Integrand size = 8, antiderivative size = 14 \[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 14, 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 = {5887, 5556, 12, 3379} \[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^2} \]
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Rule 12
Rule 3379
Rule 5556
Rule 5887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^2} \\ & = \frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^2} \]
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Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
default | \(\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
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\[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\int \frac {x}{\operatorname {acosh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x}{\text {arccosh}(a x)} \, dx=\int \frac {x}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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