Optimal. Leaf size=15 \[ -\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Rubi [A]
time = 0.01, 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 = {5428, 2718}
\begin {gather*} -\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 5428
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^3} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\cosh \left (a+\frac {b}{x^2}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {\cosh \left (a +\frac {b}{x^{2}}\right )}{2 b}\) | \(14\) |
default | \(-\frac {\cosh \left (a +\frac {b}{x^{2}}\right )}{2 b}\) | \(14\) |
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 | \(\frac {\sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (\frac {b}{x^{2}}\right )}{\sqrt {\pi }}\right )}{2 b}-\frac {\sinh \left (a \right ) \sinh \left (\frac {b}{x^{2}}\right )}{2 b}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\cosh \left (a + \frac {b}{x^{2}}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 1.13 \begin {gather*} -\frac {\cosh \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.65, size = 22, normalized size = 1.47 \begin {gather*} \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x^{2}} \right )}}{2 b} & \text {for}\: b \neq 0 \\- \frac {\sinh {\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (13) = 26\).
time = 0.42, size = 31, normalized size = 2.07 \begin {gather*} -\frac {e^{\left (\frac {a x^{2} + b}{x^{2}}\right )} + e^{\left (-\frac {a x^{2} + b}{x^{2}}\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\mathrm {cosh}\left (a+\frac {b}{x^2}\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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