Integrand size = 12, antiderivative size = 46 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^3}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2} \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5429, 3377, 2717} \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=-\frac {2 \sinh \left (a+\frac {b}{x}\right )}{b^3}+\frac {2 \cosh \left (a+\frac {b}{x}\right )}{b^2 x}-\frac {\sinh \left (a+\frac {b}{x}\right )}{b x^2} \]
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Rule 2717
Rule 3377
Rule 5429
Rubi steps \begin{align*} \text {integral}& = -\text {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 \text {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 \text {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*}
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {2 b x \cosh \left (a+\frac {b}{x}\right )-\left (b^2+2 x^2\right ) \sinh \left (a+\frac {b}{x}\right )}{b^3 x^2} \]
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Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-\frac {\left (b^{2}-2 b x +2 x^{2}\right ) {\mathrm e}^{\frac {a x +b}{x}}}{2 b^{3} x^{2}}+\frac {\left (b^{2}+2 b x +2 x^{2}\right ) {\mathrm e}^{-\frac {a x +b}{x}}}{2 b^{3} x^{2}}\) | \(65\) |
parallelrisch | \(\frac {-2 \tanh \left (\frac {a x +b}{2 x}\right )^{2} x b +4 \tanh \left (\frac {a x +b}{2 x}\right ) x^{2}+2 \tanh \left (\frac {a x +b}{2 x}\right ) b^{2}-2 b x}{x^{2} b^{3} \left (\tanh \left (\frac {a x +b}{2 x}\right )^{2}-1\right )}\) | \(79\) |
derivativedivides | \(-\frac {a^{2} \sinh \left (a +\frac {b}{x}\right )-2 a \left (\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{2} \sinh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )+2 \sinh \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(94\) |
default | \(-\frac {a^{2} \sinh \left (a +\frac {b}{x}\right )-2 a \left (\left (a +\frac {b}{x}\right ) \sinh \left (a +\frac {b}{x}\right )-\cosh \left (a +\frac {b}{x}\right )\right )+\left (a +\frac {b}{x}\right )^{2} \sinh \left (a +\frac {b}{x}\right )-2 \left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )+2 \sinh \left (a +\frac {b}{x}\right )}{b^{3}}\) | \(94\) |
meijerg | \(-\frac {4 i \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {i b \cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}-\frac {i \left (\frac {3 b^{2}}{2 x^{2}}+3\right ) \sinh \left (\frac {b}{x}\right )}{6 \sqrt {\pi }}\right )}{b^{3}}-\frac {4 \sqrt {\pi }\, \sinh \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {b^{2}}{2 x^{2}}+1\right ) \cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}-\frac {b \sinh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{3}}\) | \(104\) |
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none
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {2 \, b x \cosh \left (\frac {a x + b}{x}\right ) - {\left (b^{2} + 2 \, x^{2}\right )} \sinh \left (\frac {a x + b}{x}\right )}{b^{3} x^{2}} \]
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Time = 0.50 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\begin {cases} - \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b x^{2}} + \frac {2 \cosh {\left (a + \frac {b}{x} \right )}}{b^{2} x} - \frac {2 \sinh {\left (a + \frac {b}{x} \right )}}{b^{3}} & \text {for}\: b \neq 0 \\- \frac {\cosh {\left (a \right )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {1}{6} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (4, \frac {b}{x}\right )}{b^{4}} - \frac {e^{a} \Gamma \left (4, -\frac {b}{x}\right )}{b^{4}}\right )} - \frac {\cosh \left (a + \frac {b}{x}\right )}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (46) = 92\).
Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 4.70 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=-\frac {a^{2} e^{\left (\frac {a x + b}{x}\right )} - a^{2} e^{\left (-\frac {a x + b}{x}\right )} + 2 \, a e^{\left (\frac {a x + b}{x}\right )} - \frac {2 \, {\left (a x + b\right )} a e^{\left (\frac {a x + b}{x}\right )}}{x} + 2 \, a e^{\left (-\frac {a x + b}{x}\right )} + \frac {2 \, {\left (a x + b\right )} a e^{\left (-\frac {a x + b}{x}\right )}}{x} + \frac {{\left (a x + b\right )}^{2} e^{\left (\frac {a x + b}{x}\right )}}{x^{2}} - \frac {2 \, {\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )}^{2} e^{\left (-\frac {a x + b}{x}\right )}}{x^{2}} - \frac {2 \, {\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + 2 \, e^{\left (\frac {a x + b}{x}\right )} - 2 \, e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{3}} \]
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Time = 1.60 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43 \[ \int \frac {\cosh \left (a+\frac {b}{x}\right )}{x^4} \, dx=\frac {{\mathrm {e}}^{-a-\frac {b}{x}}\,\left (\frac {x}{b^2}+\frac {1}{2\,b}+\frac {x^2}{b^3}\right )}{x^2}-\frac {{\mathrm {e}}^{a+\frac {b}{x}}\,\left (\frac {1}{2\,b}-\frac {x}{b^2}+\frac {x^2}{b^3}\right )}{x^2} \]
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