Integrand size = 29, antiderivative size = 286 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d} \]
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Time = 0.39 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2691, 3853, 3855, 2687, 276} \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]
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Rule 276
Rule 2687
Rule 2691
Rule 2952
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^6(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^7(c+d x)+a^3 \cot ^6(c+d x) \csc ^8(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^8(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx \\ & = -\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{2} a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac {1}{4} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}+\frac {1}{16} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {1}{8} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx+\frac {a^3 \text {Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx-\frac {1}{64} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{128} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx \\ & = -\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {1}{512} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx \\ & = \frac {3 a^3 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac {\left (15 a^3\right ) \int \csc (c+d x) \, dx}{1024} \\ & = \frac {27 a^3 \text {arctanh}(\cos (c+d x))}{1024 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{d}-\frac {6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac {a^3 \cot ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac {9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d} \\ \end{align*}
Time = 10.46 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (138378240 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{12}(c+d x) (-200294400-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))+43294720 \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x)))\right )}{5248122880 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Time = 0.90 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(-\frac {5 \left (\frac {8515584 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\left (\csc ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (11 d x +11 c \right )-\frac {\cos \left (13 d x +13 c \right )}{13}+484 \cos \left (d x +c \right )+\frac {957 \cos \left (3 d x +3 c \right )}{5}-\frac {121 \cos \left (5 d x +5 c \right )}{5}-\frac {198 \cos \left (7 d x +7 c \right )}{5}-6 \cos \left (9 d x +9 c \right )\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2079 \cos \left (11 d x +11 c \right )}{5120}+\frac {6102173 \cos \left (d x +c \right )}{2560}+\frac {16039023 \cos \left (3 d x +3 c \right )}{12800}+\frac {8551543 \cos \left (5 d x +5 c \right )}{25600}-\frac {614999 \cos \left (7 d x +7 c \right )}{25600}-\frac {4851 \cos \left (9 d x +9 c \right )}{1024}\right )\right ) a^{3}}{322961408 d}\) | \(201\) |
risch | \(-\frac {a^{3} \left (135135 \,{\mathrm e}^{25 i \left (d x +c \right )}-1711710 \,{\mathrm e}^{23 i \left (d x +c \right )}+820019200 i {\mathrm e}^{14 i \left (d x +c \right )}-6418412 \,{\mathrm e}^{21 i \left (d x +c \right )}-82001920 i {\mathrm e}^{20 i \left (d x +c \right )}+119165046 \,{\mathrm e}^{19 i \left (d x +c \right )}+123002880 i {\mathrm e}^{16 i \left (d x +c \right )}+305844539 \,{\mathrm e}^{17 i \left (d x +c \right )}-105431040 i {\mathrm e}^{8 i \left (d x +c \right )}+376267892 \,{\mathrm e}^{15 i \left (d x +c \right )}+41000960 i {\mathrm e}^{18 i \left (d x +c \right )}-23429120 i {\mathrm e}^{6 i \left (d x +c \right )}-376267892 \,{\mathrm e}^{11 i \left (d x +c \right )}+468582400 i {\mathrm e}^{12 i \left (d x +c \right )}-305844539 \,{\mathrm e}^{9 i \left (d x +c \right )}+386580480 i {\mathrm e}^{10 i \left (d x +c \right )}-119165046 \,{\mathrm e}^{7 i \left (d x +c \right )}-15974400 i {\mathrm e}^{4 i \left (d x +c \right )}+6418412 \,{\mathrm e}^{5 i \left (d x +c \right )}+2662400 i {\mathrm e}^{2 i \left (d x +c \right )}+1711710 \,{\mathrm e}^{3 i \left (d x +c \right )}-204800 i-135135 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2562560 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{13}}+\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{1024 d}-\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{1024 d}\) | \(318\) |
derivativedivides | \(\frac {a^{3} \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 )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \left (\cos ^{7}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
default | \(\frac {a^{3} \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 )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{7}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{7}\left (d x +c \right )}{64 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{384 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{1536 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{1024 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{1024}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3072}-\frac {5 \cos \left (d x +c \right )}{1024}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{1024}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {6 \left (\cos ^{7}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(444\) |
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Time = 0.31 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.46 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {409600 \, a^{3} \cos \left (d x + c\right )^{13} - 2662400 \, a^{3} \cos \left (d x + c\right )^{11} + 7321600 \, a^{3} \cos \left (d x + c\right )^{9} - 5857280 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 135135 \, {\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2002 \, {\left (135 \, a^{3} \cos \left (d x + c\right )^{11} - 765 \, a^{3} \cos \left (d x + c\right )^{9} + 758 \, a^{3} \cos \left (d x + c\right )^{7} + 1782 \, a^{3} \cos \left (d x + c\right )^{5} - 765 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{10250240 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15015 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12012 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {133120 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}} + \frac {10240 \, {\left (429 \, \tan \left (d x + c\right )^{6} + 1001 \, \tan \left (d x + c\right )^{4} + 819 \, \tan \left (d x + c\right )^{2} + 231\right )} a^{3}}{\tan \left (d x + c\right )^{13}}}{30750720 \, d} \]
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Time = 0.56 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.58 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {770 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2162160 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6875958 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1401400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 80080 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 450450 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 385385 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 150150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 40040 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 94380 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 65065 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20020 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8008 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11830 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5005 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 770 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13}}}{82001920 \, d} \]
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Time = 12.68 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {45\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}-\frac {77\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}-\frac {15\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {33\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}+\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {13\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8192\,d}+\frac {77\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16384\,d}+\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {33\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{28672\,d}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16384\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{90112\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{16384\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{106496\,d}-\frac {27\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {35\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,d}-\frac {35\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2048\,d} \]
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