Optimal. Leaf size=48 \[ -\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{x}-\frac{a \sqrt{a x-1} \log (x)}{\sqrt{1-a x}} \]
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Rubi [A] time = 0.254105, antiderivative size = 72, normalized size of antiderivative = 1.5, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {5798, 5724, 29} \[ -\frac{a \sqrt{a x-1} \sqrt{a x+1} \log (x)}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5724
Rule 29
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{x^2 \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{a \sqrt{-1+a x} \sqrt{1+a x} \log (x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0302518, size = 57, normalized size = 1.19 \[ \frac{\left (a^2 x^2-1\right ) \cosh ^{-1}(a x)-a x \sqrt{a x-1} \sqrt{a x+1} \log (x)}{x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.136, size = 168, normalized size = 3.5 \begin{align*} -2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\rm arccosh} \left (ax\right )a}{{a}^{2}{x}^{2}-1}}-{\frac{{\rm arccosh} \left (ax\right )}{ \left ({a}^{2}{x}^{2}-1 \right ) x}\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }+{\frac{a}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.78803, size = 99, normalized size = 2.06 \begin{align*} -\frac{1}{2} \,{\left (a^{2} \sqrt{-\frac{1}{a^{4}}} \log \left (x^{2} - \frac{1}{a^{2}}\right ) + i \, \left (-1\right )^{-2 \, a^{2} x^{2} + 2} \log \left (-2 \, a^{2} + \frac{2}{x^{2}}\right )\right )} a - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}{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{\operatorname{acosh}{\left (a x \right )}}{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.20838, size = 116, normalized size = 2.42 \begin{align*} -\frac{1}{2} i \, a \log \left (-i \, a^{2} x^{2}\right ) + \frac{1}{2} \,{\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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