Optimal. Leaf size=38 \[ -a \log \left (1-a^2 x^2\right )+a^2 (-x) \tanh ^{-1}(a x)+a \log (x)-\frac {\tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6014, 5916, 266, 36, 29, 31, 5910, 260} \[ -a \log \left (1-a^2 x^2\right )+a^2 (-x) \tanh ^{-1}(a x)+a \log (x)-\frac {\tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 5910
Rule 5916
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x^2} \, dx &=-\left (a^2 \int \tanh ^{-1}(a x) \, dx\right )+\int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{x}-a^2 x \tanh ^{-1}(a x)+a \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+a^3 \int \frac {x}{1-a^2 x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{x}-a^2 x \tanh ^{-1}(a x)-\frac {1}{2} a \log \left (1-a^2 x^2\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}(a x)}{x}-a^2 x \tanh ^{-1}(a x)-\frac {1}{2} a \log \left (1-a^2 x^2\right )+\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 {\tanh ^{-1}(a x)}{x}-a^2 x \tanh ^{-1}(a x)+a \log (x)-a \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \[ -a \log \left (1-a^2 x^2\right )+a^2 (-x) \tanh ^{-1}(a x)+a \log (x)-\frac {\tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 51, normalized size = 1.34 \[ -\frac {2 \, a x \log \left (a^{2} x^{2} - 1\right ) - 2 \, a x \log \relax (x) + {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 145, normalized size = 3.82 \[ -a {\left (\frac {2 \, \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1} + \log \left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}}\right ) - \log \left ({\left | \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - 1 \right |}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 45, normalized size = 1.18 \[ -a^{2} x \arctanh \left (a x \right )-\frac {\arctanh \left (a x \right )}{x}+a \ln \left (a x \right )-a \ln \left (a x -1\right )-a \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 0.95 \[ -a {\left (\log \left (a x + 1\right ) + \log \left (a x - 1\right ) - \log \relax (x)\right )} - {\left (a^{2} x + \frac {1}{x}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 37, normalized size = 0.97 \[ a\,\ln \relax (x)-a\,\ln \left (a^2\,x^2-1\right )-\frac {\mathrm {atanh}\left (a\,x\right )}{x}-a^2\,x\,\mathrm {atanh}\left (a\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 41, normalized size = 1.08 \[ \begin {cases} - a^{2} x \operatorname {atanh}{\left (a x \right )} + a \log {\relax (x )} - 2 a \log {\left (x - \frac {1}{a} \right )} - 2 a \operatorname {atanh}{\left (a x \right )} - \frac {\operatorname {atanh}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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