Integrand size = 18, antiderivative size = 61 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \sin ^7(a+b x)}{7 b}-\frac {128 \sin ^9(a+b x)}{3 b}+\frac {384 \sin ^{11}(a+b x)}{11 b}-\frac {128 \sin ^{13}(a+b x)}{13 b} \]
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Time = 0.08 (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, 2644, 276} \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {128 \sin ^{13}(a+b x)}{13 b}+\frac {384 \sin ^{11}(a+b x)}{11 b}-\frac {128 \sin ^9(a+b x)}{3 b}+\frac {128 \sin ^7(a+b x)}{7 b} \]
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Rule 276
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 128 \int \cos ^7(a+b x) \sin ^6(a+b x) \, dx \\ & = \frac {128 \text {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {128 \text {Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {128 \sin ^7(a+b x)}{7 b}-\frac {128 \sin ^9(a+b x)}{3 b}+\frac {384 \sin ^{11}(a+b x)}{11 b}-\frac {128 \sin ^{13}(a+b x)}{13 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \sin ^7(a+b x)}{7 b}-\frac {128 \sin ^9(a+b x)}{3 b}+\frac {384 \sin ^{11}(a+b x)}{11 b}-\frac {128 \sin ^{13}(a+b x)}{13 b} \]
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Time = 8.56 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {128 \left (\frac {\sin \left (x b +a \right )^{13}}{13}-\frac {3 \sin \left (x b +a \right )^{11}}{11}+\frac {\sin \left (x b +a \right )^{9}}{3}-\frac {\sin \left (x b +a \right )^{7}}{7}\right )}{b}\) | \(47\) |
risch | \(\frac {5 \sin \left (x b +a \right )}{8 b}-\frac {\sin \left (13 x b +13 a \right )}{416 b}-\frac {\sin \left (11 x b +11 a \right )}{352 b}+\frac {\sin \left (9 x b +9 a \right )}{48 b}+\frac {3 \sin \left (7 x b +7 a \right )}{112 b}-\frac {3 \sin \left (5 x b +5 a \right )}{32 b}-\frac {5 \sin \left (3 x b +3 a \right )}{32 b}\) | \(97\) |
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Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {128 \, {\left (231 \, \cos \left (b x + a\right )^{12} - 567 \, \cos \left (b x + a\right )^{10} + 371 \, \cos \left (b x + a\right )^{8} - 5 \, \cos \left (b x + a\right )^{6} - 6 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 16\right )} \sin \left (b x + a\right )}{3003 \, b} \]
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Timed out. \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.31 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {231 \, \sin \left (13 \, b x + 13 \, a\right ) + 273 \, \sin \left (11 \, b x + 11 \, a\right ) - 2002 \, \sin \left (9 \, b x + 9 \, a\right ) - 2574 \, \sin \left (7 \, b x + 7 \, a\right ) + 9009 \, \sin \left (5 \, b x + 5 \, a\right ) + 15015 \, \sin \left (3 \, b x + 3 \, a\right ) - 60060 \, \sin \left (b x + a\right )}{96096 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {128 \, {\left (231 \, \sin \left (b x + a\right )^{13} - 819 \, \sin \left (b x + a\right )^{11} + 1001 \, \sin \left (b x + a\right )^{9} - 429 \, \sin \left (b x + a\right )^{7}\right )}}{3003 \, b} \]
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Time = 19.97 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {-\frac {128\,{\sin \left (a+b\,x\right )}^{13}}{13}+\frac {384\,{\sin \left (a+b\,x\right )}^{11}}{11}-\frac {128\,{\sin \left (a+b\,x\right )}^9}{3}+\frac {128\,{\sin \left (a+b\,x\right )}^7}{7}}{b} \]
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