Integrand size = 29, antiderivative size = 236 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a b \cot (c+d x)}{5 d}-\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d} \]
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Time = 0.43 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2972, 3126, 3110, 3100, 2827, 3852, 8, 3855} \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}-\frac {2 a b \cot (c+d x)}{5 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d} \]
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Rule 8
Rule 2827
Rule 2972
Rule 3100
Rule 3110
Rule 3126
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (35 a^2-6 b^2+2 a b \sin (c+d x)-3 \left (10 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{30 a^2} \\ & = \frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}-\frac {\int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (6 b \left (13 a^2-2 b^2\right )-a \left (15 a^2-2 b^2\right ) \sin (c+d x)-b \left (85 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2} \\ & = \frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}+\frac {\int \csc ^3(c+d x) \left (3 \left (15 a^4-80 a^2 b^2+12 b^4\right )+144 a^3 b \sin (c+d x)+3 b^2 \left (85 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{360 a^2} \\ & = -\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}+\frac {\int \csc ^2(c+d x) \left (288 a^3 b+45 a^2 \left (a^2+6 b^2\right ) \sin (c+d x)\right ) \, dx}{720 a^2} \\ & = -\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}+\frac {1}{5} (2 a b) \int \csc ^2(c+d x) \, dx+\frac {1}{16} \left (a^2+6 b^2\right ) \int \csc (c+d x) \, dx \\ & = -\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d}-\frac {(2 a b) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{5 d} \\ & = -\frac {\left (a^2+6 b^2\right ) \text {arctanh}(\cos (c+d x))}{16 d}-\frac {2 a b \cot (c+d x)}{5 d}-\frac {\left (15 a^4-80 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{240 a^2 d}+\frac {b \left (13 a^2-2 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a d}+\frac {\left (35 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{120 a^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{10 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{6 a d} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.35 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {-384 a b \cot \left (\frac {1}{2} (c+d x)\right )-30 \left (a^2-10 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-720 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+720 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-300 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-30 a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac {1}{2} (c+d x)\right )-1344 a b \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+768 a b \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )-a \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+12 b \sin (c+d x))+6 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (5 a^2-5 b^2+14 a b \sin (c+d x)\right )+384 a b \tan \left (\frac {1}{2} (c+d x)\right )}{1920 d} \]
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Time = 0.50 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a b \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos ^{5}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a b \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}+b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(198\) |
parallelrisch | \(\frac {5 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 a^{2} \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 a b \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+30 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+15 a^{2} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30 b^{2} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 a b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+15 a^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-240 a b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+120 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{1920 d}\) | \(283\) |
risch | \(\frac {-480 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+15 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-150 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+96 i a b +235 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}+210 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+480 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+390 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-960 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}+390 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-60 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-96 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+235 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+210 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+960 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+15 a^{2} {\mathrm e}^{i \left (d x +c \right )}-150 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d}\) | \(344\) |
norman | \(\frac {-\frac {a^{2}}{384 d}+\frac {a^{2} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {\left (a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {\left (a^{2}-6 b^{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {\left (2 a^{2}+9 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {\left (2 a^{2}+9 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {\left (3 a^{2}+30 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {\left (3 a^{2}+30 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{80 d}+\frac {3 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {3 a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {a b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (a^{2}+6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(381\) |
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Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {192 \, a b \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 30 \, {\left (a^{2} - 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 80 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right ) - 15 \, {\left ({\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 6 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left ({\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 6 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 6 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.76 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 30 \, b^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {192 \, a b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.31 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, {\left (a^{2} + 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {294 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1764 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
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Time = 11.14 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^2}{16}+\frac {3\,b^2}{8}\right )}{d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}-b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+8\,b^2\right )-\frac {a^2}{6}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}\right )}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{128}+\frac {b^2}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{128}-\frac {b^2}{64}\right )}{d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{80\,d}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
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