Integrand size = 12, antiderivative size = 13 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\sinh \left (a+\frac {b}{x}\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5429, 2717} \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\sinh \left (a+\frac {b}{x}\right )}{b} \]
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Rule 2717
Rule 5429
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \cosh (a+b x) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\sinh \left (a+\frac {b}{x}\right )}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\sinh \left (a+\frac {b}{x}\right )}{b} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {\sinh \left (a +\frac {b}{x}\right )}{b}\) | \(14\) |
default | \(-\frac {\sinh \left (a +\frac {b}{x}\right )}{b}\) | \(14\) |
parallelrisch | \(-\frac {\sinh \left (\frac {a x +b}{x}\right )}{b}\) | \(16\) |
risch | \(-\frac {{\mathrm e}^{\frac {a x +b}{x}}}{2 b}+\frac {{\mathrm e}^{-\frac {a x +b}{x}}}{2 b}\) | \(33\) |
meijerg | \(-\frac {\cosh \left (a \right ) \sinh \left (\frac {b}{x}\right )}{b}+\frac {\sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{b}\) | \(39\) |
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\sinh \left (\frac {a x + b}{x}\right )}{b} \]
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Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=\begin {cases} - \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cosh {\left (a \right )}}{x} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\sinh \left (a + \frac {b}{x}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (13) = 26\).
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.23 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {e^{\left (\frac {a x + b}{x}\right )} - e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b} \]
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Time = 1.55 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^2} \, dx=-\frac {\mathrm {sinh}\left (a+\frac {b}{x}\right )}{b} \]
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