Optimal. Leaf size=55 \[ -\frac {x}{4 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{4 a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5994, 199, 206} \[ -\frac {x}{4 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac {x}{4 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a}\\ &=-\frac {x}{4 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{4 a^2}+\frac {\tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 1.20 \[ \frac {-a^2 x^2 \log (a x+1)+\left (a^2 x^2-1\right ) \log (1-a x)+2 a x+\log (a x+1)-4 \tanh ^{-1}(a x)}{8 a^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 48, normalized size = 0.87 \[ \frac {2 \, a x - {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 154, normalized size = 2.80 \[ -\frac {1}{16} \, {\left ({\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) - \frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 68, normalized size = 1.24 \[ -\frac {\arctanh \left (a x \right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {1}{8 a^{2} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 a^{2}}+\frac {1}{8 a^{2} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 62, normalized size = 1.13 \[ \frac {\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}}{8 \, a} - \frac {\operatorname {artanh}\left (a x\right )}{2 \, {\left (a^{2} x^{2} - 1\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 37, normalized size = 0.67 \[ -\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a^2}-\frac {\frac {\mathrm {atanh}\left (a\,x\right )}{2}-\frac {a\,x}{4}}{a^2\,\left (a^2\,x^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 61, normalized size = 1.11 \[ \begin {cases} - \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{4} x^{2} - 4 a^{2}} + \frac {a x}{4 a^{4} x^{2} - 4 a^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{4} x^{2} - 4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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