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 = 0.25, antiderivative size = 72, normalized size of antiderivative = 1.50, 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 29
Rule 5724
Rule 5798
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*}
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Mathematica [A] time = 0.03, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.52, size = 80, normalized size = 1.67 \[ \frac {1}{2} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - i \, a \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 168, normalized size = 3.50 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) a}{a^{2} x^{2}-1}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \mathrm {arccosh}\left (a x \right )}{x \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.86, size = 73, normalized size = 1.52 \[ -\frac {1}{2} \, {\left (a^{2} \sqrt {-\frac {1}{a^{4}}} \log \left (x^{2} - \frac {1}{a^{2}}\right ) + i \, \left (-1\right )^{-2 \, a^{2} x^{2} + 2} \log \left (-2 \, a^{2} + \frac {2}{x^{2}}\right )\right )} a - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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