Optimal. Leaf size=8 \[ \frac {\text {arctanh}\left (x^2\right )}{2} \]
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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 212}
\begin {gather*} \frac {\text {arctanh}\left (x^2\right )}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,x^2\right )\\ &=\frac {\text {arctanh}\left (x^2\right )}{2}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).
time = 0.00, size = 23, normalized size = 2.88 \begin {gather*} -\frac {1}{4} \log \left (1-x^2\right )+\frac {1}{4} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs.
\(2(6)=12\).
time = 0.07, size = 22, normalized size = 2.75
method | result | size |
meijerg | \(\frac {\arctanh \left (x^{2}\right )}{2}\) | \(7\) |
risch | \(\frac {\ln \left (x^{2}+1\right )}{4}-\frac {\ln \left (x^{2}-1\right )}{4}\) | \(18\) |
default | \(-\frac {\ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
norman | \(-\frac {\ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
parallelrisch | \(-\frac {\ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs.
\(2 (6) = 12\).
time = 0.37, size = 17, normalized size = 2.12 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{4} \, \log \left (x^{2} - 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. 17 vs.
\(2 (6) = 12\).
time = 0.56, size = 17, normalized size = 2.12 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{4} \, \log \left (x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (5) = 10\).
time = 0.03, size = 15, normalized size = 1.88 \begin {gather*} - \frac {\log {\left (x^{2} - 1 \right )}}{4} + \frac {\log {\left (x^{2} + 1 \right )}}{4} \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. 18 vs.
\(2 (6) = 12\).
time = 0.53, size = 18, normalized size = 2.25 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{4} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 6, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (x^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 8, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (x^{2}-1\right )}{2} \end {gather*}
Warning: Unable to verify antiderivative.
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