Integrand size = 12, antiderivative size = 130 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3} \]
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Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5778, 3389, 2211, 2235, 2236} \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5778
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 \sqrt {x}}+\frac {3 \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{2 a^3}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{2 a^3} \\ & = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{4 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {-e^{-3 \text {arcsinh}(a x)}+e^{-\text {arcsinh}(a x)}+e^{\text {arcsinh}(a x)}-e^{3 \text {arcsinh}(a x)}+\sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )-\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )-\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )+\sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )}{4 a^3 \sqrt {\text {arcsinh}(a x)}} \]
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\[\int \frac {x^{2}}{\operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]
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