Integrand size = 7, antiderivative size = 26 \[ \int \csc (4 x) \sin (x) \, dx=-\frac {1}{4} \text {arctanh}(\sin (x))+\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1107, 213} \[ \int \csc (4 x) \sin (x) \, dx=\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \text {arctanh}(\sin (x)) \]
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Rule 213
Rule 1107
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sin (x)\right )-2 \text {Subst}\left (\int \frac {1}{-4+8 x^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {1}{4} \text {arctanh}(\sin (x))+\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \csc (4 x) \sin (x) \, dx=-\frac {1}{4} \text {arctanh}(\sin (x))+\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
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Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\ln \left (\sin \left (x \right )-1\right )}{8}-\frac {\ln \left (\sin \left (x \right )+1\right )}{8}+\frac {\operatorname {arctanh}\left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{4}\) | \(28\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\ln \left (i+{\mathrm e}^{i x}\right )}{4}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \csc (4 x) \sin (x) \, dx=\frac {1}{8} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (22) = 44\).
Time = 3.24 (sec) , antiderivative size = 294, normalized size of antiderivative = 11.31 \[ \int \csc (4 x) \sin (x) \, dx=\frac {27720 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {39202 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {39202 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 6.58 \[ \int \csc (4 x) \sin (x) \, dx=\frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \csc (4 x) \sin (x) \, dx=-\frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \csc (4 x) \sin (x) \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{4}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \]
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