Integrand size = 12, antiderivative size = 163 \[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5} \]
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Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5780, 5556, 3388, 2211, 2235, 2236} \[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {x}}-\frac {3 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5} \\ & = \frac {\text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5} \\ & = \frac {\text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5} \\ & = \frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{32 a^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\frac {\sqrt {5} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {5 \sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {10 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-10 \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )+5 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )-\sqrt {5} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )}{160 a^5} \]
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\[\int \frac {x^{4}}{\sqrt {\operatorname {arcsinh}\left (a x \right )}}d x\]
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Exception generated. \[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {x^{4}}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
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\[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {x^{4}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {x^{4}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {x^4}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \]
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