Integrand size = 8, antiderivative size = 65 \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{6 x^2}-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{6} a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5883, 105, 12, 94, 211} \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\frac {1}{6} a^3 \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{6 x^2} \]
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Rule 12
Rule 94
Rule 105
Rule 211
Rule 5883
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{3} a \int \frac {1}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{6 x^2}-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{6} a \int \frac {a^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{6 x^2}-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{6} a^3 \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{6 x^2}-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{6} a^4 \text {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right ) \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{6 x^2}-\frac {\text {arccosh}(a x)}{3 x^3}+\frac {1}{6} a^3 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20 \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\frac {-2 \text {arccosh}(a x)+\frac {a x \left (-1+a^2 x^2+a^2 x^2 \sqrt {-1+a^2 x^2} \arctan \left (\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a x} \sqrt {1+a x}}}{6 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15
method | result | size |
parts | \(-\frac {\operatorname {arccosh}\left (a x \right )}{3 x^{3}}-\frac {a \sqrt {a x -1}\, \sqrt {a x +1}\, \left (\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-\sqrt {a^{2} x^{2}-1}\right )}{6 \sqrt {a^{2} x^{2}-1}\, x^{2}}\) | \(75\) |
derivativedivides | \(a^{3} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-\sqrt {a^{2} x^{2}-1}\right )}{6 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}}\right )\) | \(84\) |
default | \(a^{3} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-\sqrt {a^{2} x^{2}-1}\right )}{6 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}}\right )\) | \(84\) |
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\frac {2 \, a^{3} x^{3} \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 2 \, x^{3} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + \sqrt {a^{2} x^{2} - 1} a x + 2 \, {\left (x^{3} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{6 \, x^{3}} \]
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\[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{4}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=-\frac {1}{6} \, {\left (a^{2} \arcsin \left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{x^{2}}\right )} a - \frac {\operatorname {arcosh}\left (a x\right )}{3 \, x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\frac {a^{4} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) + \frac {\sqrt {a^{2} x^{2} - 1} a^{2}}{x^{2}}}{6 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{3 \, x^{3}} \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{x^4} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x^4} \,d x \]
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