Integrand size = 18, antiderivative size = 46 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64 \cos ^7(a+b x)}{7 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^{11}(a+b x)}{11 b} \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2645, 276} \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64 \cos ^{11}(a+b x)}{11 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^7(a+b x)}{7 b} \]
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Rule 276
Rule 2645
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 64 \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx \\ & = -\frac {64 \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \cos ^7(a+b x)}{7 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^{11}(a+b x)}{11 b} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.93 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {5 \cos (a+b x)}{8 b}-\frac {5 \cos (3 (a+b x))}{24 b}+\frac {\cos (5 (a+b x))}{16 b}+\frac {5 \cos (7 (a+b x))}{112 b}-\frac {\cos (9 (a+b x))}{144 b}-\frac {\cos (11 (a+b x))}{176 b} \]
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Time = 6.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {64 \left (\frac {\cos \left (x b +a \right )^{11}}{11}-\frac {2 \cos \left (x b +a \right )^{9}}{9}+\frac {\cos \left (x b +a \right )^{7}}{7}\right )}{b}\) | \(37\) |
risch | \(-\frac {5 \cos \left (x b +a \right )}{8 b}-\frac {\cos \left (11 x b +11 a \right )}{176 b}-\frac {\cos \left (9 x b +9 a \right )}{144 b}+\frac {5 \cos \left (7 x b +7 a \right )}{112 b}+\frac {\cos \left (5 x b +5 a \right )}{16 b}-\frac {5 \cos \left (3 x b +3 a \right )}{24 b}\) | \(83\) |
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64 \, {\left (63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}\right )}}{693 \, b} \]
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Timed out. \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.50 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {63 \, \cos \left (11 \, b x + 11 \, a\right ) + 77 \, \cos \left (9 \, b x + 9 \, a\right ) - 495 \, \cos \left (7 \, b x + 7 \, a\right ) - 693 \, \cos \left (5 \, b x + 5 \, a\right ) + 2310 \, \cos \left (3 \, b x + 3 \, a\right ) + 6930 \, \cos \left (b x + a\right )}{11088 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (40) = 80\).
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 4.43 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {1024 \, {\left (\frac {11 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {55 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {297 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {2079 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac {2541 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac {462 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - 1\right )}}{693 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{11}} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64\,\left (63\,{\cos \left (a+b\,x\right )}^{11}-154\,{\cos \left (a+b\,x\right )}^9+99\,{\cos \left (a+b\,x\right )}^7\right )}{693\,b} \]
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