Integrand size = 15, antiderivative size = 3 \[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\text {Shi}(\text {arcsinh}(x)) \]
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Time = 0.05 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5819, 3379} \[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\text {Shi}(\text {arcsinh}(x)) \]
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Rule 3379
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(x)\right ) \\ & = \text {Shi}(\text {arcsinh}(x)) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\text {Shi}(\text {arcsinh}(x)) \]
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Time = 0.61 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
default | \(\operatorname {Shi}\left (\operatorname {arcsinh}\left (x \right )\right )\) | \(4\) |
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\[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\left (x\right )} \,d x } \]
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\[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\int \frac {x}{\sqrt {x^{2} + 1} \operatorname {asinh}{\left (x \right )}}\, dx \]
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\[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\left (x\right )} \,d x } \]
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\[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\left (x\right )} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx=\int \frac {x}{\mathrm {asinh}\left (x\right )\,\sqrt {x^2+1}} \,d x \]
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