Optimal. Leaf size=15 \[ -\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5321, 2637} \[ -\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5321
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx &=-\left (\frac {1}{2} \operatorname {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*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {\sinh \left (a+\frac {b}{x^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 17, normalized size = 1.13 \[ -\frac {\sinh \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 29, normalized size = 1.93 \[ -\frac {{\left (e^{\left (2 \, a + \frac {b}{x^{2}}\right )} - e^{\left (-\frac {b}{x^{2}}\right )}\right )} e^{\left (-a\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.93 \[ -\frac {\sinh \left (a +\frac {b}{x^{2}}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 13, normalized size = 0.87 \[ -\frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 13, normalized size = 0.87 \[ -\frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.91, size = 22, normalized size = 1.47 \[ \begin {cases} - \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{2 b} & \text {for}\: b \neq 0 \\- \frac {\cosh {\relax (a )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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