Integrand size = 18, antiderivative size = 28 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4373, 2672, 327, 212} \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b} \]
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Rule 212
Rule 327
Rule 2672
Rule 4373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \sin (a+b x) \tan (a+b x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b} \\ & = -\frac {\sin (a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b} \\ & = \frac {\text {arctanh}(\sin (a+b x))}{2 b}-\frac {\sin (a+b x)}{2 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {1}{2} \left (\frac {\text {arctanh}(\sin (a+b x))}{b}-\frac {\sin (a+b x)}{b}\right ) \]
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Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {-\sin \left (x b +a \right )+\ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{2 b}\) | \(29\) |
risch | \(\frac {i {\mathrm e}^{i \left (x b +a \right )}}{4 b}-\frac {i {\mathrm e}^{-i \left (x b +a \right )}}{4 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{2 b}+\frac {\ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{2 b}\) | \(68\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, b} \]
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Timed out. \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.43 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=-\frac {\log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + 2 \, \sin \left (b x + a\right )}{4 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=\frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx=-\frac {\frac {\sin \left (a+b\,x\right )}{2}-\frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2}}{b} \]
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