Optimal. Leaf size=73 \[ \frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x^2}{3 \sqrt{x+1}}+\frac{4 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{3 \sqrt{x+1}} \]
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Rubi [A] time = 0.096523, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6176, 6181, 45, 37} \[ \frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x^2}{3 \sqrt{x+1}}+\frac{4 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{3 \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 45
Rule 37
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{1}{\sqrt{1-x} x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{1+x}}\\ &=\frac{2 \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x^2}{3 \sqrt{1+x}}-\frac{\left (2 \sqrt{1+\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{\frac{1}{x}} \sqrt{1+x}}\\ &=\frac{4 \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x}{3 \sqrt{1+x}}+\frac{2 \sqrt{1+\frac{1}{x}} \sqrt{-\frac{1-x}{x}} x^2}{3 \sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.0147966, size = 26, normalized size = 0.36 \[ \frac{2 \sqrt{1-\frac{1}{x^2}} x (x+2)}{3 \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 25, normalized size = 0.3 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( x+2 \right ) }{3}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08373, size = 18, normalized size = 0.25 \begin{align*} \frac{2 \,{\left (x^{2} + x - 2\right )}}{3 \, \sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55459, size = 63, normalized size = 0.86 \begin{align*} \frac{2}{3} \,{\left (x + 2\right )} \sqrt{x + 1} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 62.8377, size = 48, normalized size = 0.66 \begin{align*} \begin{cases} \frac{2 x \sqrt{x - 1}}{3} + \frac{4 \sqrt{x - 1}}{3} & \text{for}\: \left |{x}\right | > 1 \\\frac{2 i x \sqrt{1 - x}}{3} + \frac{4 i \sqrt{1 - x}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.12459, size = 27, normalized size = 0.37 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{2}{3} i \, \sqrt{2} + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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