Optimal. Leaf size=32 \[ a \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\cosh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5662, 92, 205} \[ a \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )-\frac {\cosh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 5662
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{x^2} \, dx &=-\frac {\cosh ^{-1}(a x)}{x}+a \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\cosh ^{-1}(a x)}{x}+a^2 \operatorname {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )\\ &=-\frac {\cosh ^{-1}(a x)}{x}+a \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 1.78 \[ \frac {a \sqrt {a^2 x^2-1} \tan ^{-1}\left (\sqrt {a^2 x^2-1}\right )}{\sqrt {a x-1} \sqrt {a x+1}}-\frac {\cosh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 65, normalized size = 2.03 \[ \frac {2 \, a x \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (x - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + x \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 36, normalized size = 1.12 \[ a \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 51, normalized size = 1.59 \[ -\frac {\mathrm {arccosh}\left (a x \right )}{x}-\frac {a \sqrt {a x -1}\, \sqrt {a x +1}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{\sqrt {a^{2} x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 22, normalized size = 0.69 \[ -a \arcsin \left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\operatorname {arcosh}\left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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