Integrand size = 14, antiderivative size = 23 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3400, 3382} \[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rule 3382
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x} \, dx \\ & = \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\sqrt {a (1+\cosh (x))} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \]
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\[\int \frac {\sqrt {a +a \cosh \left (x \right )}}{x}d x\]
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Exception generated. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (x \right )} + 1\right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x} \,d x } \]
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\[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (x\right )}}{x} \,d x \]
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