Optimal. Leaf size=25 \[ a \tanh ^{-1}\left (\sqrt {\frac {a^2}{x^2}+1}\right )+x \text {csch}^{-1}\left (\frac {x}{a}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5892, 6278, 266, 63, 208} \[ a \tanh ^{-1}\left (\sqrt {\frac {a^2}{x^2}+1}\right )+x \text {csch}^{-1}\left (\frac {x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 5892
Rule 6278
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int \text {csch}^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=x \text {csch}^{-1}\left (\frac {x}{a}\right )+a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x} \, dx\\ &=x \text {csch}^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=x \text {csch}^{-1}\left (\frac {x}{a}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+\frac {a^2}{x^2}}\right )}{a}\\ &=x \text {csch}^{-1}\left (\frac {x}{a}\right )+a \tanh ^{-1}\left (\sqrt {1+\frac {a^2}{x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.10, size = 77, normalized size = 3.08 \[ \frac {a \sqrt {a^2+x^2} \left (\log \left (\frac {x}{\sqrt {a^2+x^2}}+1\right )-\log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right )\right )}{2 x \sqrt {\frac {a^2}{x^2}+1}}+x \sinh ^{-1}\left (\frac {a}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 96, normalized size = 3.84 \[ -a \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - x\right ) + {\left (x - 1\right )} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a - x\right ) - \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - a - x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 49, normalized size = 1.96 \[ a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x) + x \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right ) - \frac {a \log \left (-x + \sqrt {a^{2} + x^{2}}\right )}{\mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.24 \[ -a \left (-\frac {\arcsinh \left (\frac {a}{x}\right ) x}{a}-\arctanh \left (\frac {1}{\sqrt {1+\frac {a^{2}}{x^{2}}}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 43, normalized size = 1.72 \[ \frac {1}{2} \, a {\left (\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} - 1\right )\right )} + x \operatorname {arsinh}\left (\frac {a}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 25, normalized size = 1.00 \[ x\,\mathrm {asinh}\left (\frac {a}{x}\right )+a\,\ln \left (x+\sqrt {a^2+x^2}\right )\,\mathrm {sign}\relax (x) \]
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 \[ \int \operatorname {asinh}{\left (\frac {a}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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