Optimal. Leaf size=47 \[ \frac {2 \left (\frac {1}{x}+1\right )}{\sqrt {1-\frac {1}{x^2}}}-\sqrt {1-\frac {1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {6175, 6177, 852, 1805, 807, 266, 63, 206} \[ \frac {2 \left (\frac {1}{x}+1\right )}{\sqrt {1-\frac {1}{x^2}}}-\sqrt {1-\frac {1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 6175
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)} x}{1-x} \, dx &=-\int \frac {e^{\coth ^{-1}(x)}}{1-\frac {1}{x}} \, dx\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{(1-x)^2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^2 \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 {-1-2 x}{x^2 \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \left (1+\frac {1}{x}\right )}{\sqrt {1-\frac {1}{x^2}}}-\sqrt {1-\frac {1}{x^2}} 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}}}-\sqrt {1-\frac {1}{x^2}} x+\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}}}-\sqrt {1-\frac {1}{x^2}} 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}}}-\sqrt {1-\frac {1}{x^2}} x-2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 41, normalized size = 0.87 \[ -\frac {\sqrt {1-\frac {1}{x^2}} (x-3) x}{x-1}-2 \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 66, normalized size = 1.40 \[ -\frac {2 \, {\left (x - 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - 2 \, {\left (x - 1\right )} \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) + {\left (x^{2} - 2 \, x - 3\right )} \sqrt {\frac {x - 1}{x + 1}}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 84, normalized size = 1.79 \[ \frac {2 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - 1\right )}}{\frac {{\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x + 1} - \sqrt {\frac {x - 1}{x + 1}}} - 2 \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) + 2 \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 106, normalized size = 2.26 \[ \frac {\left (x^{2}-1\right )^{\frac {3}{2}}-2 x^{2} \sqrt {x^{2}-1}-2 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}+4 x \sqrt {x^{2}-1}+4 \ln \left (x +\sqrt {x^{2}-1}\right ) x -2 \sqrt {x^{2}-1}-2 \ln \left (x +\sqrt {x^{2}-1}\right )}{\left (-1+x \right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}\, \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 74, normalized size = 1.57 \[ \frac {2 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - 1\right )}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - \sqrt {\frac {x - 1}{x + 1}}} - 2 \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) + 2 \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 43, normalized size = 0.91 \[ -\frac {2\,x+8\,\mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )\,\sqrt {\frac {x-1}{x+1}}-6}{2\,\sqrt {\frac {x-1}{x+1}}} \]
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 \[ - \int \frac {x}{x \sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}} - \sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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