Optimal. Leaf size=12 \[ \frac {\coth ^{-1}(x)^{1+n}}{1+n} \]
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6096}
\begin {gather*} \frac {\coth ^{-1}(x)^{n+1}}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 6096
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(x)^n}{1-x^2} \, dx &=\frac {\coth ^{-1}(x)^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} \frac {\coth ^{-1}(x)^{1+n}}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 13, normalized size = 1.08
method | result | size |
default | \(\frac {\mathrm {arccoth}\left (x \right )^{1+n}}{1+n}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 1.00 \begin {gather*} \frac {\operatorname {arcoth}\left (x\right )^{n + 1}}{n + 1} \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. 62 vs.
\(2 (12) = 24\).
time = 0.35, size = 62, normalized size = 5.17 \begin {gather*} \frac {\cosh \left (n \log \left (\frac {1}{2} \, \log \left (\frac {x + 1}{x - 1}\right )\right )\right ) \log \left (\frac {x + 1}{x - 1}\right ) + \log \left (\frac {x + 1}{x - 1}\right ) \sinh \left (n \log \left (\frac {1}{2} \, \log \left (\frac {x + 1}{x - 1}\right )\right )\right )}{2 \, {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.10, size = 15, normalized size = 1.25 \begin {gather*} \begin {cases} \frac {\operatorname {acoth}^{n + 1}{\left (x \right )}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (\operatorname {acoth}{\left (x \right )} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 22, normalized size = 1.83 \begin {gather*} \frac {\left (\frac {1}{2} \, \log \left (\frac {x + 1}{x - 1}\right )\right )^{n + 1}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 22, normalized size = 1.83 \begin {gather*} \left \{\begin {array}{cl} \ln \left (\mathrm {acoth}\left (x\right )\right ) & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {acoth}\left (x\right )}^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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