Optimal. Leaf size=173 \[ \frac {5^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-5 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3^{-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {3^{-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{32 a^5}-\frac {5^{-1-n} \Gamma \left (1+n,5 \sinh ^{-1}(a x)\right )}{32 a^5} \]
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Rubi [A]
time = 0.16, 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 = {5780, 5556,
3388, 2212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 5556
Rule 5780
Rubi steps
\begin {align*} \int x^4 \sinh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \cosh (x) \sinh ^4(x) \, dx,x,\sinh ^{-1}(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,\sinh ^{-1}(a x)\right )}{a^5}\\ &=\frac {\text {Subst}\left (\int x^n \cosh (5 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int x^n \cosh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^5}-\frac {3 \text {Subst}\left (\int x^n \cosh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}\\ &=\frac {\text {Subst}\left (\int e^{-5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{5 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}+\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}+\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{32 a^5}-\frac {3 \text {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*}
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Mathematica [A]
time = 0.10, size = 145, normalized size = 0.84 \begin {gather*} \frac {5^{-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-5 \sinh ^{-1}(a x)\right )-5\ 3^{-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )+10 \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )-10 \Gamma \left (1+n,\sinh ^{-1}(a x)\right )+5\ 3^{-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )-5^{-n} \Gamma \left (1+n,5 \sinh ^{-1}(a x)\right )}{160 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.90, size = 0, normalized size = 0.00 \[\int x^{4} \arcsinh \left (a x \right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{4} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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