Optimal. Leaf size=46 \[ -\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^3}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^2}+\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^2 x} \]
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Rubi [A]
time = 0.04, 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 = {5428, 3377,
2718} \begin {gather*} -\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^3}+\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5428
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x}\right )}{x^4} \, dx &=-\text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^2}+\frac {2 \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^2}+\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {2 \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\frac {1}{x}\right )}{b^2}\\ &=-\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^3}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x^2}+\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 0.85 \begin {gather*} \frac {-\left (\left (b^2+2 x^2\right ) \cosh \left (a+\frac {b}{x}\right )\right )+2 b x \sinh \left (a+\frac {b}{x}\right )}{b^3 x^2} \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. \(93\) vs.
\(2(46)=92\).
time = 0.61, size = 94, normalized size = 2.04
method | result | size |
risch | \(-\frac {\left (b^{2}-2 b x +2 x^{2}\right ) {\mathrm e}^{\frac {a x +b}{x}}}{2 b^{3} x^{2}}-\frac {\left (b^{2}+2 b x +2 x^{2}\right ) {\mathrm e}^{-\frac {a x +b}{x}}}{2 b^{3} x^{2}}\) | \(65\) |
derivativedivides | \(-\frac {a^{2} \cosh \left (a +\frac {b}{x}\right )-2 a \left (\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{2} \cosh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )+2 \cosh \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(94\) |
default | \(-\frac {a^{2} \cosh \left (a +\frac {b}{x}\right )-2 a \left (\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{2} \cosh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )+2 \cosh \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(94\) |
meijerg | \(-\frac {4 \sqrt {\pi }\, \cosh \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {b^{2}}{2 x^{2}}+1\right ) \cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}-\frac {b \sinh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{3}}-\frac {4 i \sqrt {\pi }\, \sinh \left (a \right ) \left (\frac {i b \cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}-\frac {i \left (\frac {3 b^{2}}{2 x^{2}}+3\right ) \sinh \left (\frac {b}{x}\right )}{6 \sqrt {\pi }}\right )}{b^{3}}\) | \(104\) |
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 = 0.30, size = 47, normalized size = 1.02 \begin {gather*} -\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 {\sinh \left (a + \frac {b}{x}\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 43, normalized size = 0.93 \begin {gather*} \frac {2 \, b x \sinh \left (\frac {a x + b}{x}\right ) - {\left (b^{2} + 2 \, x^{2}\right )} \cosh \left (\frac {a x + b}{x}\right )}{b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.68, size = 46, normalized size = 1.00 \begin {gather*} \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x} \right )}}{b x^{2}} + \frac {2 \sinh {\left (a + \frac {b}{x} \right )}}{b^{2} x} - \frac {2 \cosh {\left (a + \frac {b}{x} \right )}}{b^{3}} & \text {for}\: b \neq 0 \\- \frac {\sinh {\left (a \right )}}{3 x^{3}} & \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. 214 vs.
\(2 (46) = 92\).
time = 0.46, size = 214, normalized size = 4.65 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 67, normalized size = 1.46 \begin {gather*} -\frac {{\mathrm {e}}^{a+\frac {b}{x}}\,\left (\frac {1}{2\,b}-\frac {x}{b^2}+\frac {x^2}{b^3}\right )}{x^2}-\frac {{\mathrm {e}}^{-a-\frac {b}{x}}\,\left (\frac {x}{b^2}+\frac {1}{2\,b}+\frac {x^2}{b^3}\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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