Integrand size = 12, antiderivative size = 21 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=-\text {Chi}\left (\frac {b}{x}\right ) \sinh (a)-\cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 21, 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 = {5426, 5425, 5424} \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=\sinh (a) \left (-\text {Chi}\left (\frac {b}{x}\right )\right )-\cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \]
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Rule 5424
Rule 5425
Rule 5426
Rubi steps \begin{align*} \text {integral}& = \cosh (a) \int \frac {\sinh \left (\frac {b}{x}\right )}{x} \, dx+\sinh (a) \int \frac {\cosh \left (\frac {b}{x}\right )}{x} \, dx \\ & = -\text {Chi}\left (\frac {b}{x}\right ) \sinh (a)-\cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=-\text {Chi}\left (\frac {b}{x}\right ) \sinh (a)-\cosh (a) \text {Shi}\left (\frac {b}{x}\right ) \]
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Time = 0.94 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (\frac {b}{x}\right )}{2}+\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-\frac {b}{x}\right )}{2}\) | \(27\) |
meijerg | \(-\cosh \left (a \right ) \operatorname {Shi}\left (\frac {b}{x}\right )-\frac {\sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {2 \,\operatorname {Chi}\left (\frac {b}{x}\right )-2 \ln \left (\frac {b}{x}\right )-2 \gamma }{\sqrt {\pi }}+\frac {2 \gamma -2 \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{2}\) | \(62\) |
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none
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=-\frac {1}{2} \, {\left ({\rm Ei}\left (\frac {b}{x}\right ) - {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \left (a\right ) - \frac {1}{2} \, {\left ({\rm Ei}\left (\frac {b}{x}\right ) + {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \left (a\right ) \]
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Time = 0.46 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=- \sinh {\left (a \right )} \operatorname {Chi}\left (\frac {b}{x}\right ) - \cosh {\left (a \right )} \operatorname {Shi}{\left (\frac {b}{x} \right )} \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=\frac {1}{2} \, {\rm Ei}\left (-\frac {b}{x}\right ) e^{\left (-a\right )} - \frac {1}{2} \, {\rm Ei}\left (\frac {b}{x}\right ) e^{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.10 \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=\frac {b {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} - b {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{2 \, b} \]
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Timed out. \[ \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x} \, dx=-\mathrm {sinh}\left (a\right )\,\mathrm {coshint}\left (\frac {b}{x}\right )-\mathrm {cosh}\left (a\right )\,\mathrm {sinhint}\left (\frac {b}{x}\right ) \]
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