Optimal. Leaf size=40 \[ \frac{2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0724067, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {5677, 5675} \[ \frac{2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5677
Rule 5675
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0420324, size = 40, normalized size = 1. \[ \frac{2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 36, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}{a\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} \sqrt{\operatorname{arsinh}\left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \frac{1}{\sqrt{c \left (a^{2} x^{2} + 1\right )} \sqrt{\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 \frac{1}{\sqrt{a^{2} c x^{2} + c} \sqrt{\operatorname{arsinh}\left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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