Optimal. Leaf size=29 \[ \frac {\sinh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x} \]
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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 = {5320, 3296, 2637} \[ \frac {\sinh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5320
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x}+\frac {\operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x}+\frac {\sinh \left (a+\frac {b}{x}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.00 \[ \frac {x \sinh \left (a+\frac {b}{x}\right )-b \cosh \left (a+\frac {b}{x}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 34, normalized size = 1.17 \[ -\frac {b \cosh \left (\frac {a x + b}{x}\right ) - x \sinh \left (\frac {a x + b}{x}\right )}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 95, normalized size = 3.28 \[ \frac {a e^{\left (\frac {a x + b}{x}\right )} + a e^{\left (-\frac {a x + b}{x}\right )} - \frac {{\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + e^{\left (\frac {a x + b}{x}\right )} - e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 44, normalized size = 1.52 \[ -\frac {\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )-a \cosh \left (a +\frac {b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 48, normalized size = 1.66 \[ -\frac {1}{4} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (3, \frac {b}{x}\right )}{b^{3}} - \frac {e^{a} \Gamma \left (3, -\frac {b}{x}\right )}{b^{3}}\right )} - \frac {\sinh \left (a + \frac {b}{x}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 29, normalized size = 1.00 \[ \frac {\mathrm {sinh}\left (a+\frac {b}{x}\right )}{b^2}-\frac {\mathrm {cosh}\left (a+\frac {b}{x}\right )}{b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.77, size = 29, normalized size = 1.00 \[ \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x} \right )}}{b x} + \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\sinh {\relax (a )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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