Optimal. Leaf size=113 \[ \frac {3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \Gamma \left (n+1,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac {3^{-n-1} \Gamma \left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^4} \]
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Rubi [A] time = 0.25, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5779, 3312, 3308, 2181} \[ \frac {3^{-n-1} \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \sinh ^{-1}(a x)^n \left (-\sinh ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \text {Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac {3^{-n-1} \text {Gamma}\left (n+1,3 \sinh ^{-1}(a x)\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 3312
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^3 \sinh ^{-1}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^n \sinh ^3(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {i \operatorname {Subst}\left (\int \left (\frac {3}{4} i x^n \sinh (x)-\frac {1}{4} i x^n \sinh (3 x)\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int x^n \sinh (3 x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}-\frac {3 \operatorname {Subst}\left (\int x^n \sinh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac {\operatorname {Subst}\left (\int e^{3 x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac {3 \operatorname {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \operatorname {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{8 a^4}\\ &=\frac {3^{-1-n} \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-3 \sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{8 a^4}-\frac {3 \Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{8 a^4}+\frac {3^{-1-n} \Gamma \left (1+n,3 \sinh ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 100, normalized size = 0.88 \[ \frac {3^{-n-1} \left (-\sinh ^{-1}(a x)\right )^{-n} \left (\left (-\sinh ^{-1}(a x)\right )^n \left (\Gamma \left (n+1,3 \sinh ^{-1}(a x)\right )-3^{n+2} \Gamma \left (n+1,\sinh ^{-1}(a x)\right )\right )+\sinh ^{-1}(a x)^n \Gamma \left (n+1,-3 \sinh ^{-1}(a x)\right )-3^{n+2} \sinh ^{-1}(a x)^n \Gamma \left (n+1,-\sinh ^{-1}(a x)\right )\right )}{8 a^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {x^{3} \arcsinh \left (a x \right )^{n}}{\sqrt {a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {x^{3} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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