Integrand size = 29, antiderivative size = 180 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {15}{4} a b x+\frac {\left (5 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (2 a^2-b^2\right ) \cos (c+d x)}{d}-\frac {\left (a^2-b^2\right ) \cos ^3(c+d x)}{3 d}+\frac {b^2 \cos ^5(c+d x)}{5 d}-\frac {15 a b \cot (c+d x)}{4 d}+\frac {5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}+\frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d} \]
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Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2990, 2671, 294, 327, 209, 466, 1824, 212} \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (5 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (a^2-b^2\right ) \cos ^3(c+d x)}{3 d}-\frac {\left (2 a^2-b^2\right ) \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {15 a b \cot (c+d x)}{4 d}+\frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}+\frac {5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}-\frac {15 a b x}{4}+\frac {b^2 \cos ^5(c+d x)}{5 d} \]
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Rule 209
Rule 212
Rule 294
Rule 327
Rule 466
Rule 1824
Rule 2671
Rule 2990
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cos ^4(c+d x) \cot ^2(c+d x) \, dx+\int \cos ^3(c+d x) \cot ^3(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {x^6 \left (a^2+b^2-b^2 x^2\right )}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {(2 a b) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {\text {Subst}\left (\int \frac {a^2+2 a^2 x^2+2 a^2 x^4-2 b^2 x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac {(5 a b) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = \frac {5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}+\frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {\text {Subst}\left (\int \left (-2 \left (2 a^2-b^2\right )-2 \left (a^2-b^2\right ) x^2+2 b^2 x^4+\frac {5 a^2-2 b^2}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{2 d}-\frac {(15 a b) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{4 d} \\ & = -\frac {\left (2 a^2-b^2\right ) \cos (c+d x)}{d}-\frac {\left (a^2-b^2\right ) \cos ^3(c+d x)}{3 d}+\frac {b^2 \cos ^5(c+d x)}{5 d}-\frac {15 a b \cot (c+d x)}{4 d}+\frac {5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}+\frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {(15 a b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{4 d}+\frac {\left (5 a^2-2 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d} \\ & = -\frac {15}{4} a b x+\frac {\left (5 a^2-2 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {\left (2 a^2-b^2\right ) \cos (c+d x)}{d}-\frac {\left (a^2-b^2\right ) \cos ^3(c+d x)}{3 d}+\frac {b^2 \cos ^5(c+d x)}{5 d}-\frac {15 a b \cot (c+d x)}{4 d}+\frac {5 a b \cos ^2(c+d x) \cot (c+d x)}{4 d}+\frac {a b \cos ^4(c+d x) \cot (c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d} \\ \end{align*}
Time = 6.53 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.39 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {15 a b (c+d x)}{4 d}-\frac {\left (18 a^2-11 b^2\right ) \cos (c+d x)}{8 d}-\frac {\left (4 a^2-7 b^2\right ) \cos (3 (c+d x))}{48 d}+\frac {b^2 \cos (5 (c+d x))}{80 d}-\frac {a b \cot \left (\frac {1}{2} (c+d x)\right )}{d}-\frac {a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {\left (5 a^2-2 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {\left (-5 a^2+2 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a b \sin (2 (c+d x))}{d}-\frac {a b \sin (4 (c+d x))}{16 d}+\frac {a b \tan \left (\frac {1}{2} (c+d x)\right )}{d} \]
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Time = 0.64 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+b^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(187\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+b^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(187\) |
parallelrisch | \(\frac {\left (-2400 a^{2}+960 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-250 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {71 \cos \left (2 d x +2 c \right )}{50}-\frac {\cos \left (3 d x +3 c \right )}{2}-\frac {\cos \left (5 d x +5 c \right )}{50}+\frac {71}{50}\right ) a^{2} \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1200 b \left (\cos \left (d x +c \right )-\frac {3 \cos \left (3 d x +3 c \right )}{16}-\frac {\cos \left (5 d x +5 c \right )}{80}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-3600 a b x d +1320 \cos \left (d x +c \right ) b^{2}+140 \cos \left (3 d x +3 c \right ) b^{2}+12 \cos \left (5 d x +5 c \right ) b^{2}+1472 b^{2}}{960 d}\) | \(198\) |
risch | \(-\frac {15 a b x}{4}-\frac {{\mathrm e}^{3 i \left (d x +c \right )} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{3 i \left (d x +c \right )} b^{2}}{96 d}-\frac {i a b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 d}-\frac {9 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {11 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {9 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 d}-\frac {i a \left (i a \,{\mathrm e}^{3 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}+4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {{\mathrm e}^{-3 i \left (d x +c \right )} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{-3 i \left (d x +c \right )} b^{2}}{96 d}+\frac {i a b \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{80 d}-\frac {a b \sin \left (4 d x +4 c \right )}{16 d}\) | \(351\) |
norman | \(\frac {\frac {a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{2}}{8 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (8 a^{2}-6 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (205 a^{2}-96 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (259 a^{2}-112 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (355 a^{2}-184 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {\left (805 a^{2}-448 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {17 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {10 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {10 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {17 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {15 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {75 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {75 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {75 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {75 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {15 a b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\left (5 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(419\) |
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Time = 0.36 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.36 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {12 \, b^{2} \cos \left (d x + c\right )^{7} - 225 \, a b d x \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 225 \, a b d x - 20 \, {\left (5 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (5 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) + 15 \, {\left ({\left (5 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 5 \, a^{2} + 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left ({\left (5 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 5 \, a^{2} + 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (2 \, a b \cos \left (d x + c\right )^{5} + 5 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.06 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {5 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 15 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a b - 2 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{2}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (164) = 328\).
Time = 0.40 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.92 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 450 \, {\left (d x + c\right )} a b + 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, {\left (5 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {15 \, {\left (30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {4 \, {\left (135 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 180 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 150 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 600 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 360 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 800 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 150 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 280 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 140 \, a^{2} + 92 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
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Time = 11.50 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.69 \[ \int \cos ^3(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {5\,a^2}{2}-b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {49\,a^2}{2}-24\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {165\,a^2}{2}-48\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {127\,a^2}{6}-\frac {184\,b^2}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {223\,a^2}{3}-\frac {112\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {335\,a^2}{3}-\frac {224\,b^2}{3}\right )+\frac {a^2}{2}+38\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+60\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+40\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-14\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {15\,a\,b\,\mathrm {atan}\left (\frac {225\,a^2\,b^2}{4\,\left (-\frac {75\,a^3\,b}{2}+\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2}{4}+15\,a\,b^3\right )}-\frac {15\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-\frac {75\,a^3\,b}{2}+\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2}{4}+15\,a\,b^3}+\frac {75\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (-\frac {75\,a^3\,b}{2}+\frac {225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2}{4}+15\,a\,b^3\right )}\right )}{2\,d}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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