Optimal. Leaf size=43 \[ -\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sinh ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 43, 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 = {5661, 5723, 29} \[ -\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sinh ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 5661
Rule 5723
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x^3} \, dx &=-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a \int \frac {\sinh ^{-1}(a x)}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx\\ &=-\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 1.00 \[ -\frac {a \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sinh ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 67, normalized size = 1.56 \[ \frac {2 \, a^{2} x^{2} \log \relax (x) - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 98, normalized size = 2.28 \[ -{\left (a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) - a \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 67, normalized size = 1.56 \[ -a^{2} \arcsinh \left (a x \right )-\frac {a \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}}{x}-\frac {\arcsinh \left (a x \right )^{2}}{2 x^{2}}+a^{2} \ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 39, normalized size = 0.91 \[ a^{2} \log \relax (x) - \frac {\sqrt {a^{2} x^{2} + 1} a \operatorname {arsinh}\left (a x\right )}{x} - \frac {\operatorname {arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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