Optimal. Leaf size=173 \[ \frac {5^{-n-1} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {\cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {\Gamma \left (n+1,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (n+1,3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {5^{-n-1} \Gamma \left (n+1,5 \cosh ^{-1}(a x)\right )}{32 a^5} \]
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Rubi [A] time = 0.25, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2181
Rule 3308
Rule 5448
Rule 5670
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*}
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Mathematica [A] time = 0.21, size = 144, normalized size = 0.83 \[ \frac {5^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-5 \cosh ^{-1}(a x)\right )+5\ 3^{-n} \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \cosh ^{-1}(a x)\right )+10 \cosh ^{-1}(a x)^n \left (-\cosh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\cosh ^{-1}(a x)\right )+10 \Gamma \left (n+1,\cosh ^{-1}(a x)\right )+5\ 3^{-n} \Gamma \left (n+1,3 \cosh ^{-1}(a x)\right )+5^{-n} \Gamma \left (n+1,5 \cosh ^{-1}(a x)\right )}{160 a^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arcosh}\left (a x\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{4} \mathrm {arccosh}\left (a x \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arcosh}\left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {acosh}\left (a\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {acosh}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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