Optimal. Leaf size=27 \[ a \log (x)-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {5723, 29} \[ a \log (x)-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 5723
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx &=-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}+a \int \frac {1}{x} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}+a \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 29, normalized size = 1.07 \[ a \log (a x)-\frac {\sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 39, normalized size = 1.44 \[ \frac {a x \log \relax (x) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 71, normalized size = 2.63 \[ -a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) + a \log \left ({\left | x \right |}\right ) + \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 56, normalized size = 2.07 \[ -2 a \arcsinh \left (a x \right )+\frac {\left (a x -\sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right )}{x}+a \ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 25, normalized size = 0.93 \[ a \log \relax (x) - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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