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 = 0.02, 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 = {5429, 3377,
2718} \begin {gather*} \frac {\cosh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5429
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\text {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 {\text {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 = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {x \cosh \left (a+\frac {b}{x}\right )-b \sinh \left (a+\frac {b}{x}\right )}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.21, size = 44, normalized size = 1.52
method | result | size |
derivativedivides | \(-\frac {\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )-a \sinh \left (a +\frac {b}{x}\right )}{b^{2}}\) | \(44\) |
default | \(-\frac {\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )-a \sinh \left (a +\frac {b}{x}\right )}{b^{2}}\) | \(44\) |
risch | \(-\frac {\left (-x +b \right ) {\mathrm e}^{\frac {a x +b}{x}}}{2 b^{2} x}+\frac {\left (x +b \right ) {\mathrm e}^{-\frac {a x +b}{x}}}{2 b^{2} x}\) | \(47\) |
meijerg | \(\frac {2 \sqrt {\pi }\, \cosh \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}-\frac {b \sinh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{2}}-\frac {\sinh \left (a \right ) \left (\frac {\cosh \left (\frac {b}{x}\right ) b}{x}-\sinh \left (\frac {b}{x}\right )\right )}{b^{2}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.29, size = 47, normalized size = 1.62 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 33, normalized size = 1.14 \begin {gather*} \frac {x \cosh \left (\frac {a x + b}{x}\right ) - b \sinh \left (\frac {a x + b}{x}\right )}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 29, normalized size = 1.00 \begin {gather*} \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 {\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. 93 vs.
\(2 (29) = 58\).
time = 0.41, size = 93, normalized size = 3.21 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \frac {\mathrm {cosh}\left (a+\frac {b}{x}\right )}{b^2}-\frac {\mathrm {sinh}\left (a+\frac {b}{x}\right )}{b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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