Optimal. Leaf size=62 \[ \frac{3 \sinh \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \sinh \left (a+\frac{b}{x}\right )}{b^4}-\frac{6 \cosh \left (a+\frac{b}{x}\right )}{b^3 x}-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3} \]
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Rubi [A] time = 0.0795417, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5320, 3296, 2637} \[ \frac{3 \sinh \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \sinh \left (a+\frac{b}{x}\right )}{b^4}-\frac{6 \cosh \left (a+\frac{b}{x}\right )}{b^3 x}-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x}\right )}{x^5} \, dx &=-\operatorname{Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3}+\frac{3 \operatorname{Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\frac{1}{x}\right )}{b}\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3}+\frac{3 \sinh \left (a+\frac{b}{x}\right )}{b^2 x^2}-\frac{6 \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\frac{1}{x}\right )}{b^2}\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3}-\frac{6 \cosh \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{3 \sinh \left (a+\frac{b}{x}\right )}{b^2 x^2}+\frac{6 \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\frac{1}{x}\right )}{b^3}\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b x^3}-\frac{6 \cosh \left (a+\frac{b}{x}\right )}{b^3 x}+\frac{6 \sinh \left (a+\frac{b}{x}\right )}{b^4}+\frac{3 \sinh \left (a+\frac{b}{x}\right )}{b^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.058288, size = 48, normalized size = 0.77 \[ \frac{3 x \left (b^2+2 x^2\right ) \sinh \left (a+\frac{b}{x}\right )-b \left (b^2+6 x^2\right ) \cosh \left (a+\frac{b}{x}\right )}{b^4 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 165, normalized size = 2.7 \begin{align*} -{\frac{1}{{b}^{4}} \left ( \left ( a+{\frac{b}{x}} \right ) ^{3}\cosh \left ( a+{\frac{b}{x}} \right ) -3\, \left ( a+{\frac{b}{x}} \right ) ^{2}\sinh \left ( a+{\frac{b}{x}} \right ) +6\, \left ( a+{\frac{b}{x}} \right ) \cosh \left ( a+{\frac{b}{x}} \right ) -6\,\sinh \left ( a+{\frac{b}{x}} \right ) -3\,a \left ( \left ( a+{\frac{b}{x}} \right ) ^{2}\cosh \left ( a+{\frac{b}{x}} \right ) -2\, \left ( a+{\frac{b}{x}} \right ) \sinh \left ( a+{\frac{b}{x}} \right ) +2\,\cosh \left ( a+{\frac{b}{x}} \right ) \right ) +3\,{a}^{2} \left ( \left ( a+{\frac{b}{x}} \right ) \cosh \left ( a+{\frac{b}{x}} \right ) -\sinh \left ( a+{\frac{b}{x}} \right ) \right ) -{a}^{3}\cosh \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 = 1.23791, size = 65, normalized size = 1.05 \begin{align*} -\frac{1}{8} \, b{\left (\frac{e^{\left (-a\right )} \Gamma \left (5, \frac{b}{x}\right )}{b^{5}} - \frac{e^{a} \Gamma \left (5, -\frac{b}{x}\right )}{b^{5}}\right )} - \frac{\sinh \left (a + \frac{b}{x}\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7854, size = 116, normalized size = 1.87 \begin{align*} -\frac{{\left (b^{3} + 6 \, b x^{2}\right )} \cosh \left (\frac{a x + b}{x}\right ) - 3 \,{\left (b^{2} x + 2 \, x^{3}\right )} \sinh \left (\frac{a x + b}{x}\right )}{b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.01725, size = 61, normalized size = 0.98 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x} \right )}}{b x^{3}} + \frac{3 \sinh{\left (a + \frac{b}{x} \right )}}{b^{2} x^{2}} - \frac{6 \cosh{\left (a + \frac{b}{x} \right )}}{b^{3} x} + \frac{6 \sinh{\left (a + \frac{b}{x} \right )}}{b^{4}} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{4 x^{4}} & \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{\sinh \left (a + \frac{b}{x}\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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