Optimal. Leaf size=70 \[ \frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{3 \sqrt{\frac{1}{x}+1}}+\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{3 \sqrt{\frac{1}{x}+1}} \]
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Rubi [A] time = 0.0811716, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 78, 37} \[ \frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{3 \sqrt{\frac{1}{x}+1}}+\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{3 \sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} \sqrt{1+x} \, dx &=\frac{\sqrt{1+x} \int e^{\coth ^{-1}(x)} \sqrt{1+\frac{1}{x}} \sqrt{x} \, dx}{\sqrt{1+\frac{1}{x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{1+x}\right ) \operatorname{Subst}\left (\int \frac{1+x}{\sqrt{1-x} x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1+\frac{1}{x}}}\\ &=\frac{2 \sqrt{-\frac{1-x}{x}} x \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}-\frac{\left (5 \sqrt{\frac{1}{x}} \sqrt{1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1+\frac{1}{x}}}\\ &=\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}+\frac{2 \sqrt{-\frac{1-x}{x}} x \sqrt{1+x}}{3 \sqrt{1+\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0145598, size = 34, normalized size = 0.49 \[ \frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} (x+5)}{3 \sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 25, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( x+5 \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 = 1.05387, size = 20, normalized size = 0.29 \begin{align*} \frac{2 \,{\left (x^{2} + 4 \, x - 5\right )}}{3 \, \sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61203, size = 63, normalized size = 0.9 \begin{align*} \frac{2}{3} \,{\left (x + 5\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 = 23.2935, size = 39, normalized size = 0.56 \begin{align*} 2 \left (\begin{cases} 2 \sqrt{2} \left (\frac{\sqrt{2} \left (x - 1\right )^{\frac{3}{2}}}{12} + \frac{\sqrt{2} \sqrt{x - 1}}{2}\right ) & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.13628, size = 27, normalized size = 0.39 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{8}{3} i \, \sqrt{2} + 4 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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