Integrand size = 10, antiderivative size = 54 \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {\text {Shi}(\text {arcsinh}(a x))}{4 a^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5778, 3379} \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {\text {Shi}(\text {arcsinh}(a x))}{4 a^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{a \text {arcsinh}(a x)} \]
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Rule 3379
Rule 5778
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a 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 x)\right )}{a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{a \text {arcsinh}(a x)}-\frac {\text {Shi}(\text {arcsinh}(a x))}{4 a^3}+\frac {3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {\frac {4 a^2 x^2 \sqrt {1+a^2 x^2}}{\text {arcsinh}(a x)}+\text {Shi}(\text {arcsinh}(a x))-3 \text {Shi}(3 \text {arcsinh}(a x))}{4 a^3} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{4 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 \,\operatorname {arcsinh}\left (a x \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{4}}{a^{3}}\) | \(56\) |
default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{4 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{4}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 \,\operatorname {arcsinh}\left (a x \right )}+\frac {3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{4}}{a^{3}}\) | \(56\) |
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^2} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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