Optimal. Leaf size=126 \[ \frac {2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{3 \sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
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Rubi [A] time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6176, 6181, 96, 94, 93, 206} \[ \frac {2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{3 \sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 206
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)} \sqrt {x}}{\sqrt {1-\frac {1}{x}}} \, dx}{\sqrt {1-x}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{5/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\sqrt {1-\frac {1}{x}} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\left (2 \sqrt {1-\frac {1}{x}}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\left (4 \sqrt {1-\frac {1}{x}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 0.55 \[ \frac {2 \sqrt {\frac {x-1}{x}} x \left (\sqrt {\frac {1}{x}+1} (x+4)-3 \sqrt {2} \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{x+1}}\right )\right )}{3 \sqrt {1-x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 72, normalized size = 0.57 \[ \frac {2 \, {\left (3 \, \sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x^{2} + 5 \, x + 4\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 65, normalized size = 0.52 \[ -\frac {2 \sqrt {1-x}\, \left (\sqrt {-1-x}\, x -3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )+4 \sqrt {-1-x}\right )}{3 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \]
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 \[ \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \sqrt {1 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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