Integrand size = 12, antiderivative size = 15 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5429, 2717} \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Rule 2717
Rule 5429
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \cosh (a+b x) \, dx,x,\frac {1}{x^2}\right )\right ) \\ & = -\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {\sinh \left (a +\frac {b}{x^{2}}\right )}{2 b}\) | \(14\) |
default | \(-\frac {\sinh \left (a +\frac {b}{x^{2}}\right )}{2 b}\) | \(14\) |
parallelrisch | \(-\frac {\sinh \left (\frac {a \,x^{2}+b}{x^{2}}\right )}{2 b}\) | \(18\) |
risch | \(-\frac {{\mathrm e}^{\frac {a \,x^{2}+b}{x^{2}}}}{4 b}+\frac {{\mathrm e}^{-\frac {a \,x^{2}+b}{x^{2}}}}{4 b}\) | \(37\) |
meijerg | \(-\frac {\cosh \left (a \right ) \sinh \left (\frac {b}{x^{2}}\right )}{2 b}+\frac {\sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }}\right )}{2 b}\) | \(40\) |
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Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\sinh \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b} \]
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Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=\begin {cases} - \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{2 b} & \text {for}\: b \neq 0 \\- \frac {\cosh {\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} - e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, b} \]
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Time = 1.53 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx=-\frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{2\,b} \]
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