Integrand size = 12, antiderivative size = 188 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x^4 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5} \]
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Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5778, 3389, 2211, 2235, 2236} \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5778
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {x}}-\frac {9 \sinh (3 x)}{16 \sqrt {x}}+\frac {5 \sinh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = -\frac {2 x^4 \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 )}{4 a^5}+\frac {5 \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5} \\ & = -\frac {2 x^4 \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 )}{8 a^5}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {5 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {5 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {9 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}-\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5} \\ & = -\frac {2 x^4 \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 )}{4 a^5}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4 a^5}-\frac {5 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {9 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {9 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{16 a^5} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.41 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {e^{-5 \text {arcsinh}(a x)} \left (-1+3 e^{2 \text {arcsinh}(a x)}-2 e^{4 \text {arcsinh}(a x)}-2 e^{6 \text {arcsinh}(a x)}+3 e^{8 \text {arcsinh}(a x)}-e^{10 \text {arcsinh}(a x)}+\sqrt {5} e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )-3 \sqrt {3} e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )+2 e^{5 \text {arcsinh}(a x)} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+2 e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )-3 \sqrt {3} e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )+\sqrt {5} e^{5 \text {arcsinh}(a x)} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )\right )}{16 a^5 \sqrt {\text {arcsinh}(a x)}} \]
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\[\int \frac {x^{4}}{\operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]
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