Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\sinh ^{-1}(a x)^n}{x \sqrt{a^2 x^2+1}},x\right ) \]
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Rubi [A] time = 0.102227, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx &=\int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx\\ \end{align*}
Mathematica [A] time = 5.66551, size = 0, normalized size = 0. \[ \int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{n}}{x}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{n}}{a^{2} x^{3} + x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{n}{\left (a x \right )}}{x \sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{n}}{\sqrt{a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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