Integrand size = 21, antiderivative size = 175 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 a^3 x}{2}-\frac {25 a^3 \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2788, 3855, 3852, 8, 3853, 2715, 2713} \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {25 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {13 a^3 x}{2} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2788
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (8 a^9+6 a^9 \csc (c+d x)-6 a^9 \csc ^2(c+d x)-8 a^9 \csc ^3(c+d x)+3 a^9 \csc ^5(c+d x)+a^9 \csc ^6(c+d x)-3 a^9 \sin ^2(c+d x)-a^9 \sin ^3(c+d x)\right ) \, dx}{a^6} \\ & = 8 a^3 x+a^3 \int \csc ^6(c+d x) \, dx-a^3 \int \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (6 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = 8 a^3 x-\frac {6 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{4} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^3\right ) \int \csc (c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^3 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (6 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = \frac {13 a^3 x}{2}-\frac {2 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {13 a^3 x}{2}-\frac {25 a^3 \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 6.62 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.55 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (6240 (c+d x)+720 \cos (c+d x)-80 \cos (3 (c+d x))+2624 \cot \left (\frac {1}{2} (c+d x)\right )+690 \csc ^2\left (\frac {1}{2} (c+d x)\right )-45 \csc ^4\left (\frac {1}{2} (c+d x)\right )-3000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-690 \sec ^2\left (\frac {1}{2} (c+d x)\right )+45 \sec ^4\left (\frac {1}{2} (c+d x)\right )+304 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-19 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-3 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+720 \sin (2 (c+d x))-2624 \tan \left (\frac {1}{2} (c+d x)\right )+6 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Time = 0.51 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.28
method | result | size |
parallelrisch | \(\frac {a^{3} \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-15000 \left (\sin \left (3 d x +3 c \right )-\frac {\sin \left (5 d x +5 c \right )}{5}-2 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-31200 d x \sin \left (3 d x +3 c \right )+6240 d x \sin \left (5 d x +5 c \right )+62400 d x \sin \left (d x +c \right )+7440 \sin \left (2 d x +2 c \right )-11525 \sin \left (3 d x +3 c \right )-7360 \sin \left (4 d x +4 c \right )+2305 \sin \left (5 d x +5 c \right )+560 \sin \left (6 d x +6 c \right )-40 \sin \left (8 d x +8 c \right )+8200 \cos \left (d x +c \right )-17960 \cos \left (3 d x +3 c \right )+7048 \cos \left (5 d x +5 c \right )-360 \cos \left (7 d x +7 c \right )+23050 \sin \left (d x +c \right )\right )}{491520 d}\) | \(224\) |
risch | \(\frac {13 a^{3} x}{2}-\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {a^{3} \left (-720 i {\mathrm e}^{8 i \left (d x +c \right )}+345 \,{\mathrm e}^{9 i \left (d x +c \right )}+2880 i {\mathrm e}^{6 i \left (d x +c \right )}-330 \,{\mathrm e}^{7 i \left (d x +c \right )}-3680 i {\mathrm e}^{4 i \left (d x +c \right )}+2560 i {\mathrm e}^{2 i \left (d x +c \right )}+330 \,{\mathrm e}^{3 i \left (d x +c \right )}-656 i-345 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {25 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {25 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(268\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(287\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(287\) |
norman | \(\frac {\frac {4 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3}}{160 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {17 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {31 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {201 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {657 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {657 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {201 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {31 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {17 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {3 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {13 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {39 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {39 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {333 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {461 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {25 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(388\) |
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Time = 0.28 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.59 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {360 \, a^{3} \cos \left (d x + c\right )^{7} - 2392 \, a^{3} \cos \left (d x + c\right )^{5} + 3640 \, a^{3} \cos \left (d x + c\right )^{3} - 1560 \, a^{3} \cos \left (d x + c\right ) + 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 156 \, a^{3} d x \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{5} + 312 \, a^{3} d x \cos \left (d x + c\right )^{2} + 125 \, a^{3} \cos \left (d x + c\right )^{3} - 156 \, a^{3} d x - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.39 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {20 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + 16 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 45 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6240 \, {\left (d x + c\right )} a^{3} + 3000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {320 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac {6850 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
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Time = 10.57 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.33 \[ \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {25\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}+\frac {13\,a^3\,\mathrm {atan}\left (\frac {169\,a^6}{\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}+\frac {-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {769\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {373\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {1744\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {589\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6}+\frac {402\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {a^3}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}-\frac {43\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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