Integrand size = 15, antiderivative size = 23 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d} \]
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Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4482, 3265, 212} \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d} \]
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Rule 212
Rule 3265
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin (c+d x)}{1+\sin ^2(c+d x)} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {arctanh}\left (\frac {\cos (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\cos (c)-(-i+\sin (c)) \tan \left (\frac {d x}{2}\right )}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\cos (c)-(i+\sin (c)) \tan \left (\frac {d x}{2}\right )}{\sqrt {2}}\right )}{\sqrt {2} d} \]
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Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{2 d}\) | \(21\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{2 d}\) | \(21\) |
risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-2 \sqrt {2}\, {\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+2 \sqrt {2}\, {\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\sqrt {2} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - 2}\right )}{4 \, d} \]
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\[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=\int \frac {1}{\sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 176, normalized size of antiderivative = 7.65 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\sqrt {2} \log \left (-\frac {2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) - \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} - 3}{2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sqrt {2} + 3}\right ) + \sqrt {2} \log \left (-\frac {2 \, {\left (\sqrt {2} - 1\right )} \cos \left (d x + c\right ) - \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} - 3}{2 \, {\left (\sqrt {2} + 1\right )} \cos \left (d x + c\right ) + \cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sqrt {2} + 3}\right )}{8 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 6 \right |}}\right )}{4 \, d} \]
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Time = 23.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\csc (c+d x)+\sin (c+d x)} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {2\,\sqrt {2}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1}\right )}{2\,d} \]
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