Optimal. Leaf size=29 \[ \frac {\cosh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5321, 3296, 2638} \[ \frac {\cosh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5321
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\operatorname {Subst}\left (\int x \cosh (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x}+\frac {\operatorname {Subst}\left (\int \sinh (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=\frac {\cosh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.00 \[ \frac {x \cosh \left (a+\frac {b}{x}\right )-b \sinh \left (a+\frac {b}{x}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 33, normalized size = 1.14 \[ \frac {x \cosh \left (\frac {a x + b}{x}\right ) - b \sinh \left (\frac {a x + b}{x}\right )}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 93, normalized size = 3.21 \[ \frac {a e^{\left (\frac {a x + b}{x}\right )} - a e^{\left (-\frac {a x + b}{x}\right )} - \frac {{\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} + \frac {{\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + e^{\left (\frac {a x + b}{x}\right )} + e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 44, normalized size = 1.52 \[ -\frac {\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )-a \sinh \left (a +\frac {b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 47, normalized size = 1.62 \[ \frac {1}{4} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (3, \frac {b}{x}\right )}{b^{3}} + \frac {e^{a} \Gamma \left (3, -\frac {b}{x}\right )}{b^{3}}\right )} - \frac {\cosh \left (a + \frac {b}{x}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 29, normalized size = 1.00 \[ \frac {\mathrm {cosh}\left (a+\frac {b}{x}\right )}{b^2}-\frac {\mathrm {sinh}\left (a+\frac {b}{x}\right )}{b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 29, normalized size = 1.00 \[ \begin {cases} - \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b x} + \frac {\cosh {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\cosh {\relax (a )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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