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 = 0.0530366, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 5321
Rule 3296
Rule 2637
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*}
Mathematica [A] time = 0.0417849, 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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Maple [B] time = 0.013, size = 94, normalized size = 2. \begin{align*} -{\frac{1}{{b}^{3}} \left ( \left ( a+{\frac{b}{x}} \right ) ^{2}\sinh \left ( a+{\frac{b}{x}} \right ) -2\, \left ( a+{\frac{b}{x}} \right ) \cosh \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.19481, size = 65, normalized size = 1.41 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70482, size = 96, normalized size = 2.09 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.4698, size = 46, normalized size = 1. \begin{align*} \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{\left (a \right )}}{3 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (a + \frac{b}{x}\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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