Optimal. Leaf size=46 \[ -\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^3}+\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5321, 3296, 2637} \[ -\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^3}+\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5321
Rubi steps
\begin {align*} \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2}+\frac {2 \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2}-\frac {2 \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\frac {1}{x}\right )}{b^2}\\ &=\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^3}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.85 \[ \frac {2 b x \cosh \left (a+\frac {b}{x}\right )-\left (b^2+2 x^2\right ) \sinh \left (a+\frac {b}{x}\right )}{b^3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 43, normalized size = 0.93 \[ \frac {2 \, b x \cosh \left (\frac {a x + b}{x}\right ) - {\left (b^{2} + 2 \, x^{2}\right )} \sinh \left (\frac {a x + b}{x}\right )}{b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 216, normalized size = 4.70 \[ -\frac {a^{2} e^{\left (\frac {a x + b}{x}\right )} - a^{2} e^{\left (-\frac {a x + b}{x}\right )} + 2 \, a e^{\left (\frac {a x + b}{x}\right )} - \frac {2 \, {\left (a x + b\right )} a e^{\left (\frac {a x + b}{x}\right )}}{x} + 2 \, a e^{\left (-\frac {a x + b}{x}\right )} + \frac {2 \, {\left (a x + b\right )} a e^{\left (-\frac {a x + b}{x}\right )}}{x} + \frac {{\left (a x + b\right )}^{2} e^{\left (\frac {a x + b}{x}\right )}}{x^{2}} - \frac {2 \, {\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )}^{2} e^{\left (-\frac {a x + b}{x}\right )}}{x^{2}} - \frac {2 \, {\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + 2 \, e^{\left (\frac {a x + b}{x}\right )} - 2 \, e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 94, normalized size = 2.04 \[ -\frac {\left (a +\frac {b}{x}\right )^{2} \sinh \left (a +\frac {b}{x}\right )-2 \cosh \left (a +\frac {b}{x}\right ) \left (a +\frac {b}{x}\right )+2 \sinh \left (a +\frac {b}{x}\right )-2 a \left (\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )\right )+a^{2} \sinh \left (a +\frac {b}{x}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.35, size = 48, normalized size = 1.04 \[ \frac {1}{6} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (4, \frac {b}{x}\right )}{b^{4}} - \frac {e^{a} \Gamma \left (4, -\frac {b}{x}\right )}{b^{4}}\right )} - \frac {\cosh \left (a + \frac {b}{x}\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 66, normalized size = 1.43 \[ \frac {{\mathrm {e}}^{-a-\frac {b}{x}}\,\left (\frac {x}{b^2}+\frac {1}{2\,b}+\frac {x^2}{b^3}\right )}{x^2}-\frac {{\mathrm {e}}^{a+\frac {b}{x}}\,\left (\frac {1}{2\,b}-\frac {x}{b^2}+\frac {x^2}{b^3}\right )}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.13, size = 46, normalized size = 1.00 \[ \begin {cases} - \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b x^{2}} + \frac {2 \cosh {\left (a + \frac {b}{x} \right )}}{b^{2} x} - \frac {2 \sinh {\left (a + \frac {b}{x} \right )}}{b^{3}} & \text {for}\: b \neq 0 \\- \frac {\cosh {\relax (a )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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