Optimal. Leaf size=33 \[ -\frac {a \sqrt {a^2 x^2+1}}{2 x}-\frac {\sinh ^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5661, 264} \[ -\frac {a \sqrt {a^2 x^2+1}}{2 x}-\frac {\sinh ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 5661
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{x^3} \, dx &=-\frac {\sinh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2}}{2 x}-\frac {\sinh ^{-1}(a x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.85 \[ -\frac {a x \sqrt {a^2 x^2+1}+\sinh ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 36, normalized size = 1.09 \[ -\frac {\sqrt {a^{2} x^{2} + 1} a x + \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 50, normalized size = 1.52 \[ \frac {a {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 37, normalized size = 1.12 \[ a^{2} \left (-\frac {\arcsinh \left (a x \right )}{2 a^{2} x^{2}}-\frac {\sqrt {a^{2} x^{2}+1}}{2 a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 0.82 \[ -\frac {\sqrt {a^{2} x^{2} + 1} a}{2 \, x} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, x^{2}} \]
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 \[ \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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