Optimal. Leaf size=60 \[ \frac{x^{m+1} \sinh ^{-1}(a x)}{m+1}-\frac{a x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )}{m^2+3 m+2} \]
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Rubi [A] time = 0.0195273, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5661, 364} \[ \frac{x^{m+1} \sinh ^{-1}(a x)}{m+1}-\frac{a x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right )}{m^2+3 m+2} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 364
Rubi steps
\begin{align*} \int x^m \sinh ^{-1}(a x) \, dx &=\frac{x^{1+m} \sinh ^{-1}(a x)}{1+m}-\frac{a \int \frac{x^{1+m}}{\sqrt{1+a^2 x^2}} \, dx}{1+m}\\ &=\frac{x^{1+m} \sinh ^{-1}(a x)}{1+m}-\frac{a x^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}\\ \end{align*}
Mathematica [A] time = 0.0222289, size = 55, normalized size = 0.92 \[ \frac{x^{m+1} \left ((m+2) \sinh ^{-1}(a x)-a x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},-a^2 x^2\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.546, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\it Arcsinh} \left ( ax \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{arsinh}\left (a x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{asinh}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arsinh}\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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