Integrand size = 7, antiderivative size = 15 \[ \int \sec (2 x) \sin (x) \, dx=\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4442, 213} \[ \int \sec (2 x) \sin (x) \, dx=\frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{\sqrt {2}} \]
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Rule 213
Rule 4442
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cos (x)\right ) \\ & = \frac {\text {arctanh}\left (\sqrt {2} \cos (x)\right )}{\sqrt {2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(15)=30\).
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \sec (2 x) \sin (x) \, dx=\frac {\text {arctanh}\left (\sqrt {2}-\tan \left (\frac {x}{2}\right )\right )+\text {arctanh}\left (\sqrt {2}+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {2}} \]
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Time = 0.46 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\cos \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{2}\) | \(13\) |
risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{4}\) | \(47\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20 \[ \int \sec (2 x) \sin (x) \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) \]
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\[ \int \sec (2 x) \sin (x) \, dx=\int \frac {\sin {\left (x \right )}}{\cos {\left (2 x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 8.60 \[ \int \sec (2 x) \sin (x) \, dx=\frac {1}{8} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \left (x\right ) + 2 \, {\left (\sqrt {2} \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sin \left (2 \, x\right ) \sin \left (x\right ) - 2 \, {\left (\sqrt {2} \cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 2 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.27 \[ \int \sec (2 x) \sin (x) \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \sec (2 x) \sin (x) \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\cos \left (x\right )\right )}{2} \]
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