Integrand size = 10, antiderivative size = 173 \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\frac {5^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-5 \text {arcsinh}(a x))}{32 a^5}-\frac {3^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{32 a^5}+\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{16 a^5}-\frac {\Gamma (1+n,\text {arcsinh}(a x))}{16 a^5}+\frac {3^{-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{32 a^5}-\frac {5^{-1-n} \Gamma (1+n,5 \text {arcsinh}(a x))}{32 a^5} \]
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Time = 0.15 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5780, 5556, 3388, 2212} \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\frac {5^{-n-1} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-5 \text {arcsinh}(a x))}{32 a^5}-\frac {3^{-n} \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-3 \text {arcsinh}(a x))}{32 a^5}+\frac {\text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))}{16 a^5}-\frac {\Gamma (n+1,\text {arcsinh}(a x))}{16 a^5}+\frac {3^{-n} \Gamma (n+1,3 \text {arcsinh}(a x))}{32 a^5}-\frac {5^{-n-1} \Gamma (n+1,5 \text {arcsinh}(a x))}{32 a^5} \]
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Rule 2212
Rule 3388
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \cosh (x) \sinh ^4(x) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{8} x^n \cosh (x)-\frac {3}{16} x^n \cosh (3 x)+\frac {1}{16} x^n \cosh (5 x)\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5} \\ & = \frac {\text {Subst}\left (\int x^n \cosh (5 x) \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int x^n \cosh (x) \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int x^n \cosh (3 x) \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5} \\ & = \frac {\text {Subst}\left (\int e^{-5 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{5 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}-\frac {3 \text {Subst}\left (\int e^{3 x} x^n \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5} \\ & = \frac {5^{-1-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-5 \text {arcsinh}(a x))}{32 a^5}-\frac {3^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))}{32 a^5}+\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{16 a^5}-\frac {\Gamma (1+n,\text {arcsinh}(a x))}{16 a^5}+\frac {3^{-n} \Gamma (1+n,3 \text {arcsinh}(a x))}{32 a^5}-\frac {5^{-1-n} \Gamma (1+n,5 \text {arcsinh}(a x))}{32 a^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.84 \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\frac {5^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-5 \text {arcsinh}(a x))-5\ 3^{-n} (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-3 \text {arcsinh}(a x))+10 (-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))-10 \Gamma (1+n,\text {arcsinh}(a x))+5\ 3^{-n} \Gamma (1+n,3 \text {arcsinh}(a x))-5^{-n} \Gamma (1+n,5 \text {arcsinh}(a x))}{160 a^5} \]
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\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{n}d x\]
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\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int x^{4} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
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\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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\[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{n} \,d x } \]
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Timed out. \[ \int x^4 \text {arcsinh}(a x)^n \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]
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