Optimal. Leaf size=21 \[ \frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6175, 6180, 37} \[ \frac {\sqrt {\frac {x-1}{x}}}{\sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6175
Rule 6180
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac {e^{\coth ^{-1}(x)}}{\left (1+\frac {1}{x}\right )^2 x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {\frac {-1+x}{x}}}{\sqrt {1+\frac {1}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.86 \[ \frac {\sqrt {1-\frac {1}{x^2}} x}{x+1} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.64, size = 11, normalized size = 0.52 \[ \sqrt {\frac {x - 1}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 11, normalized size = 0.52 \[ \sqrt {\frac {x - 1}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 1.00 \[ \frac {-1+x}{\left (1+x \right ) \sqrt {\frac {-1+x}{1+x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 0.52 \[ \sqrt {\frac {x - 1}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 11, normalized size = 0.52 \[ \sqrt {1-\frac {2}{x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.38, size = 8, normalized size = 0.38 \[ \sqrt {\frac {x - 1}{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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