Integrand size = 29, antiderivative size = 178 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {135 a^4 x}{16}+\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d} \]
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Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2951, 3855, 3852, 8, 3853, 2715, 2713} \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {4 a^4 \cot (c+d x)}{d}+\frac {a^4 \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {23 a^4 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {89 a^4 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {135 a^4 x}{16} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-14 a^{10}-8 a^{10} \csc (c+d x)+3 a^{10} \csc ^2(c+d x)+4 a^{10} \csc ^3(c+d x)+a^{10} \csc ^4(c+d x)+14 a^{10} \sin ^2(c+d x)+8 a^{10} \sin ^3(c+d x)-3 a^{10} \sin ^4(c+d x)-4 a^{10} \sin ^5(c+d x)-a^{10} \sin ^6(c+d x)\right ) \, dx}{a^6} \\ & = -14 a^4 x+a^4 \int \csc ^4(c+d x) \, dx-a^4 \int \sin ^6(c+d x) \, dx+\left (3 a^4\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^4\right ) \int \sin ^4(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \sin ^5(c+d x) \, dx-\left (8 a^4\right ) \int \csc (c+d x) \, dx+\left (8 a^4\right ) \int \sin ^3(c+d x) \, dx+\left (14 a^4\right ) \int \sin ^2(c+d x) \, dx \\ & = -14 a^4 x+\frac {8 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {7 a^4 \cos (c+d x) \sin (c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{6} \left (5 a^4\right ) \int \sin ^4(c+d x) \, dx+\left (2 a^4\right ) \int \csc (c+d x) \, dx-\frac {1}{4} \left (9 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (7 a^4\right ) \int 1 \, dx-\frac {a^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^4\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (4 a^4\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (8 a^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -7 a^4 x+\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {47 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{8} \left (5 a^4\right ) \int \sin ^2(c+d x) \, dx-\frac {1}{8} \left (9 a^4\right ) \int 1 \, dx \\ & = -\frac {65 a^4 x}{8}+\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 a^4\right ) \int 1 \, dx \\ & = -\frac {135 a^4 x}{16}+\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d} \\ \end{align*}
Time = 7.20 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.29 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (1+\sin (c+d x))^4 \left (-8100 (c+d x)-3360 \cos (c+d x)+240 \cos (3 (c+d x))+48 \cos (5 (c+d x))-1760 \cot \left (\frac {1}{2} (c+d x)\right )-480 \csc ^2\left (\frac {1}{2} (c+d x)\right )+5760 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5760 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 \sec ^2\left (\frac {1}{2} (c+d x)\right )+320 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-20 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-2415 \sin (2 (c+d x))-135 \sin (4 (c+d x))+5 \sin (6 (c+d x))+1760 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \]
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Time = 0.59 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(\frac {a^{4} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-34560 \left (\sin \left (d x +c \right )-\frac {\sin \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48600 d x \sin \left (d x +c \right )+16200 d x \sin \left (3 d x +3 c \right )-35712 \sin \left (d x +c \right )-15072 \sin \left (2 d x +2 c \right )+11904 \sin \left (3 d x +3 c \right )+3936 \sin \left (4 d x +4 c \right )-96 \sin \left (6 d x +6 c \right )-48 \sin \left (8 d x +8 c \right )-14295 \cos \left (d x +c \right )+13875 \cos \left (3 d x +3 c \right )-1995 \cos \left (5 d x +5 c \right )-150 \cos \left (7 d x +7 c \right )+5 \cos \left (9 d x +9 c \right )\right )}{61440 d}\) | \(200\) |
risch | \(-\frac {135 a^{4} x}{16}+\frac {a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{40 d}-\frac {161 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {7 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{4 d}-\frac {7 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{4 d}-\frac {9 i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{128 d}+\frac {9 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{128 d}+\frac {a^{4} {\mathrm e}^{-5 i \left (d x +c \right )}}{40 d}+\frac {161 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {2 a^{4} \left (-9 i {\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i {\mathrm e}^{2 i \left (d x +c \right )}-11 i-6 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{4 d}\) | \(292\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(320\) |
default | \(\frac {a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(320\) |
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Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.38 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {40 \, a^{4} \cos \left (d x + c\right )^{9} - 390 \, a^{4} \cos \left (d x + c\right )^{7} - 405 \, a^{4} \cos \left (d x + c\right )^{5} + 2700 \, a^{4} \cos \left (d x + c\right )^{3} - 2025 \, a^{4} \cos \left (d x + c\right ) - 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (64 \, a^{4} \cos \left (d x + c\right )^{7} - 64 \, a^{4} \cos \left (d x + c\right )^{5} - 675 \, a^{4} d x \cos \left (d x + c\right )^{2} - 320 \, a^{4} \cos \left (d x + c\right )^{3} + 675 \, a^{4} d x + 480 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.65 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {128 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 320 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 720 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 160 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4}}{960 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.82 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2025 \, {\left (d x + c\right )} a^{4} - 1440 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 450 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {10 \, {\left (264 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3840 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4608 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 768 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 10.67 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.66 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {6\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {135\,a^4\,\mathrm {atan}\left (\frac {18225\,a^8}{64\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {405\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}-\frac {-74\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-\frac {346\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+280\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+153\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+572\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+379\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+592\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {1312\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1836\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+184\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {376\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+17\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+160\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {15\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
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