Optimal. Leaf size=62 \[ \frac {6 \sinh \left (a+\frac {b}{x}\right )}{b^4}-\frac {6 \cosh \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {3 \sinh \left (a+\frac {b}{x}\right )}{b^2 x^2}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 2637
Rule 3296
Rule 5320
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.38, size = 53, normalized size = 0.85 \[ -\frac {{\left (b^{3} + 6 \, b x^{2}\right )} \cosh \left (\frac {a x + b}{x}\right ) - 3 \, {\left (b^{2} x + 2 \, x^{3}\right )} \sinh \left (\frac {a x + b}{x}\right )}{b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 386, normalized size = 6.23 \[ \frac {a^{3} e^{\left (\frac {a x + b}{x}\right )} + a^{3} e^{\left (-\frac {a x + b}{x}\right )} + 3 \, a^{2} e^{\left (\frac {a x + b}{x}\right )} - \frac {3 \, {\left (a x + b\right )} a^{2} e^{\left (\frac {a x + b}{x}\right )}}{x} - 3 \, a^{2} e^{\left (-\frac {a x + b}{x}\right )} - \frac {3 \, {\left (a x + b\right )} a^{2} e^{\left (-\frac {a x + b}{x}\right )}}{x} + 6 \, a e^{\left (\frac {a x + b}{x}\right )} + \frac {3 \, {\left (a x + b\right )}^{2} a e^{\left (\frac {a x + b}{x}\right )}}{x^{2}} - \frac {6 \, {\left (a x + b\right )} a e^{\left (\frac {a x + b}{x}\right )}}{x} + 6 \, a e^{\left (-\frac {a x + b}{x}\right )} + \frac {3 \, {\left (a x + b\right )}^{2} a e^{\left (-\frac {a x + b}{x}\right )}}{x^{2}} + \frac {6 \, {\left (a x + b\right )} a e^{\left (-\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )}^{3} e^{\left (\frac {a x + b}{x}\right )}}{x^{3}} + \frac {3 \, {\left (a x + b\right )}^{2} e^{\left (\frac {a x + b}{x}\right )}}{x^{2}} - \frac {6 \, {\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )}^{3} e^{\left (-\frac {a x + b}{x}\right )}}{x^{3}} - \frac {3 \, {\left (a x + b\right )}^{2} e^{\left (-\frac {a x + b}{x}\right )}}{x^{2}} - \frac {6 \, {\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + 6 \, e^{\left (\frac {a x + b}{x}\right )} - 6 \, e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 165, normalized size = 2.66 \[ -\frac {\left (a +\frac {b}{x}\right )^{3} \cosh \left (a +\frac {b}{x}\right )-3 \sinh \left (a +\frac {b}{x}\right ) \left (a +\frac {b}{x}\right )^{2}+6 \left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-6 \sinh \left (a +\frac {b}{x}\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \cosh \left (a +\frac {b}{x}\right )-2 \sinh \left (a +\frac {b}{x}\right ) \left (a +\frac {b}{x}\right )+2 \cosh \left (a +\frac {b}{x}\right )\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )\right )-a^{3} \cosh \left (a +\frac {b}{x}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 48, normalized size = 0.77 \[ -\frac {1}{8} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (5, \frac {b}{x}\right )}{b^{5}} - \frac {e^{a} \Gamma \left (5, -\frac {b}{x}\right )}{b^{5}}\right )} - \frac {\sinh \left (a + \frac {b}{x}\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 85, normalized size = 1.37 \[ \frac {{\mathrm {e}}^{a+\frac {b}{x}}\,\left (\frac {3\,x}{2\,b^2}-\frac {1}{2\,b}-\frac {3\,x^2}{b^3}+\frac {3\,x^3}{b^4}\right )}{x^3}-\frac {{\mathrm {e}}^{-a-\frac {b}{x}}\,\left (\frac {3\,x}{2\,b^2}+\frac {1}{2\,b}+\frac {3\,x^2}{b^3}+\frac {3\,x^3}{b^4}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.90, size = 61, normalized size = 0.98 \[ \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x} \right )}}{b x^{3}} + \frac {3 \sinh {\left (a + \frac {b}{x} \right )}}{b^{2} x^{2}} - \frac {6 \cosh {\left (a + \frac {b}{x} \right )}}{b^{3} x} + \frac {6 \sinh {\left (a + \frac {b}{x} \right )}}{b^{4}} & \text {for}\: b \neq 0 \\- \frac {\sinh {\relax (a )}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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