Optimal. Leaf size=49 \[ \frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{2 a}-\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5774, 3388,
2212} \begin {gather*} \frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \text {Gamma}\left (n+1,-\sinh ^{-1}(a x)\right )}{2 a}-\frac {\text {Gamma}\left (n+1,\sinh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 5774
Rubi steps
\begin {align*} \int \sinh ^{-1}(a x)^n \, dx &=\frac {\text {Subst}\left (\int x^n \cosh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^{-x} x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int e^x x^n \, dx,x,\sinh ^{-1}(a x)\right )}{2 a}\\ &=\frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )}{2 a}-\frac {\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.92 \begin {gather*} \frac {\left (-\sinh ^{-1}(a x)\right )^{-n} \sinh ^{-1}(a x)^n \Gamma \left (1+n,-\sinh ^{-1}(a x)\right )-\Gamma \left (1+n,\sinh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 1.64, size = 40, normalized size = 0.82
method | result | size |
default | \(\frac {\arcsinh \left (a x \right )^{1+n} \hypergeom \left (\left [\frac {1}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {3}{2}\right ], \frac {\arcsinh \left (a x \right )^{2}}{4}\right )}{a \left (1+n \right )}\) | \(40\) |
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 \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 = -0.02 \begin {gather*} \int {\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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