Optimal. Leaf size=40 \[ -\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}-\frac {a x^3}{12}+\frac {x}{4 a} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5994} \[ -\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}-\frac {a x^3}{12}+\frac {x}{4 a} \]
Antiderivative was successfully verified.
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Rule 5994
Rubi steps
\begin {align*} \int x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}+\frac {\int \left (1-a^2 x^2\right ) \, dx}{4 a}\\ &=\frac {x}{4 a}-\frac {a x^3}{12}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.72 \[ -\frac {1}{4} a^2 x^4 \tanh ^{-1}(a x)+\frac {\log (1-a x)}{8 a^2}-\frac {\log (a x+1)}{8 a^2}-\frac {a x^3}{12}+\frac {1}{2} x^2 \tanh ^{-1}(a x)+\frac {x}{4 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 52, normalized size = 1.30 \[ -\frac {2 \, a^{3} x^{3} - 6 \, a x + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{24 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 160, normalized size = 4.00 \[ -\frac {1}{3} \, a {\left (\frac {\frac {3 \, {\left (a x + 1\right )}}{a x - 1} - 1}{a^{3} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{{\left (a x - 1\right )}^{2} a^{3} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 1.42 \[ -\frac {a^{2} \arctanh \left (a x \right ) x^{4}}{4}+\frac {\arctanh \left (a x \right ) x^{2}}{2}-\frac {x^{3} a}{12}+\frac {x}{4 a}+\frac {\ln \left (a x -1\right )}{8 a^{2}}-\frac {\ln \left (a x +1\right )}{8 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 37, normalized size = 0.92 \[ -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{4 \, a^{2}} - \frac {a^{2} x^{3} - 3 \, x}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.86, size = 44, normalized size = 1.10 \[ \frac {x^2\,\mathrm {atanh}\left (a\,x\right )}{2}-\frac {\frac {\mathrm {atanh}\left (a\,x\right )}{4}-\frac {a\,x}{4}}{a^2}-\frac {a\,x^3}{12}-\frac {a^2\,x^4\,\mathrm {atanh}\left (a\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 46, normalized size = 1.15 \[ \begin {cases} - \frac {a^{2} x^{4} \operatorname {atanh}{\left (a x \right )}}{4} - \frac {a x^{3}}{12} + \frac {x^{2} \operatorname {atanh}{\left (a x \right )}}{2} + \frac {x}{4 a} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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