Optimal. Leaf size=15 \[ -\frac {1}{3 x^3}-\frac {1}{x}+\tanh ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 331, 212}
\begin {gather*} -\frac {1}{3 x^3}-\frac {1}{x}+\tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 331
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (x-x^3\right )} \, dx &=\int \frac {1}{x^4 \left (1-x^2\right )} \, dx\\ &=-\frac {1}{3 x^3}+\int \frac {1}{x^2 \left (1-x^2\right )} \, dx\\ &=-\frac {1}{3 x^3}-\frac {1}{x}+\int \frac {1}{1-x^2} \, dx\\ &=-\frac {1}{3 x^3}-\frac {1}{x}+\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(31\) vs. \(2(15)=30\).
time = 0.00, size = 31, normalized size = 2.07 \begin {gather*} -\frac {1}{3 x^3}-\frac {1}{x}-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 24, normalized size = 1.60
method | result | size |
meijerg | \(-\frac {i \left (-\frac {2 i}{x}-\frac {2 i}{3 x^{3}}+2 i \arctanh \left (x \right )\right )}{2}\) | \(22\) |
default | \(\frac {\ln \left (x +1\right )}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {1}{3 x^{3}}-\frac {1}{x}\) | \(24\) |
norman | \(\frac {-\frac {1}{3}-x^{2}}{x^{3}}-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(25\) |
risch | \(\frac {-\frac {1}{3}-x^{2}}{x^{3}}-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 1.67 \begin {gather*} -\frac {3 \, x^{2} + 1}{3 \, x^{3}} + \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (13) = 26\).
time = 2.57, size = 30, normalized size = 2.00 \begin {gather*} \frac {3 \, x^{3} \log \left (x + 1\right ) - 3 \, x^{3} \log \left (x - 1\right ) - 6 \, x^{2} - 2}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 24, normalized size = 1.60 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} - \frac {3 x^{2} + 1}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (13) = 26\).
time = 1.04, size = 27, normalized size = 1.80 \begin {gather*} -\frac {3 \, x^{2} + 1}{3 \, x^{3}} + \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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Mupad [B]
time = 4.92, size = 13, normalized size = 0.87 \begin {gather*} \mathrm {atanh}\left (x\right )-\frac {x^2+\frac {1}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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