Integrand size = 27, antiderivative size = 97 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]
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Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 90} \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{10}(c+d x)}{10 d}+\frac {2 a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d} \]
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Rule 12
Rule 90
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^{12} (a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {a^7 \text {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^7 \text {Subst}\left (\int \left (\frac {a^5}{x^{12}}+\frac {a^4}{x^{11}}-\frac {2 a^3}{x^{10}}-\frac {2 a^2}{x^9}+\frac {a}{x^8}+\frac {1}{x^7}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d} \]
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Time = 0.43 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(68\) |
default | \(-\frac {a \left (\frac {\left (\csc ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\csc ^{10}\left (d x +c \right )\right )}{10}-\frac {2 \left (\csc ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\csc ^{8}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}\right )}{d}\) | \(68\) |
parallelrisch | \(-\frac {a \left (\sec ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (947200+1126400 \cos \left (2 d x +2 c \right )-2541 \sin \left (9 d x +9 c \right )+12705 \sin \left (7 d x +7 c \right )+257565 \sin \left (5 d x +5 c \right )+366366 \sin \left (d x +c \right )+371910 \sin \left (3 d x +3 c \right )+506880 \cos \left (4 d x +4 c \right )+231 \sin \left (11 d x +11 c \right )\right )}{58133053440 d}\) | \(116\) |
risch | \(\frac {32 a \left (1980 i {\mathrm e}^{15 i \left (d x +c \right )}+1155 \,{\mathrm e}^{16 i \left (d x +c \right )}+4400 i {\mathrm e}^{13 i \left (d x +c \right )}+1155 \,{\mathrm e}^{14 i \left (d x +c \right )}+7400 i {\mathrm e}^{11 i \left (d x +c \right )}+1848 \,{\mathrm e}^{12 i \left (d x +c \right )}+4400 i {\mathrm e}^{9 i \left (d x +c \right )}-1848 \,{\mathrm e}^{10 i \left (d x +c \right )}+1980 i {\mathrm e}^{7 i \left (d x +c \right )}-1155 \,{\mathrm e}^{8 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{3465 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}\) | \(147\) |
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Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.32 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1980 \, a \cos \left (d x + c\right )^{4} - 880 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{4} - 5 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 160 \, a}{13860 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]
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Time = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]
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Time = 9.59 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.72 \[ \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]
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