Optimal. Leaf size=33 \[ \frac {2 \left (\frac {1}{x}+1\right )}{\sqrt {1-\frac {1}{x^2}}}-\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6175, 6178, 852, 1805, 266, 63, 206} \[ \frac {2 \left (\frac {1}{x}+1\right )}{\sqrt {1-\frac {1}{x^2}}}-\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 852
Rule 1805
Rule 6175
Rule 6178
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\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{(1-x)^2 x} \, dx,x,\frac {1}{x}\right )\\ &=\operatorname {Subst}\left (\int \frac {(1+x)^2}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \left (1+\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \left (1+\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {2 \left (1+\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{x^2}}\right )\\ &=\frac {2 \left (1+\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}-\tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.15 \[ \frac {2 \sqrt {1-\frac {1}{x^2}} x}{x-1}-\log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.54, size = 61, normalized size = 1.85 \[ -\frac {{\left (x - 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - {\left (x - 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) - 2 \, {\left (x + 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 45, normalized size = 1.36 \[ \frac {2}{\sqrt {\frac {x - 1}{x + 1}}} - \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) + \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 106, normalized size = 3.21 \[ \frac {\left (x^{2}-1\right )^{\frac {3}{2}}-x^{2} \sqrt {x^{2}-1}-\ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}+2 x \sqrt {x^{2}-1}+2 \ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}-\ln \left (x +\sqrt {x^{2}-1}\right )}{\left (-1+x \right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 44, normalized size = 1.33 \[ \frac {2}{\sqrt {\frac {x - 1}{x + 1}}} - \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) + \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 28, normalized size = 0.85 \[ \frac {2}{\sqrt {\frac {x-1}{x+1}}}-2\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right ) \]
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 \[ - \int \frac {1}{x \sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}} - \sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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