Optimal. Leaf size=22 \[ \tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6310, 6315, 94,
212} \begin {gather*} \tanh ^{-1}\left (\sqrt {\frac {1}{x}+1} \sqrt {\frac {x-1}{x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 212
Rule 6310
Rule 6315
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)}}{1+x} \, dx &=\int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right ) x} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ &=\tanh ^{-1}\left (\sqrt {1+\frac {1}{x}} \sqrt {\frac {-1+x}{x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.82 \begin {gather*} \log \left (x \left (1+\sqrt {\frac {-1+x^2}{x^2}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 35, normalized size = 1.59
method | result | size |
trager | \(\ln \left (\sqrt {-\frac {1-x}{1+x}}\, x +\sqrt {-\frac {1-x}{1+x}}+x \right )\) | \(34\) |
default | \(\frac {\left (-1+x \right ) \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 31, normalized size = 1.41 \begin {gather*} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 31, normalized size = 1.41 \begin {gather*} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.34, size = 29, normalized size = 1.32 \begin {gather*} - \log {\left (-1 + \frac {1}{\sqrt {1 - \frac {2}{x + 1}}} \right )} + \log {\left (1 + \frac {1}{\sqrt {1 - \frac {2}{x + 1}}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 0.95 \begin {gather*} -\frac {\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{\mathrm {sgn}\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.64 \begin {gather*} 2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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