c)+160 \sin (7 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}-\frac {1155 i \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(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{8 d (a+i a \tan (c+d x))^8}+\frac {1155 i \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(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{8 d (a+i a \tan (c+d x))^8}+\frac {\sec (c) \sec ^8(c+d x) \left (-\frac {236}{3} i \cos (8 c)+\frac {236}{3} \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) (-160 \cos (7 c)-160 i \sin (7 c)) (\cos (d x)+i \sin (d x))^8 \sin (d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {32}{3} \cos (5 c)+\frac {32}{3} i \sin (5 c)\right ) (\cos (d x)+i \sin (d x))^8 \sin (3 d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {1}{16} \cos (8 c)+\frac {1}{16} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4 (a+i a \tan (c+d x))^8}-\frac {\left (\frac {1}{96}+\frac {i}{96}\right ) \sec ^8(c+d x) \left (-407 i \cos \left (\frac {15 c}{2}\right )+343 \cos \left (\frac {17 c}{2}\right )+407 \sin \left (\frac {15 c}{2}\right )+343 i \sin \left (\frac {17 c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8}{d \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 (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (-\frac {1}{16} \cos (8 c)-\frac {1}{16} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4 (a+i a \tan (c+d x))^8}+\frac {\left (\frac {1}{96}+\frac {i}{96}\right ) \sec ^8(c+d x) \left (407 \cos \left (\frac {15 c}{2}\right )-343 i \cos \left (\frac {17 c}{2}\right )+407 i \sin \left (\frac {15 c}{2}\right )+343 \sin \left (\frac {17 c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^8}{d \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 (a+i a \tan (c+d x))^8}+\frac {236 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{3 d \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 ) (a+i a \tan (c+d x))^8}+\frac {4 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{3 d \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 (a+i a \tan (c+d x))^8}+\frac {4 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (-\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{3 d \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 (a+i a \tan (c+d x))^8}+\frac {236 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (-\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{3 d \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 ) (a+i a \tan (c+d x))^8} \]
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Time = 1.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {160 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{8} d}+\frac {32 i {\mathrm e}^{-3 i \left (d x +c \right )}}{3 a^{8} d}-\frac {i \left (1545 \,{\mathrm e}^{7 i \left (d x +c \right )}+5153 \,{\mathrm e}^{5 i \left (d x +c \right )}+5855 \,{\mathrm e}^{3 i \left (d x +c \right )}+2295 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \,a^{8} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {1155 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 a^{8} d}-\frac {1155 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 a^{8} d}\) | \(147\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {1}{4}-\frac {4 i}{3}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {121}{16}-2 i\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {123}{16}+38 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {1155 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {256}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {256}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \left (\frac {121}{16}-2 i\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\frac {1}{4}+\frac {4 i}{3}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {123}{16}-38 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1155 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a^{8} d}\) | \(219\) |
default | \(\frac {\frac {2 \left (\frac {1}{4}-\frac {4 i}{3}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {2 \left (-\frac {121}{16}-2 i\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2 \left (-\frac {123}{16}+38 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {1155 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {256}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {256}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \left (\frac {121}{16}-2 i\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2 \left (\frac {1}{4}+\frac {4 i}{3}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2 \left (-\frac {123}{16}-38 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1155 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a^{8} d}\) | \(219\) |
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Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.30 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {3465 \, {\left (e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, e^{\left (5 i \, d x + 5 i \, c\right )} + e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3465 \, {\left (e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, e^{\left (5 i \, d x + 5 i \, c\right )} + e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 6930 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 25410 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 33726 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 18414 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 2816 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 256 i}{24 \, {\left (a^{8} d e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, a^{8} d e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, a^{8} d e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, a^{8} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{8} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
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\[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\int \frac {\sec ^{13}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (179) = 358\).
Time = 0.39 (sec) , antiderivative size = 786, normalized size of antiderivative = 3.83 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \]
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Time = 1.88 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.95 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {3465 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {3465 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {1024 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}} - \frac {2 \, {\left (369 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1728 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 393 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5568 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 393 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5696 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 369 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1856 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a^{8}}}{24 \, d} \]
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Time = 8.50 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {\frac {33847\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6\,a^8}-\frac {12041\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,a^8}-\frac {3585\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{a^8}+\frac {3505\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4\,a^8}+\frac {4293\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,27565{}\mathrm {i}}{12\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,4575{}\mathrm {i}}{a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,25993{}\mathrm {i}}{6\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,5639{}\mathrm {i}}{3\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,1147{}\mathrm {i}}{4\,a^8}-\frac {1360{}\mathrm {i}}{3\,a^8}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,7{}\mathrm {i}+13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,18{}\mathrm {i}-22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,22{}\mathrm {i}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,13{}\mathrm {i}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}+1\right )}+\frac {1155\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^8\,d} \]
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