Integrand size = 27, antiderivative size = 138 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \]
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Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2687, 14, 2691, 3853, 3855} \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rule 14
Rule 2687
Rule 2691
Rule 2917
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx \\ & = -\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} (5 a) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a \text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} (5 a) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a \text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} (5 a) \int \csc ^3(c+d x) \, dx \\ & = -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} (5 a) \int \csc (c+d x) \, dx \\ & = \frac {5 a \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(138)=276\).
Time = 0.19 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.18 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {2 a \cot (c+d x)}{63 d}+\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {15 a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {7 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{63 d}-\frac {5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac {19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {15 a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {7 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d} \]
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Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(166\) |
default | \(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(166\) |
parallelrisch | \(-\frac {\left (180 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {9 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {27 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-12 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+36 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-54 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\left (-54+24 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {27 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {9 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )+36 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{4608 d}\) | \(220\) |
risch | \(-\frac {a \left (315 \,{\mathrm e}^{17 i \left (d x +c \right )}+80640 i {\mathrm e}^{10 i \left (d x +c \right )}+8022 \,{\mathrm e}^{15 i \left (d x +c \right )}+16128 i {\mathrm e}^{14 i \left (d x +c \right )}+10458 \,{\mathrm e}^{13 i \left (d x +c \right )}+6912 i {\mathrm e}^{4 i \left (d x +c \right )}+18270 \,{\mathrm e}^{11 i \left (d x +c \right )}+48384 i {\mathrm e}^{8 i \left (d x +c \right )}+2304 i {\mathrm e}^{2 i \left (d x +c \right )}-18270 \,{\mathrm e}^{7 i \left (d x +c \right )}+48384 i {\mathrm e}^{6 i \left (d x +c \right )}-10458 \,{\mathrm e}^{5 i \left (d x +c \right )}+26880 i {\mathrm e}^{12 i \left (d x +c \right )}-8022 \,{\mathrm e}^{3 i \left (d x +c \right )}-256 i-315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4032 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}\) | \(232\) |
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Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (124) = 248\).
Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.88 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {512 \, a \cos \left (d x + c\right )^{9} - 2304 \, a \cos \left (d x + c\right )^{7} + 315 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a \cos \left (d x + c\right )^{7} + 73 \, a \cos \left (d x + c\right )^{5} - 55 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {21 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {256 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{16128 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (124) = 248\).
Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.86 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {28 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 108 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 504 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1512 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1512 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 672 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 28 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]
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Time = 10.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.07 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d} \]
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