Integrand size = 29, antiderivative size = 183 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b \left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d} \]
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Time = 0.39 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2968, 3127, 3126, 3110, 3100, 2827, 3852, 8, 3855} \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b \left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d} \]
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Rule 8
Rule 2827
Rule 2968
Rule 3100
Rule 3110
Rule 3126
Rule 3127
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {1}{20} \int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (-2 \left (2 a^2-3 b^2\right )-11 a b \sin (c+d x)-13 b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{60} \int \csc ^3(c+d x) \left (9 b \left (5 a^2-2 b^2\right )+4 a \left (2 a^2+15 b^2\right ) \sin (c+d x)+39 b^3 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{120} \int \csc ^2(c+d x) \left (8 a \left (2 a^2+15 b^2\right )+15 b \left (3 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{8} \left (b \left (3 a^2+4 b^2\right )\right ) \int \csc (c+d x) \, dx-\frac {1}{15} \left (a \left (2 a^2+15 b^2\right )\right ) \int \csc ^2(c+d x) \, dx \\ & = \frac {b \left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {\left (a \left (2 a^2+15 b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{15 d} \\ & = \frac {b \left (3 a^2+4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d} \\ \end{align*}
Time = 1.63 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.88 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {32 \left (2 a^3+15 a b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+30 \left (3 a^2 b-4 b^3\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+480 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-480 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-90 a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right )+120 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+45 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-16 a^3 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+960 a b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-3 a^3 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+a \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (-45 a b+\left (a^2-60 b^2\right ) \sin (c+d x)\right )-64 a^3 \tan \left (\frac {1}{2} (c+d x)\right )-480 a b^2 \tan \left (\frac {1}{2} (c+d x)\right )+6 a^3 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{960 d} \]
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Time = 0.40 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+3 a^{2} b \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+b^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )+3 a^{2} b \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+b^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
parallelrisch | \(\frac {-6 a^{3} \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 a^{2} b \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -10 a^{3} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 a \,b^{2} \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-120 b^{3} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+60 a^{3} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+360 a \,b^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-60 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-360 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -480 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{960 d}\) | \(260\) |
risch | \(\frac {-240 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+60 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+16 i a^{3}+360 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-270 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-120 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-720 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+120 i a \,b^{2}-240 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-80 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+270 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+120 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+480 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-80 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+45 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-60 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(341\) |
norman | \(\frac {-\frac {a^{3}}{160 d}+\frac {a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {\left (3 a^{2} b +7 b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (9 a^{2} b +24 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {\left (15 a^{2} b +44 b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {a \left (7 a^{2}+30 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}+\frac {a \left (7 a^{2}+30 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a \left (7 a^{2}+30 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a \left (7 a^{2}+30 b^{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{240 d}-\frac {3 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {3 a^{2} b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {b \left (9 a^{2}+8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {b \left (9 a^{2}+8 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {b \left (3 a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(412\) |
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Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.50 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {16 \, {\left (2 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left ({\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b + 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left ({\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b + 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.86 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {45 \, a^{2} b {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 60 \, b^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a b^{2}}{\tan \left (d x + c\right )^{3}} + \frac {16 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{240 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.58 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {822 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1096 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
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Time = 9.82 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.32 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3}{96}+\frac {a\,b^2}{8}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{8}+\frac {b^3}{2}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{3}+4\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^3+12\,a\,b^2\right )+\frac {a^3}{5}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{32\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{16}+\frac {3\,a\,b^2}{8}\right )}{d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d} \]
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