Integrand size = 11, antiderivative size = 15 \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\frac {\text {arcsinh}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4441, 221} \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\frac {\text {arcsinh}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
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Rule 221
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {1+2 x^2}} \, dx,x,\sinh (x)\right ) \\ & = \frac {\text {arcsinh}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\frac {\text {arcsinh}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(12)=24\).
Time = 0.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 4.20
method | result | size |
default | \(\frac {\sqrt {\left (2 \left (\cosh ^{2}\left (x \right )\right )-1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \ln \left (\sqrt {2}\, \left (\sinh ^{2}\left (x \right )\right )+\frac {\sqrt {2}}{4}+\sqrt {2 \left (\sinh ^{4}\left (x \right )\right )+\sinh ^{2}\left (x \right )}\right ) \sqrt {2}}{4 \sinh \left (x \right ) \sqrt {2 \left (\cosh ^{2}\left (x \right )\right )-1}}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (12) = 24\).
Time = 0.26 (sec) , antiderivative size = 482, normalized size of antiderivative = 32.13 \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\frac {1}{8} \, \sqrt {2} \log \left (-\frac {\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 3 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \, {\left (14 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 5 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} - 15 \, \cosh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} - 4\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 4\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} - 6 \, \cosh \left (x\right )^{3} + 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} - 4 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4}{\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right )^{5} \sinh \left (x\right ) + 15 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{2} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6}}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) \]
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\[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\int \frac {\cosh {\left (x \right )}}{\sqrt {\cosh {\left (2 x \right )}}}\, dx \]
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\[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\int { \frac {\cosh \left (x\right )}{\sqrt {\cosh \left (2 \, x\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.87 \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (\log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \]
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Timed out. \[ \int \frac {\cosh (x)}{\sqrt {\cosh (2 x)}} \, dx=\int \frac {\mathrm {cosh}\left (x\right )}{\sqrt {\mathrm {cosh}\left (2\,x\right )}} \,d x \]
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