Integrand size = 29, antiderivative size = 173 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15 a^3 x}{16}-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2951, 3855, 3852, 8, 2718, 2715, 2713} \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin ^5(c+d x) \cos (c+d x)}{2 d}-\frac {11 a^3 \sin ^3(c+d x) \cos (c+d x)}{8 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {15 a^3 x}{16} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2951
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (3 a^9 \csc (c+d x)+a^9 \csc ^2(c+d x)-8 a^9 \sin (c+d x)-6 a^9 \sin ^2(c+d x)+6 a^9 \sin ^3(c+d x)+8 a^9 \sin ^4(c+d x)-3 a^9 \sin ^6(c+d x)-a^9 \sin ^7(c+d x)\right ) \, dx}{a^6} \\ & = a^3 \int \csc ^2(c+d x) \, dx-a^3 \int \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (3 a^3\right ) \int \sin ^6(c+d x) \, dx-\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (6 a^3\right ) \int \sin ^3(c+d x) \, dx-\left (8 a^3\right ) \int \sin (c+d x) \, dx+\left (8 a^3\right ) \int \sin ^4(c+d x) \, dx \\ & = -\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{d}-\frac {2 a^3 \cos (c+d x) \sin ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{2} \left (5 a^3\right ) \int \sin ^4(c+d x) \, dx-\left (3 a^3\right ) \int 1 \, dx+\left (6 a^3\right ) \int \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {a^3 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (6 a^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -3 a^3 x-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{8} \left (15 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int 1 \, dx \\ & = -\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d}-\frac {1}{16} \left (15 a^3\right ) \int 1 \, dx \\ & = -\frac {15 a^3 x}{16}-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {11 a^3 \cos (c+d x) \sin ^3(c+d x)}{8 d}+\frac {a^3 \cos (c+d x) \sin ^5(c+d x)}{2 d} \\ \end{align*}
Time = 6.74 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {(a+a \sin (c+d x))^3 \left (-2100 (c+d x)+9065 \cos (c+d x)+875 \cos (3 (c+d x))+49 \cos (5 (c+d x))-5 \cos (7 (c+d x))-1120 \cot \left (\frac {1}{2} (c+d x)\right )-6720 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+6720 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+455 \sin (2 (c+d x))+245 \sin (4 (c+d x))+35 \sin (6 (c+d x))+1120 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{2240 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Time = 0.47 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(-\frac {a^{3} \left (4200 d x \sin \left (d x +c \right )-13440 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )+4025 \cos \left (d x +c \right )-19968 \sin \left (d x +c \right )-8190 \sin \left (2 d x +2 c \right )-826 \sin \left (4 d x +4 c \right )-54 \sin \left (6 d x +6 c \right )+35 \cos \left (7 d x +7 c \right )+5 \sin \left (8 d x +8 c \right )+210 \cos \left (5 d x +5 c \right )+210 \cos \left (3 d x +3 c \right )\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{8960 d}\) | \(149\) |
derivativedivides | \(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) | \(180\) |
default | \(\frac {-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+3 a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) | \(180\) |
risch | \(-\frac {15 a^{3} x}{16}+\frac {13 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}+\frac {259 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {259 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}-\frac {13 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{64 d}+\frac {7 a^{3} \cos \left (5 d x +5 c \right )}{320 d}+\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {25 a^{3} \cos \left (3 d x +3 c \right )}{64 d}\) | \(225\) |
norman | \(\frac {\frac {16 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {136 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {176 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3}}{2 d}-\frac {9 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {21 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {29 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {29 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {9 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {15 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {105 a^{3} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {315 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {525 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {525 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {315 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {105 a^{3} x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {15 a^{3} x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {72 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {576 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {232 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {312 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(454\) |
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Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {280 \, a^{3} \cos \left (d x + c\right )^{7} - 70 \, a^{3} \cos \left (d x + c\right )^{5} - 175 \, a^{3} \cos \left (d x + c\right )^{3} + 840 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 840 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 525 \, a^{3} \cos \left (d x + c\right ) + {\left (80 \, a^{3} \cos \left (d x + c\right )^{7} - 336 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, a^{3} \cos \left (d x + c\right )^{3} + 525 \, a^{3} d x - 1680 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{560 \, d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.08 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {320 \, a^{3} \cos \left (d x + c\right )^{7} - 224 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 280 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{2240 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.68 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {525 \, {\left (d x + c\right )} a^{3} - 1680 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {280 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 4480 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 980 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 20160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 38080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 49280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32256 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 980 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12992 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 525 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2496 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{560 \, d} \]
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Time = 10.87 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.48 \[ \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {15\,a^3\,\mathrm {atan}\left (\frac {225\,a^6}{64\,\left (\frac {45\,a^6}{4}+\frac {225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}-\frac {45\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {45\,a^6}{4}+\frac {225\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}+\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{4}-32\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-144\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\frac {111\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{4}-272\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+35\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-352\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {113\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}-\frac {1152\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+28\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {464\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {624\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+a^3}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
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