Integrand size = 12, antiderivative size = 154 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Shi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))}{b^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \]
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Time = 0.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5859, 5829, 5773, 5819, 3379, 5778, 3382} \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))}{b^3}-\frac {a^2 \sqrt {(a+b x)^2+1}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Shi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3}+\frac {2 a (a+b x) \sqrt {(a+b x)^2+1}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{b^3 \text {arcsinh}(a+b x)} \]
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Rule 3379
Rule 3382
Rule 5773
Rule 5778
Rule 5819
Rule 5829
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2}{b^2 \text {arcsinh}(x)^2}-\frac {2 a x}{b^2 \text {arcsinh}(x)^2}+\frac {x^2}{b^2 \text {arcsinh}(x)^2}\right ) \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 x}+\frac {3 \sinh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)}-\frac {2 a \text {Chi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {\text {Shi}(\text {arcsinh}(a+b x))}{4 b^3}+\frac {a^2 \text {Shi}(\text {arcsinh}(a+b x))}{b^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\frac {-\frac {4 b^2 x^2 \sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)}-8 a \text {Chi}(2 \text {arcsinh}(a+b x))+\left (-1+4 a^2\right ) \text {Shi}(\text {arcsinh}(a+b x))+3 \text {Shi}(3 \text {arcsinh}(a+b x))}{4 b^3} \]
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Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {a \left (2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{\operatorname {arcsinh}\left (b x +a \right )}+\frac {a^{2} \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{3}}\) | \(146\) |
default | \(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4}-\frac {a \left (2 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{\operatorname {arcsinh}\left (b x +a \right )}+\frac {a^{2} \left (\operatorname {Shi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\operatorname {arcsinh}\left (b x +a \right )}}{b^{3}}\) | \(146\) |
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
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