Optimal. Leaf size=35 \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2 \]
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Rubi [A] time = 0.05, 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 = {6175, 6178, 266, 47, 63, 206} \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2 \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 266
Rule 6175
Rule 6178
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\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \operatorname {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} \operatorname {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} \operatorname {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 = 0.02, size = 39, normalized size = 1.11 \[ \frac {1}{2} \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right )-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2 \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 51, normalized size = 1.46 \[ -\frac {1}{2} \, {\left (x^{2} + x\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 110, normalized size = 3.14 \[ -\frac {\sqrt {\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {\frac {x - 1}{x + 1}}}}{{\left (\sqrt {\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {\frac {x - 1}{x + 1}}}\right )}^{2} - 4} + \frac {1}{4} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {\frac {x - 1}{x + 1}}} + 2\right ) - \frac {1}{4} \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {\frac {x - 1}{x + 1}}} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 48, normalized size = 1.37 \[ -\frac {\left (-1+x \right ) \left (x \sqrt {x^{2}-1}-\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 83, normalized size = 2.37 \[ \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {x - 1}{x + 1}}}{\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
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 \[ \mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\sqrt {\frac {x-1}{x+1}}+{\left (\frac {x-1}{x+1}\right )}^{3/2}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1} \]
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 \[ - \int \frac {x}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\, dx - \int \left (- \frac {1}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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