Optimal. Leaf size=27 \[ -\frac {\sinh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 272, 65,
214} \begin {gather*} -a \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {\sinh ^{-1}(a x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 5776
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{x^2} \, dx &=-\frac {\sinh ^{-1}(a x)}{x}+a \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}(a x)}{x}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sinh ^{-1}(a x)}{x}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a}\\ &=-\frac {\sinh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {\sinh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 30, normalized size = 1.11
method | result | size |
derivativedivides | \(a \left (-\frac {\arcsinh \left (a x \right )}{a x}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\) | \(30\) |
default | \(a \left (-\frac {\arcsinh \left (a x \right )}{a x}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 22, normalized size = 0.81 \begin {gather*} -a \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\operatorname {arsinh}\left (a x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (25) = 50\).
time = 0.39, size = 90, normalized size = 3.33 \begin {gather*} -\frac {a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - {\left (x - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - x \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{x} \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}{\left (a x \right )}}{x^{2}}\, 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. 56 vs.
\(2 (25) = 50\).
time = 0.41, size = 56, normalized size = 2.07 \begin {gather*} -\frac {1}{2} \, a {\left (\log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right )\right )} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{x} \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 = -0.04 \begin {gather*} \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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