Optimal. Leaf size=90 \[ -\frac {\sqrt {\frac {1}{x}+1} x \sqrt {1-\frac {1}{x}}}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac {\sqrt {\frac {1}{x}+1} x \sqrt {1-\frac {1}{x}}}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 206
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\left (1-\frac {1}{x}\right )^{3/2} x^{3/2}} \, dx}{(1-x)^{3/2}}\\ &=-\frac {\left (1-\frac {1}{x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x)^2 \sqrt {x}} \, dx,x,\frac {1}{x}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )}{2 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 58, normalized size = 0.64 \[ \frac {\frac {2}{\sqrt {\frac {1}{x+1}}}+\sqrt {2} (x-1) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{x+1}}\right )}{2 \sqrt {-\frac {(x-1)^2}{x^2}} x} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 76, normalized size = 0.84 \[ -\frac {\sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) + 2 \, {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 44, normalized size = 0.49 \[ \frac {{\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) + \frac {2 \, \sqrt {-x - 1}}{x - 1}\right )} \mathrm {sgn}\relax (x)}{2 \, \mathrm {sgn}\left (-x - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 79, normalized size = 0.88 \[ -\frac {\sqrt {1-x}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )+2 \sqrt {-1-x}\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (-1+x \right ) \sqrt {-1-x}} \]
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 \[ \int \frac {1}{{\left (-x + 1\right )}^{\frac {3}{2}} \sqrt {\frac {x - 1}{x + 1}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,{\left (1-x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\frac {x - 1}{x + 1}} \left (1 - x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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