Optimal. Leaf size=119 \[ \frac {2^{-2 (3+n)} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-4 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-4-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-4-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^4}+\frac {2^{-2 (3+n)} \Gamma \left (1+n,4 \sinh ^{-1}(a x)\right )}{a^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5780, 5556,
3389, 2212} \begin {gather*} \frac {2^{-2 (n+3)} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-4 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \text {Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^4}+\frac {2^{-2 (n+3)} \text {Gamma}\left (n+1,4 \sinh ^{-1}(a x)\right )}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 5556
Rule 5780
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \cosh (x) \sinh ^3(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{4} x^n \sinh (2 x)+\frac {1}{8} x^n \sinh (4 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int x^n \sinh (4 x) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\text {Subst}\left (\int e^{-4 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{4 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^4}+\frac {\text {Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int e^{2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {4^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-4 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-4-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-4-n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )}{a^4}+\frac {4^{-3-n} \Gamma \left (1+n,4 \sinh ^{-1}(a x)\right )}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 99, normalized size = 0.83 \begin {gather*} \frac {4^{-3-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \left (\sinh ^{-1}(a x)^n \Gamma \left (1+n,-4 \sinh ^{-1}(a x)\right )-2^{2+n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-2 \sinh ^{-1}(a x)\right )+\left (-\sinh ^{-1}(a x)\right )^n \left (-2^{2+n} \Gamma \left (1+n,2 \sinh ^{-1}(a x)\right )+\Gamma \left (1+n,4 \sinh ^{-1}(a x)\right )\right )\right )}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.73, size = 0, normalized size = 0.00 \[\int x^{3} \arcsinh \left (a x \right )^{n}\, dx\]
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^{3} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,{\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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