Optimal. Leaf size=107 \[ \frac{2 \sqrt{-\frac{1-x}{x}} x (x+1)^{3/2}}{5 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{28 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{86 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left (\frac{1}{x}+1\right )^{3/2} x} \]
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Rubi [A] time = 0.107356, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6176, 6181, 89, 78, 37} \[ \frac{2 \sqrt{-\frac{1-x}{x}} x (x+1)^{3/2}}{5 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{28 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{86 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left (\frac{1}{x}+1\right )^{3/2} x} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx &=\frac{(1+x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1+\frac{1}{x}\right )^{3/2} x^{3/2} \, dx}{\left (1+\frac{1}{x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{(1+x)^2}{\sqrt{1-x} x^{7/2}} \, dx,x,\frac{1}{x}\right )}{\left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{2 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac{1}{x}\right )^{3/2}}-\frac{\left (2 \left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{7+\frac{5 x}{2}}{\sqrt{1-x} x^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 \left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{28 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{2 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac{1}{x}\right )^{3/2}}-\frac{\left (43 \left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 \left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{28 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{86 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{15 \left (1+\frac{1}{x}\right )^{3/2} x}+\frac{2 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{5 \left (1+\frac{1}{x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0158638, size = 41, normalized size = 0.38 \[ \frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} \left (3 x^2+14 x+43\right )}{15 \sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 32, normalized size = 0.3 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 3\,{x}^{2}+14\,x+43 \right ) }{15}{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02345, size = 30, normalized size = 0.28 \begin{align*} \frac{2 \,{\left (3 \, x^{3} + 11 \, x^{2} + 29 \, x - 43\right )}}{15 \, \sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58061, size = 81, normalized size = 0.76 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + 14 \, x + 43\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16821, size = 36, normalized size = 0.34 \begin{align*} \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{8}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{64}{15} i \, \sqrt{2} + 8 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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