Optimal. Leaf size=173 \[ \frac{5^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{5^{-n-1} \text{Gamma}\left (n+1,5 \sinh ^{-1}(a x)\right )}{32 a^5} \]
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Rubi [A] time = 0.216363, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5669, 5448, 3307, 2181} \[ \frac{5^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{\sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac{\text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{5^{-n-1} \text{Gamma}\left (n+1,5 \sinh ^{-1}(a x)\right )}{32 a^5} \]
Antiderivative was successfully verified.
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Rule 5669
Rule 5448
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^4 \sinh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \cosh (x) \sinh ^4(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{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,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int x^n \cosh (5 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int x^n \cosh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac{3 \operatorname{Subst}\left (\int x^n \cosh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{\operatorname{Subst}\left (\int e^{-5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac{\operatorname{Subst}\left (\int e^{5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac{\operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac{5^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{3^{-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac{\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac{\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac{5^{-1-n} \Gamma \left (1+n,5 \sinh ^{-1}(a x)\right )}{32 a^5}\\ \end{align*}
Mathematica [A] time = 0.146366, size = 145, normalized size = 0.84 \[ \frac{5^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \sinh ^{-1}(a x)\right )-5\ 3^{-n} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )+10 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )-10 \text{Gamma}\left (n+1,\sinh ^{-1}(a x)\right )+5\ 3^{-n} \text{Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )-5^{-n} \text{Gamma}\left (n+1,5 \sinh ^{-1}(a x)\right )}{160 a^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.164, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{arsinh}\left (a x\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{asinh}^{n}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arsinh}\left (a x\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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