Integrand size = 29, antiderivative size = 210 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (3 a^2+10 b^2\right ) \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d} \]
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Time = 0.37 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2990, 2687, 14, 4451, 466, 1828, 1171, 393, 205, 212} \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (3 a^2+10 b^2\right ) \text {arctanh}(\cos (c+d x))}{256 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}-\frac {2 a b \cot ^7(c+d x)}{7 d} \]
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Rule 14
Rule 205
Rule 212
Rule 393
Rule 466
Rule 1171
Rule 1828
Rule 2687
Rule 2990
Rule 4451
Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^5(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 )^6} \, dx,x,\cos (c+d x)\right )}{d}+\frac {(2 a b) \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\text {Subst}\left (\int \frac {a^2+10 a^2 x^2+10 a^2 x^4-10 b^2 x^6}{\left (1-x^2\right )^5} \, dx,x,\cos (c+d x)\right )}{10 d}+\frac {(2 a b) \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}-\frac {\text {Subst}\left (\int \frac {13 a^2-10 b^2+80 \left (a^2-b^2\right ) x^2-80 b^2 x^4}{\left (1-x^2\right )^4} \, dx,x,\cos (c+d x)\right )}{80 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\text {Subst}\left (\int \frac {5 \left (3 a^2-22 b^2\right )-480 b^2 x^2}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{480 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\left (3 a^2+10 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{128 d} \\ & = -\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d}+\frac {\left (3 a^2+10 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{256 d} \\ & = \frac {\left (3 a^2+10 b^2\right ) \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d} \\ \end{align*}
Time = 1.78 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {-80640 \left (3 a^2+10 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+80640 \left (3 a^2+10 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^{10}(c+d x) \left (630 \left (1879 a^2+290 b^2\right ) \cos (c+d x)+1260 \left (519 a^2-62 b^2\right ) \cos (3 (c+d x))+218484 a^2 \cos (5 (c+d x))-24360 b^2 \cos (5 (c+d x))+9135 a^2 \cos (7 (c+d x))-77070 b^2 \cos (7 (c+d x))-945 a^2 \cos (9 (c+d x))-3150 b^2 \cos (9 (c+d x))+537600 a b \sin (2 (c+d x))+522240 a b \sin (4 (c+d x))+207360 a b \sin (6 (c+d x))+25600 a b \sin (8 (c+d x))-2560 a b \sin (10 (c+d x))\right )}{20643840 d} \]
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Time = 0.72 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.48
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(311\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(311\) |
parallelrisch | \(\frac {5040 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-315 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15120 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-50400 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}+630 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-3360 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-10080 b^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30240 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+30240 a b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-5040 b^{2} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+630 a^{2} \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13440 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1260 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+126 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}-126 a^{2} \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2520 a^{2} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10080 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}-13440 a b \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3360 b^{2} \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 b^{2} \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2520 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1260 a^{2} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 a^{2} \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2160 a b \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-560 a b \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+560 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -2160 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{1290240 d}\) | \(441\) |
risch | \(-\frac {-9135 a^{2} {\mathrm e}^{17 i \left (d x +c \right )}-829440 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-92160 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-51200 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+322560 i a b \,{\mathrm e}^{16 i \left (d x +c \right )}+1075200 i a b \,{\mathrm e}^{12 i \left (d x +c \right )}+215040 i a b \,{\mathrm e}^{14 i \left (d x +c \right )}-645120 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}-218484 a^{2} {\mathrm e}^{15 i \left (d x +c \right )}+24360 b^{2} {\mathrm e}^{15 i \left (d x +c \right )}-653940 a^{2} {\mathrm e}^{13 i \left (d x +c \right )}+78120 b^{2} {\mathrm e}^{13 i \left (d x +c \right )}+5120 i a b +945 a^{2} {\mathrm e}^{i \left (d x +c \right )}-9135 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-1183770 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-182700 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-1183770 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}-182700 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+945 a^{2} {\mathrm e}^{19 i \left (d x +c \right )}+77070 b^{2} {\mathrm e}^{17 i \left (d x +c \right )}+3150 b^{2} {\mathrm e}^{19 i \left (d x +c \right )}-653940 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}+78120 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-218484 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+24360 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+77070 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+3150 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{128 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{128 d}\) | \(484\) |
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Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (194) = 388\).
Time = 0.39 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.17 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 420 \, {\left (21 \, a^{2} - 58 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 5376 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 2940 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right ) - 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a b \cos \left (d x + c\right )^{9} - 9 \, a b \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]
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none
Time = 0.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.30 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, b^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a b}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (194) = 388\).
Time = 0.47 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.23 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 630 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2160 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3360 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 13440 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10080 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5040 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44286 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 147620 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10080 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13440 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5040 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3360 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 630 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]
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Time = 11.95 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.88 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{256}+\frac {5\,b^2}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (2\,a^2+4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+\frac {8\,b^2}{3}\right )+\frac {a^2}{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (a^2+8\,b^2\right )-\frac {12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{7}+\frac {32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-24\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9}\right )}{1024\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{512}+\frac {b^2}{256}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{1024}+\frac {b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2}{2048}+\frac {b^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^2}{4096}-\frac {b^2}{2048}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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