Optimal. Leaf size=29 \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5892, 6284, 261} \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5892
Rule 6284
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx &=\int \frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x^2} \, dx\\ &=-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x}-a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x^3} \, dx\\ &=\frac {\sqrt {1+\frac {a^2}{x^2}}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\sinh ^{-1}\left (\frac {a}{x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 49, normalized size = 1.69 \[ -\frac {a \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) - x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 39, normalized size = 1.34 \[ -\frac {\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right )}{x} + \frac {\sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.07 \[ -\frac {\frac {a \arcsinh \left (\frac {a}{x}\right )}{x}-\sqrt {1+\frac {a^{2}}{x^{2}}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 30, normalized size = 1.03 \[ -\frac {\frac {a \operatorname {arsinh}\left (\frac {a}{x}\right )}{x} - \sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 27, normalized size = 0.93 \[ \frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.93, size = 20, normalized size = 0.69 \[ \begin {cases} - \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x} + \frac {\sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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