Optimal. Leaf size=53 \[ -\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \]
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Rubi [A] time = 0.09, antiderivative size = 53, 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, 807, 266, 47, 63, 206} \[ \frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3-\frac {1}{2} \sqrt {1-\frac {1}{x^2}} x^2+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 206
Rule 266
Rule 807
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} (1-x)^2 \, dx &=\int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right )^2 x^2 \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(1-x) \sqrt {1-x^2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\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}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3-\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}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\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}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3+\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.89 \[ \frac {1}{6} \sqrt {1-\frac {1}{x^2}} x \left (2 x^2-3 x-2\right )+\frac {1}{2} \log \left (\left (\sqrt {1-\frac {1}{x^2}}+1\right ) x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 61, normalized size = 1.15 \[ \frac {1}{6} \, {\left (2 \, x^{3} - x^{2} - 5 \, x - 2\right )} \sqrt {\frac {x - 1}{x + 1}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 107, normalized size = 2.02 \[ -\frac {\frac {8 \, {\left (x - 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x + 1} + \frac {3 \, {\left (x - 1\right )}^{2} \sqrt {\frac {x - 1}{x + 1}}}{{\left (x + 1\right )}^{2}} - 3 \, \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {\frac {x - 1}{x + 1}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 1.13 \[ \frac {\left (-1+x \right ) \left (2 \left (\left (1+x \right ) \left (-1+x \right )\right )^{\frac {3}{2}}-3 x \sqrt {x^{2}-1}+3 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{6 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {\left (1+x \right ) \left (-1+x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 112, normalized size = 2.11 \[ -\frac {3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{2}} + 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}} - 3 \, \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x - 1}{x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 90, normalized size = 1.70 \[ \mathrm {atanh}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{3/2}}{3}-\sqrt {\frac {x-1}{x+1}}+{\left (\frac {x-1}{x+1}\right )}^{5/2}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1} \]
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 \[ \int \frac {\left (x - 1\right )^{2}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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