Optimal. Leaf size=30 \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)-\frac {\coth ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5917, 266, 36, 29, 31} \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)-\frac {\coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5917
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{x^2} \, dx &=-\frac {\coth ^{-1}(a x)}{x}+a \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)}{x}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\coth ^{-1}(a x)}{x}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {\coth ^{-1}(a x)}{x}+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)-\frac {\coth ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 39, normalized size = 1.30 \[ -\frac {a x \log \left (a^{2} x^{2} - 1\right ) - 2 \, a x \log \relax (x) + \log \left (\frac {a x + 1}{a x - 1}\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.17 \[ -\frac {\mathrm {arccoth}\left (a x \right )}{x}+a \ln \left (a x \right )-\frac {a \ln \left (a x -1\right )}{2}-\frac {a \ln \left (a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 1.00 \[ -\frac {1}{2} \, a {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\operatorname {arcoth}\left (a x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 27, normalized size = 0.90 \[ a\,\ln \relax (x)-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}-\frac {\mathrm {acoth}\left (a\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 26, normalized size = 0.87 \[ a \log {\relax (x )} - a \log {\left (a x + 1 \right )} + a \operatorname {acoth}{\left (a x \right )} - \frac {\operatorname {acoth}{\left (a x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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