Integrand size = 6, antiderivative size = 49 \[ \int x \text {arccosh}(a x) \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{4 a}-\frac {\text {arccosh}(a x)}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x) \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 92, 54} \[ \int x \text {arccosh}(a x) \, dx=-\frac {\text {arccosh}(a x)}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{4 a} \]
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Rule 54
Rule 92
Rule 5883
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arccosh}(a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{4 a}+\frac {1}{2} x^2 \text {arccosh}(a x)-\frac {\int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a} \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{4 a}-\frac {\text {arccosh}(a x)}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.24 \[ \int x \text {arccosh}(a x) \, dx=-\frac {a x \sqrt {-1+a x} \sqrt {1+a x}-2 a^2 x^2 \text {arccosh}(a x)+2 \text {arctanh}\left (\sqrt {\frac {-1+a x}{1+a x}}\right )}{4 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{2}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a x \sqrt {a^{2} x^{2}-1}+\ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{4 \sqrt {a^{2} x^{2}-1}}}{a^{2}}\) | \(76\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{2}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a x \sqrt {a^{2} x^{2}-1}+\ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{4 \sqrt {a^{2} x^{2}-1}}}{a^{2}}\) | \(76\) |
parts | \(\frac {x^{2} \operatorname {arccosh}\left (a x \right )}{2}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (x \sqrt {a^{2} x^{2}-1}\, \operatorname {csgn}\left (a \right ) a +\ln \left (\left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2} x^{2}-1}+a x \right ) \operatorname {csgn}\left (a \right )\right )\right ) \operatorname {csgn}\left (a \right )}{4 a^{2} \sqrt {a^{2} x^{2}-1}}\) | \(82\) |
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Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int x \text {arccosh}(a x) \, dx=-\frac {\sqrt {a^{2} x^{2} - 1} a x - {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, a^{2}} \]
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\[ \int x \text {arccosh}(a x) \, dx=\int x \operatorname {acosh}{\left (a x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14 \[ \int x \text {arccosh}(a x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcosh}\left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x}{a^{2}} + \frac {\log \left (2 \, a^{2} x + 2 \, \sqrt {a^{2} x^{2} - 1} a\right )}{a^{3}}\right )} \]
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Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int x \text {arccosh}(a x) \, dx=\frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x}{a^{2}} - \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{a^{2} {\left | a \right |}}\right )} \]
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Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int x \text {arccosh}(a x) \, dx=x\,\mathrm {acosh}\left (a\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,a^2\,x}\right )-\frac {x\,\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{4\,a} \]
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