Integrand size = 20, antiderivative size = 31 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b} \]
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Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2645, 14} \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {64 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^7(a+b x)}{7 b} \]
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Rule 14
Rule 2645
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 64 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx \\ & = -\frac {64 \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {32 \cos ^7(a+b x) (-11+7 \cos (2 (a+b x)))}{63 b} \]
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Time = 23.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\frac {64 \cos \left (x b +a \right )^{9}}{9}-\frac {64 \cos \left (x b +a \right )^{7}}{7}}{b}\) | \(27\) |
risch | \(-\frac {3 \cos \left (x b +a \right )}{2 b}+\frac {\cos \left (9 x b +9 a \right )}{36 b}+\frac {3 \cos \left (7 x b +7 a \right )}{28 b}-\frac {2 \cos \left (3 x b +3 a \right )}{3 b}\) | \(55\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {64 \, {\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \]
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Timed out. \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\text {Timed out} \]
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none
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{252 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 5.87 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {256 \, {\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {27 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {315 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {105 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {63 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - 1\right )}}{63 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{9}} \]
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Time = 20.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64\,\left (9\,{\cos \left (a+b\,x\right )}^7-7\,{\cos \left (a+b\,x\right )}^9\right )}{63\,b} \]
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