Optimal. Leaf size=35 \[ -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6310, 6313,
272, 43, 65, 212} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 212
Rule 272
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} (1-x) \, dx &=-\int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right ) x \, dx\\ &=\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right )\\ &=-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.11 \begin {gather*} -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \log \left (\left (1+\sqrt {1-\frac {1}{x^2}}\right ) x\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 48, normalized size = 1.37
method | result | size |
default | \(-\frac {\left (-1+x \right ) \left (x \sqrt {x^{2}-1}-\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}}\) | \(48\) |
risch | \(-\frac {x \left (-1+x \right )}{2 \sqrt {\frac {-1+x}{1+x}}}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right )}\) | \(56\) |
trager | \(-\frac {\left (1+x \right ) \sqrt {-\frac {1-x}{1+x}}\, x}{2}+\frac {\ln \left (\sqrt {-\frac {1-x}{1+x}}\, x +\sqrt {-\frac {1-x}{1+x}}+x \right )}{2}\) | \(57\) |
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. 83 vs.
\(2 (27) = 54\).
time = 0.26, size = 83, normalized size = 2.37 \begin {gather*} \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {x - 1}{x + 1}}}{\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \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 = 51, normalized size = 1.46 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} + x\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\, dx - \int \left (- \frac {1}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 38, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {x^{2} - 1} x}{2 \, \mathrm {sgn}\left (x + 1\right )} - \frac {\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right )}{2 \, \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 = 63, normalized size = 1.80 \begin {gather*} \mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\sqrt {\frac {x-1}{x+1}}+{\left (\frac {x-1}{x+1}\right )}^{3/2}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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