Integrand size = 29, antiderivative size = 227 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=3 a b^2 x-\frac {3 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d} \]
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Time = 0.48 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2972, 3126, 3110, 3102, 2814, 3855} \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {3 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}+3 a b^2 x-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d} \]
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Rule 2814
Rule 2972
Rule 3102
Rule 3110
Rule 3126
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x))^3 \left (24 a^2+3 a b \sin (c+d x)-\left (20 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{20 a^2} \\ & = \frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^2 \left (81 a^2 b-6 a \left (2 a^2-b^2\right ) \sin (c+d x)-3 b \left (28 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{60 a^2} \\ & = \frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (-6 a^2 \left (4 a^2-29 b^2\right )-3 a b \left (37 a^2-2 b^2\right ) \sin (c+d x)-3 b^2 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2} \\ & = -\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)+3 b^3 \left (83 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2} \\ & = -\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {\int \csc (c+d x) \left (45 a^2 b \left (3 a^2-4 b^2\right )+360 a^3 b^2 \sin (c+d x)\right ) \, dx}{120 a^2} \\ & = 3 a b^2 x-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d}+\frac {1}{8} \left (3 b \left (3 a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx \\ & = 3 a b^2 x-\frac {3 b \left (3 a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{8 d}-\frac {b^3 \left (83 a^2+2 b^2\right ) \cos (c+d x)}{40 a^2 d}-\frac {a \left (4 a^2-29 b^2\right ) \cot (c+d x)}{20 d}+\frac {27 b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac {2 \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^4}{20 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^4}{5 a d} \\ \end{align*}
Time = 2.13 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.78 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {960 a b^2 c+960 a b^2 d x-320 b^3 \cos (c+d x)-32 \left (a^3-20 a b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+150 a^2 b \csc ^2\left (\frac {1}{2} (c+d x)\right )-40 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 a^2 b \csc ^4\left (\frac {1}{2} (c+d x)\right )-360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+480 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-480 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-150 a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right )+40 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+15 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-112 a^3 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+320 a b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+64 a^3 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+7 a^3 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-20 a b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-a^3 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+32 a^3 \tan \left (\frac {1}{2} (c+d x)\right )-640 a b^2 \tan \left (\frac {1}{2} (c+d x)\right )}{320 d} \]
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Time = 0.61 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(192\) |
default | \(\frac {-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(192\) |
parallelrisch | \(\frac {\left (5760 a^{2} b -7680 b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} \left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )-90 b \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {2 \cos \left (2 d x +2 c \right )}{3}+\frac {5 \cos \left (3 d x +3 c \right )}{3}+\frac {\cos \left (4 d x +4 c \right )}{6}+\frac {1}{2}\right ) a^{2} \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-640 a \,b^{2} \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (3 d x +3 c \right )-960 b^{3} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )+\frac {3 \cos \left (2 d x +2 c \right )}{4}-\frac {\cos \left (3 d x +3 c \right )}{3}-\frac {3}{4}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15360 a \,b^{2} d x}{5120 d}\) | \(257\) |
risch | \(3 a \,b^{2} x -\frac {b^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {160 i a \,b^{2}-40 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-75 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+20 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+240 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+30 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-40 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-80 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-8 i a^{3}-720 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-30 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+40 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-560 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+880 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+75 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-20 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(367\) |
norman | \(\frac {-\frac {a^{3}}{160 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {\left (18 a^{2} b -23 b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {\left (63 a^{2} b -88 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {\left (117 a^{2} b -172 b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+3 a \,b^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9 a \,b^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9 a \,b^{2} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \,b^{2} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 a \left (a^{2}-90 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {3 a \left (a^{2}-90 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a \left (a^{2}-10 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a \left (a^{2}-10 b^{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a \left (a^{2}+120 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a \left (a^{2}+120 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 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 (15 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {b \left (15 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 {3 b \left (3 a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(490\) |
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Time = 0.29 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.47 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {560 \, a b^{2} \cos \left (d x + c\right )^{3} + 16 \, {\left (a^{3} - 20 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 240 \, a b^{2} \cos \left (d x + c\right ) + 15 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b - 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 10 \, {\left (24 \, a b^{2} d x \cos \left (d x + c\right )^{4} - 8 \, b^{3} \cos \left (d x + c\right )^{5} - 48 \, a b^{2} d x \cos \left (d x + c\right )^{2} + 24 \, a b^{2} d x - 5 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.80 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {80 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a b^{2} - 15 \, a^{2} b {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 20 \, b^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \]
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Time = 0.43 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 960 \, {\left (d x + c\right )} a b^{2} + 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {640 \, b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 120 \, {\left (3 \, a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {822 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1096 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{320 \, d} \]
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Time = 14.54 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.44 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Too large to display} \]
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