Optimal. Leaf size=6 \[ \tanh ^{-1}(x)-x \]
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Rubi [A] time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1584, 321, 206} \begin {gather*} \tanh ^{-1}(x)-x \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^3}{x-x^3} \, dx &=\int \frac {x^2}{1-x^2} \, dx\\ &=-x+\int \frac {1}{1-x^2} \, dx\\ &=-x+\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 22, normalized size = 3.67 \begin {gather*} -x-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{x-x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 16, normalized size = 2.67 \begin {gather*} -x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 18, normalized size = 3.00 \begin {gather*} -x + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 17, normalized size = 2.83 \begin {gather*} -x -\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 16, normalized size = 2.67 \begin {gather*} -x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 6, normalized size = 1.00 \begin {gather*} \mathrm {atanh}\relax (x)-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 14, normalized size = 2.33 \begin {gather*} - x - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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