Optimal. Leaf size=43 \[ -\frac {a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac {\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x) \]
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Rubi [A]
time = 0.05, 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 = {5776, 5800, 29}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 5776
Rule 5800
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 = 0.02, size = 43, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.17, size = 72, normalized size = 1.67
method | result | size |
derivativedivides | \(a^{2} \left (-2 \arcsinh \left (a x \right )-\frac {\arcsinh \left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\) | \(72\) |
default | \(a^{2} \left (-2 \arcsinh \left (a x \right )-\frac {\arcsinh \left (a x \right ) \left (-2 a^{2} x^{2}+2 a x \sqrt {a^{2} x^{2}+1}+\arcsinh \left (a x \right )\right )}{2 a^{2} x^{2}}+\ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right )\right )\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 39, normalized size = 0.91 \begin {gather*} a^{2} \log \left (x\right ) - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 67, normalized size = 1.56 \begin {gather*} \frac {2 \, a^{2} x^{2} \log \left (x\right ) - 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}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (39) = 78\).
time = 0.46, size = 98, normalized size = 2.28 \begin {gather*} -{\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}} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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