Integrand size = 6, antiderivative size = 39 \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{a} \]
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Time = 0.13 (sec) , antiderivative size = 39, 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 = {5880, 5953, 3382} \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\frac {\text {Chi}(\text {arccosh}(a x))}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)} \]
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Rule 3382
Rule 5880
Rule 5953
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)} \, dx \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a} \\ & = -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{a} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\frac {1-a x+\sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))}{a \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)} \]
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Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{a}\) | \(33\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{a}\) | \(33\) |
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\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int \frac {1}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^2} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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