Integrand size = 12, antiderivative size = 25 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=-\frac {1}{2} \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{2} \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5427, 5425, 5424} \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=-\frac {1}{2} \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{2} \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \]
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Rule 5424
Rule 5425
Rule 5427
Rubi steps \begin{align*} \text {integral}& = \cosh (a) \int \frac {\cosh \left (\frac {b}{x^2}\right )}{x} \, dx+\sinh (a) \int \frac {\sinh \left (\frac {b}{x^2}\right )}{x} \, dx \\ & = -\frac {1}{2} \cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\frac {1}{2} \sinh (a) \text {Shi}\left (\frac {b}{x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=\frac {1}{2} \left (-\cosh (a) \text {Chi}\left (\frac {b}{x^2}\right )-\sinh (a) \text {Shi}\left (\frac {b}{x^2}\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {{\mathrm e}^{2 a} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (-\frac {b}{x^{2}}\right )}{4}+\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (\frac {b}{x^{2}}\right )}{4}\) | \(33\) |
meijerg | \(-\frac {\sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {2 \gamma -4 \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}+\frac {2 \,\operatorname {Chi}\left (\frac {b}{x^{2}}\right )-2 \ln \left (\frac {b}{x^{2}}\right )-2 \gamma }{\sqrt {\pi }}\right )}{4}-\frac {\operatorname {Shi}\left (\frac {b}{x^{2}}\right ) \sinh \left (a \right )}{2}\) | \(62\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=-\frac {1}{4} \, {\left ({\rm Ei}\left (\frac {b}{x^{2}}\right ) + {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \cosh \left (a\right ) - \frac {1}{4} \, {\left ({\rm Ei}\left (\frac {b}{x^{2}}\right ) - {\rm Ei}\left (-\frac {b}{x^{2}}\right )\right )} \sinh \left (a\right ) \]
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\[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=\int \frac {\cosh {\left (a + \frac {b}{x^{2}} \right )}}{x}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=-\frac {1}{4} \, {\rm Ei}\left (-\frac {b}{x^{2}}\right ) e^{\left (-a\right )} - \frac {1}{4} \, {\rm Ei}\left (\frac {b}{x^{2}}\right ) e^{a} \]
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\[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=\int { \frac {\cosh \left (a + \frac {b}{x^{2}}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x} \, dx=-\frac {\mathrm {cosh}\left (a\right )\,\mathrm {coshint}\left (\frac {b}{x^2}\right )}{2}-\frac {\mathrm {sinh}\left (a\right )\,\mathrm {sinhint}\left (\frac {b}{x^2}\right )}{2} \]
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