Integrand size = 8, antiderivative size = 64 \[ \int x \text {arccosh}(a x)^2 \, dx=\frac {x^2}{4}-\frac {x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{2 a}-\frac {\text {arccosh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^2 \]
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Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 5939, 5893, 30} \[ \int x \text {arccosh}(a x)^2 \, dx=-\frac {\text {arccosh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^2-\frac {x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{2 a}+\frac {x^2}{4} \]
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Rule 30
Rule 5883
Rule 5893
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arccosh}(a x)^2-a \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{2 a}+\frac {1}{2} x^2 \text {arccosh}(a x)^2+\frac {\int x \, dx}{2}-\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 a} \\ & = \frac {x^2}{4}-\frac {x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{2 a}-\frac {\text {arccosh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^2 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int x \text {arccosh}(a x)^2 \, dx=\frac {a^2 x^2-2 a x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)+\left (-1+2 a^2 x^2\right ) \text {arccosh}(a x)^2}{4 a^2} \]
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Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}}{2}-\frac {a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{2}-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{4}}{a^{2}}\) | \(58\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}}{2}-\frac {a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{2}-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{4}}{a^{2}}\) | \(58\) |
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Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.14 \[ \int x \text {arccosh}(a x)^2 \, dx=\frac {a^{2} x^{2} - 2 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{4 \, a^{2}} \]
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\[ \int x \text {arccosh}(a x)^2 \, dx=\int x \operatorname {acosh}^{2}{\left (a x \right )}\, dx \]
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\[ \int x \text {arccosh}(a x)^2 \, dx=\int { x \operatorname {arcosh}\left (a x\right )^{2} \,d x } \]
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Exception generated. \[ \int x \text {arccosh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \text {arccosh}(a x)^2 \, dx=\int x\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]
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