Optimal. Leaf size=17 \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
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Rubi [A] time = 0.0380282, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5675} \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx &=\frac{\sinh ^{-1}(a x)^{1+n}}{a (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0098283, size = 17, normalized size = 1. \[ \frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{1+n}}{a \left ( 1+n \right ) }} \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 [B] time = 2.52692, size = 212, normalized size = 12.47 \begin{align*} \frac{\cosh \left (n \log \left (\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )\right )\right ) \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) \sinh \left (n \log \left (\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )\right )\right )}{a n + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05296, size = 34, normalized size = 2. \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge n = -1 \\0^{n} x & \text{for}\: a = 0 \\\frac{\log{\left (\operatorname{asinh}{\left (a x \right )} \right )}}{a} & \text{for}\: n = -1 \\\frac{\operatorname{asinh}{\left (a x \right )} \operatorname{asinh}^{n}{\left (a x \right )}}{a n + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39808, size = 39, normalized size = 2.29 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{n + 1}}{a{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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