Optimal. Leaf size=8 \[ \frac {1}{2} \coth ^{-1}(x)^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6096}
\begin {gather*} \frac {1}{2} \coth ^{-1}(x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6096
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(x)}{1-x^2} \, dx &=\frac {1}{2} \coth ^{-1}(x)^2\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {1}{2} \coth ^{-1}(x)^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 13, normalized size = 1.62
method | result | size |
default | \(\arctanh \left (x \right ) \mathrm {arccoth}\left (x \right )-\frac {\arctanh \left (x \right )^{2}}{2}\) | \(13\) |
risch | \(\frac {\ln \left (1+x \right )^{2}}{8}-\frac {\ln \left (-1+x \right ) \ln \left (1+x \right )}{4}+\frac {\ln \left (-1+x \right )^{2}}{8}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \operatorname {arcoth}\left (x\right )^{2} \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. 14 vs.
\(2 (6) = 12\).
time = 0.35, size = 14, normalized size = 1.75 \begin {gather*} \frac {1}{8} \, \log \left (\frac {x + 1}{x - 1}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.45, size = 5, normalized size = 0.62 \begin {gather*} \frac {\operatorname {acoth}^{2}{\left (x \right )}}{2} \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. 14 vs.
\(2 (6) = 12\).
time = 0.39, size = 14, normalized size = 1.75 \begin {gather*} \frac {1}{8} \, \log \left (\frac {x + 1}{x - 1}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 21, normalized size = 2.62 \begin {gather*} \frac {{\left (\ln \left (1-\frac {1}{x}\right )-\ln \left (\frac {1}{x}+1\right )\right )}^2}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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