Integrand size = 12, antiderivative size = 60 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=-\frac {\text {Chi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{b^3}+\frac {\text {Chi}(3 \text {arcsinh}(a+b x))}{4 b^3}-\frac {a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3} \]
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Time = 0.40 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5859, 5830, 6873, 12, 6874, 3382, 5556, 3379} \[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{b^3}-\frac {\text {Chi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {\text {Chi}(3 \text {arcsinh}(a+b x))}{4 b^3}-\frac {a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3} \]
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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 {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\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 )^2}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) (a-\sinh (x))^2}{b^2 x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) (a-\sinh (x))^2}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2 \cosh (x)}{x}-\frac {2 a \cosh (x) \sinh (x)}{x}+\frac {\cosh (x) \sinh ^2(x)}{x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{b^3}+\frac {\text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = \frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{b^3}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^3}+\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^3}-\frac {a \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = -\frac {\text {Chi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{b^3}+\frac {\text {Chi}(3 \text {arcsinh}(a+b x))}{4 b^3}-\frac {a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\frac {\left (-1+4 a^2\right ) \text {Chi}(\text {arcsinh}(a+b x))+\text {Chi}(3 \text {arcsinh}(a+b x))-4 a \text {Shi}(2 \text {arcsinh}(a+b x))}{4 b^3} \]
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}+\frac {\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-a \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )+a^{2} \operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b^{3}}\) | \(49\) |
default | \(\frac {-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}+\frac {\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-a \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )+a^{2} \operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{b^{3}}\) | \(49\) |
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\int \frac {x^{2}}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a+b x)} \, dx=\int \frac {x^2}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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