Optimal. Leaf size=119 \[ \frac {2^{-2 (n+3)} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-4 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \Gamma \left (n+1,2 \sinh ^{-1}(a x)\right )}{a^4}+\frac {2^{-2 (n+3)} \Gamma \left (n+1,4 \sinh ^{-1}(a x)\right )}{a^4} \]
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Rubi [A] time = 0.17, 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 = {5669, 5448, 3308, 2181} \[ \frac {2^{-2 (n+3)} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-4 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 \sinh ^{-1}(a x)\right )}{a^4}-\frac {2^{-n-4} \text {Gamma}\left (n+1,2 \sinh ^{-1}(a x)\right )}{a^4}+\frac {2^{-2 (n+3)} \text {Gamma}\left (n+1,4 \sinh ^{-1}(a x)\right )}{a^4} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 5448
Rule 5669
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}(a x)^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \cosh (x) \sinh ^3(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {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 {\operatorname {Subst}\left (\int x^n \sinh (4 x) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\operatorname {Subst}\left (\int x^n \sinh (2 x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-4 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^4}+\frac {\operatorname {Subst}\left (\int e^{4 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{16 a^4}+\frac {\operatorname {Subst}\left (\int e^{-2 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {\operatorname {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 = 0.07, size = 99, normalized size = 0.83 \[ \frac {4^{-n-3} \left (-\sinh ^{-1}(a x)\right )^{-n} \left (\left (-\sinh ^{-1}(a x)\right )^n \left (\Gamma \left (n+1,4 \sinh ^{-1}(a x)\right )-2^{n+2} \Gamma \left (n+1,2 \sinh ^{-1}(a x)\right )\right )+\sinh ^{-1}(a x)^n \Gamma \left (n+1,-4 \sinh ^{-1}(a x)\right )-2^{n+2} \sinh ^{-1}(a x)^n \Gamma \left (n+1,-2 \sinh ^{-1}(a x)\right )\right )}{a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {arsinh}\left (a x\right )^{n}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
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 \[ \int x^{3} \arcsinh \left (a x \right )^{n}\, dx \]
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 \[ \int x^{3} \operatorname {arsinh}\left (a x\right )^{n}\,{d x} \]
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 \[ \int x^3\,{\mathrm {asinh}\left (a\,x\right )}^n \,d x \]
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 \[ \int x^{3} \operatorname {asinh}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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