Integrand size = 18, antiderivative size = 46 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^5(a+b x)}{5 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^9(a+b x)}{9 b} \]
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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, 2644, 276} \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^5(a+b x)}{5 b} \]
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Rule 276
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 32 \int \cos ^5(a+b x) \sin ^4(a+b x) \, dx \\ & = \frac {32 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {32 \sin ^5(a+b x)}{5 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^5(a+b x)}{5 b}-\frac {64 \sin ^7(a+b x)}{7 b}+\frac {32 \sin ^9(a+b x)}{9 b} \]
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Time = 2.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\frac {32 \sin \left (x b +a \right )^{9}}{9}-\frac {64 \sin \left (x b +a \right )^{7}}{7}+\frac {32 \sin \left (x b +a \right )^{5}}{5}}{b}\) | \(37\) |
risch | \(\frac {3 \sin \left (x b +a \right )}{4 b}+\frac {\sin \left (9 x b +9 a \right )}{72 b}+\frac {\sin \left (7 x b +7 a \right )}{56 b}-\frac {\sin \left (5 x b +5 a \right )}{10 b}-\frac {\sin \left (3 x b +3 a \right )}{6 b}\) | \(69\) |
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Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (35 \, \cos \left (b x + a\right )^{8} - 50 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{315 \, b} \]
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Timed out. \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {35 \, \sin \left (9 \, b x + 9 \, a\right ) + 45 \, \sin \left (7 \, b x + 7 \, a\right ) - 252 \, \sin \left (5 \, b x + 5 \, a\right ) - 420 \, \sin \left (3 \, b x + 3 \, a\right ) + 1890 \, \sin \left (b x + a\right )}{2520 \, b} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (35 \, \sin \left (b x + a\right )^{9} - 90 \, \sin \left (b x + a\right )^{7} + 63 \, \sin \left (b x + a\right )^{5}\right )}}{315 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32\,\left (35\,{\sin \left (a+b\,x\right )}^9-90\,{\sin \left (a+b\,x\right )}^7+63\,{\sin \left (a+b\,x\right )}^5\right )}{315\,b} \]
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