Integrand size = 18, antiderivative size = 31 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2644, 14} \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \]
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Rule 14
Rule 2644
Rule 4373
Rubi steps \begin{align*} \text {integral}& = 8 \int \cos ^3(a+b x) \sin ^2(a+b x) \, dx \\ & = \frac {8 \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {8 \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \]
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Time = 0.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {-\frac {8 \sin \left (x b +a \right )^{5}}{5}+\frac {8 \sin \left (x b +a \right )^{3}}{3}}{b}\) | \(27\) |
risch | \(\frac {\sin \left (x b +a \right )}{b}-\frac {\sin \left (5 x b +5 a \right )}{10 b}-\frac {\sin \left (3 x b +3 a \right )}{6 b}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8 \, {\left (3 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{15 \, b} \]
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Timed out. \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {3 \, \sin \left (5 \, b x + 5 \, a\right ) + 5 \, \sin \left (3 \, b x + 3 \, a\right ) - 30 \, \sin \left (b x + a\right )}{30 \, b} \]
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Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=-\frac {8 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )}}{15 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx=\frac {8\,\left (5\,{\sin \left (a+b\,x\right )}^3-3\,{\sin \left (a+b\,x\right )}^5\right )}{15\,b} \]
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