Integrand size = 18, antiderivative size = 61 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256 \cos ^9(a+b x)}{9 b}+\frac {768 \cos ^{11}(a+b x)}{11 b}-\frac {768 \cos ^{13}(a+b x)}{13 b}+\frac {256 \cos ^{15}(a+b x)}{15 b} \]
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Time = 0.07 (sec) , antiderivative size = 61, 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} \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {256 \cos ^{15}(a+b x)}{15 b}-\frac {768 \cos ^{13}(a+b x)}{13 b}+\frac {768 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^9(a+b x)}{9 b} \]
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Rule 276
Rule 2645
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 256 \int \cos ^8(a+b x) \sin ^7(a+b x) \, dx \\ & = -\frac {256 \text {Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {256 \text {Subst}\left (\int \left (x^8-3 x^{10}+3 x^{12}-x^{14}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {256 \cos ^9(a+b x)}{9 b}+\frac {768 \cos ^{11}(a+b x)}{11 b}-\frac {768 \cos ^{13}(a+b x)}{13 b}+\frac {256 \cos ^{15}(a+b x)}{15 b} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {35 \cos (a+b x)}{64 b}-\frac {35 \cos (3 (a+b x))}{192 b}+\frac {21 \cos (5 (a+b x))}{320 b}+\frac {3 \cos (7 (a+b x))}{64 b}-\frac {7 \cos (9 (a+b x))}{576 b}-\frac {7 \cos (11 (a+b x))}{704 b}+\frac {\cos (13 (a+b x))}{832 b}+\frac {\cos (15 (a+b x))}{960 b} \]
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Time = 19.92 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\frac {256 \cos \left (x b +a \right )^{15}}{15}-\frac {768 \cos \left (x b +a \right )^{13}}{13}+\frac {768 \cos \left (x b +a \right )^{11}}{11}-\frac {256 \cos \left (x b +a \right )^{9}}{9}}{b}\) | \(47\) |
risch | \(-\frac {35 \cos \left (x b +a \right )}{64 b}+\frac {\cos \left (15 x b +15 a \right )}{960 b}+\frac {\cos \left (13 x b +13 a \right )}{832 b}-\frac {7 \cos \left (11 x b +11 a \right )}{704 b}-\frac {7 \cos \left (9 x b +9 a \right )}{576 b}+\frac {3 \cos \left (7 x b +7 a \right )}{64 b}+\frac {21 \cos \left (5 x b +5 a \right )}{320 b}-\frac {35 \cos \left (3 x b +3 a \right )}{192 b}\) | \(111\) |
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {256 \, {\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \]
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Timed out. \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {429 \, \cos \left (15 \, b x + 15 \, a\right ) + 495 \, \cos \left (13 \, b x + 13 \, a\right ) - 4095 \, \cos \left (11 \, b x + 11 \, a\right ) - 5005 \, \cos \left (9 \, b x + 9 \, a\right ) + 19305 \, \cos \left (7 \, b x + 7 \, a\right ) + 27027 \, \cos \left (5 \, b x + 5 \, a\right ) - 75075 \, \cos \left (3 \, b x + 3 \, a\right ) - 225225 \, \cos \left (b x + a\right )}{411840 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (53) = 106\).
Time = 0.30 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.43 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {8192 \, {\left (\frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {105 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {455 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {5070 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {30030 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {70070 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {115830 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {109395 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac {75075 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} + \frac {27027 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} + \frac {6435 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{11}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{11}} - 1\right )}}{6435 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{15}} \]
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Time = 20.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {-\frac {256\,{\cos \left (a+b\,x\right )}^{15}}{15}+\frac {768\,{\cos \left (a+b\,x\right )}^{13}}{13}-\frac {768\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {256\,{\cos \left (a+b\,x\right )}^9}{9}}{b} \]
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