Optimal. Leaf size=173 \[ \frac{5^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \text{Gamma}\left (n+1,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{5^{-n-1} \text{Gamma}\left (n+1,5 \cosh ^{-1}(a x)\right )}{32 a^5} \]
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Rubi [A] time = 0.247297, 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 = {5670, 5448, 3308, 2181} \[ \frac{5^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \text{Gamma}\left (n+1,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{5^{-n-1} \text{Gamma}\left (n+1,5 \cosh ^{-1}(a x)\right )}{32 a^5} \]
Antiderivative was successfully verified.
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Rule 5670
Rule 5448
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^4 \cosh ^{-1}(a x)^n \, dx &=\frac{\operatorname{Subst}\left (\int x^n \cosh ^4(x) \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{8} x^n \sinh (x)+\frac{3}{16} x^n \sinh (3 x)+\frac{1}{16} x^n \sinh (5 x)\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int x^n \sinh (5 x) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int x^n \sinh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{3 \operatorname{Subst}\left (\int x^n \sinh (3 x) \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}\\ &=-\frac{\operatorname{Subst}\left (\int e^{-5 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}+\frac{\operatorname{Subst}\left (\int e^{5 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac{\operatorname{Subst}\left (\int e^{-x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\operatorname{Subst}\left (\int e^x x^n \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}+\frac{3 \operatorname{Subst}\left (\int e^{3 x} x^n \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=\frac{5^{-1-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{3^{-n} \left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{\left (-\cosh ^{-1}(a x)\right )^{-n} \cosh ^{-1}(a x)^n \Gamma \left (1+n,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{\Gamma \left (1+n,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac{3^{-n} \Gamma \left (1+n,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{5^{-1-n} \Gamma \left (1+n,5 \cosh ^{-1}(a x)\right )}{32 a^5}\\ \end{align*}
Mathematica [A] time = 0.193687, size = 144, normalized size = 0.83 \[ \frac{5^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-5 \cosh ^{-1}(a x)\right )+5\ 3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 \cosh ^{-1}(a x)\right )+10 \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-\cosh ^{-1}(a x)\right )+10 \text{Gamma}\left (n+1,\cosh ^{-1}(a x)\right )+5\ 3^{-n} \text{Gamma}\left (n+1,3 \cosh ^{-1}(a x)\right )+5^{-n} \text{Gamma}\left (n+1,5 \cosh ^{-1}(a x)\right )}{160 a^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ({\rm arccosh} \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{arcosh}\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{arcosh}\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{acosh}^{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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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