Integrand size = 10, antiderivative size = 30 \[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=-\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{2 b^2} \]
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Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5859, 5830, 6873, 12, 6874, 3382, 5556, 3379} \[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{2 b^2}-\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2} \]
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Rule 12
Rule 3379
Rule 3382
Rule 5556
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-\frac {a}{b}+\frac {x}{b}}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) (-a+\sinh (x))}{b x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) (-a+\sinh (x))}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a \cosh (x)}{x}+\frac {\cosh (x) \sinh (x)}{x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2}-\frac {a \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = -\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^2} \\ & = -\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^2} \\ & = -\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{2 b^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=-\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{b^2}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{2 b^2} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2}-a \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b^{2}}\) | \(27\) |
default | \(\frac {\frac {\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2}-a \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b^{2}}\) | \(27\) |
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\[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\int \frac {x}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \frac {x}{\text {arcsinh}(a+b x)} \, dx=\int \frac {x}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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