Optimal. Leaf size=28 \[ \frac{\sqrt{a x-1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{1-a x}} \]
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Rubi [A] time = 0.162425, antiderivative size = 41, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5713, 5674} \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5674
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0591602, size = 47, normalized size = 1.68 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{-(a x-1) (a x+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 48, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ({\rm arccosh} \left (ax\right ) \right ) }{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}} \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} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02138, size = 117, normalized size = 4.18 \begin{align*} -\frac{\sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} \log \left (\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )\right )}{a^{3} x^{2} - a} \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{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{acosh}{\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} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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