Integrand size = 29, antiderivative size = 238 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-a^3 x+\frac {125 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]
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Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2952, 3554, 8, 2691, 3855, 2687, 30, 3853} \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {125 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]
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Rule 8
Rule 30
Rule 2687
Rule 2691
Rule 2952
Rule 3554
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^4(c+d x) \, dx-\frac {1}{2} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \, dx+\frac {1}{8} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx-a^3 \int 1 \, dx \\ & = -a^3 x+\frac {15 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx \\ & = -a^3 x+\frac {125 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.17 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-215040 c-215040 d x-118784 \cot \left (\frac {1}{2} (c+d x)\right )-108780 \csc ^2\left (\frac {1}{2} (c+d x)\right )+210000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-210000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+108780 \sec ^2\left (\frac {1}{2} (c+d x)\right )-17010 \sec ^4\left (\frac {1}{2} (c+d x)\right )+700 \sec ^6\left (\frac {1}{2} (c+d x)\right )+105 \sec ^8\left (\frac {1}{2} (c+d x)\right )+71936 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (17010-4496 \sin (c+d x))-15 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (7+24 \sin (c+d x))+4 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (-175+732 \sin (c+d x))+118784 \tan \left (\frac {1}{2} (c+d x)\right )-5856 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+720 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{215040 d} \]
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Time = 0.46 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(-\frac {3805 \left (\frac {1228800 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{761}+\left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {1817 \cos \left (3 d x +3 c \right )}{3805}+\frac {1861 \cos \left (5 d x +5 c \right )}{3805}-\frac {777 \cos \left (7 d x +7 c \right )}{3805}\right ) \left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {3072 \left (\sec ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{25}+\frac {29 \cos \left (5 d x +5 c \right )}{75}-\frac {29 \cos \left (7 d x +7 c \right )}{525}\right ) \left (\csc ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{761}+\frac {6291456 d x}{3805}\right ) a^{3}}{6291456 d}\) | \(153\) |
risch | \(-a^{3} x +\frac {a^{3} \left (27195 \,{\mathrm e}^{15 i \left (d x +c \right )}-65135 \,{\mathrm e}^{13 i \left (d x +c \right )}+63595 \,{\mathrm e}^{11 i \left (d x +c \right )}+161280 i {\mathrm e}^{12 i \left (d x +c \right )}-133175 \,{\mathrm e}^{9 i \left (d x +c \right )}-286720 i {\mathrm e}^{10 i \left (d x +c \right )}-133175 \,{\mathrm e}^{7 i \left (d x +c \right )}+519680 i {\mathrm e}^{8 i \left (d x +c \right )}+63595 \,{\mathrm e}^{5 i \left (d x +c \right )}-544768 i {\mathrm e}^{6 i \left (d x +c \right )}-65135 \,{\mathrm e}^{3 i \left (d x +c \right )}+254464 i {\mathrm e}^{4 i \left (d x +c \right )}+27195 \,{\mathrm e}^{i \left (d x +c \right )}-118784 i {\mathrm e}^{2 i \left (d x +c \right )}+14848 i\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\) | \(232\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(296\) |
default | \(\frac {a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{16 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(296\) |
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Time = 0.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.52 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {26880 \, a^{3} d x \cos \left (d x + c\right )^{8} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{6} - 54390 \, a^{3} \cos \left (d x + c\right )^{7} + 161280 \, a^{3} d x \cos \left (d x + c\right )^{4} + 127750 \, a^{3} \cos \left (d x + c\right )^{5} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{2} - 96250 \, a^{3} \cos \left (d x + c\right )^{3} + 26880 \, a^{3} d x + 26250 \, a^{3} \cos \left (d x + c\right ) - 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 256 \, {\left (116 \, a^{3} \cos \left (d x + c\right )^{7} - 406 \, a^{3} \cos \left (d x + c\right )^{5} + 350 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1792 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 35 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {11520 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.27 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 215040 \, {\left (d x + c\right )} a^{3} - 210000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {570750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{215040 \, d} \]
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Time = 12.65 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.63 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {23\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {17\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {128\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+125\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{125\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-128\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {125\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{128\,d}-\frac {73\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {73\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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