Optimal. Leaf size=62 \[ -\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^3}-\frac {6 \cosh \left (a+\frac {b}{x}\right )}{b^3 x}+\frac {6 \sinh \left (a+\frac {b}{x}\right )}{b^4}+\frac {3 \sinh \left (a+\frac {b}{x}\right )}{b^2 x^2} \]
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Rubi [A]
time = 0.06, 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 = {5428, 3377,
2717} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rule 5428
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x^5} \, dx &=-\text {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 \text {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 \text {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 \text {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 = 0.04, size = 48, normalized size = 0.77 \begin {gather*} \frac {-b \left (b^2+6 x^2\right ) \cosh \left (a+\frac {b}{x}\right )+3 x \left (b^2+2 x^2\right ) \sinh \left (a+\frac {b}{x}\right )}{b^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs.
\(2(62)=124\).
time = 0.65, size = 165, normalized size = 2.66
method | result | size |
risch | \(-\frac {\left (b^{3}-3 b^{2} x +6 x^{2} b -6 x^{3}\right ) {\mathrm e}^{\frac {a x +b}{x}}}{2 b^{4} x^{3}}-\frac {\left (b^{3}+3 b^{2} x +6 x^{2} b +6 x^{3}\right ) {\mathrm e}^{-\frac {a x +b}{x}}}{2 b^{4} x^{3}}\) | \(81\) |
meijerg | \(\frac {8 i \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {i b \left (\frac {5 b^{2}}{2 x^{2}}+15\right ) \cosh \left (\frac {b}{x}\right )}{20 \sqrt {\pi }\, x}-\frac {i \left (\frac {15 b^{2}}{2 x^{2}}+15\right ) \sinh \left (\frac {b}{x}\right )}{20 \sqrt {\pi }}\right )}{b^{4}}-\frac {8 \sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 b^{2}}{2 x^{2}}+3\right ) \cosh \left (\frac {b}{x}\right )}{4 \sqrt {\pi }}+\frac {b \left (\frac {b^{2}}{2 x^{2}}+3\right ) \sinh \left (\frac {b}{x}\right )}{4 \sqrt {\pi }\, x}\right )}{b^{4}}\) | \(124\) |
derivativedivides | \(-\frac {-a^{3} \cosh \left (a +\frac {b}{x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \cosh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )+2 \cosh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{3} \cosh \left (a +\frac {b}{x}\right )-3 \left (a +\frac {b}{x}\right )^{2} \sinh \left (a +\frac {b}{x}\right )+6 \left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-6 \sinh \left (a +\frac {b}{x}\right )}{b^{4}}\) | \(165\) |
default | \(-\frac {-a^{3} \cosh \left (a +\frac {b}{x}\right )+3 a^{2} \left (\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )\right )-3 a \left (\left (a +\frac {b}{x}\right )^{2} \cosh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )+2 \cosh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{3} \cosh \left (a +\frac {b}{x}\right )-3 \left (a +\frac {b}{x}\right )^{2} \sinh \left (a +\frac {b}{x}\right )+6 \left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-6 \sinh \left (a +\frac {b}{x}\right )}{b^{4}}\) | \(165\) |
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 = 48, normalized size = 0.77 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 53, normalized size = 0.85 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.97, size = 61, normalized size = 0.98 \begin {gather*} \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 {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. 386 vs.
\(2 (62) = 124\).
time = 0.43, size = 386, normalized size = 6.23 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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