Optimal. Leaf size=144 \[ \frac{2 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x^2}{7 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{8 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x}{7 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{46 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{92 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left (\frac{1}{x}+1\right )^{3/2} x} \]
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Rubi [A] time = 0.127649, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {6176, 6181, 89, 78, 45, 37} \[ \frac{2 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x^2}{7 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{8 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x}{7 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{46 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left (\frac{1}{x}+1\right )^{3/2}}+\frac{92 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left (\frac{1}{x}+1\right )^{3/2} x} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(x)} 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^{5/2} \, 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^{9/2}} \, dx,x,\frac{1}{x}\right )}{\left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{2 \sqrt{-\frac{1-x}{x}} x^2 (1+x)^{3/2}}{7 \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{10+\frac{7 x}{2}}{\sqrt{1-x} x^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 \left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{8 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{2 \sqrt{-\frac{1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}-\frac{\left (23 \left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^{5/2}} \, dx,x,\frac{1}{x}\right )}{7 \left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{46 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{8 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{2 \sqrt{-\frac{1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}-\frac{\left (46 \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 )}{21 \left (1+\frac{1}{x}\right )^{3/2}}\\ &=\frac{46 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{92 \sqrt{-\frac{1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac{1}{x}\right )^{3/2} x}+\frac{8 \sqrt{-\frac{1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}+\frac{2 \sqrt{-\frac{1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac{1}{x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0211074, size = 46, normalized size = 0.32 \[ \frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} \left (3 x^3+12 x^2+23 x+46\right )}{21 \sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 37, normalized size = 0.3 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( 3\,{x}^{3}+12\,{x}^{2}+23\,x+46 \right ) }{21}{\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 = 1.07647, size = 36, normalized size = 0.25 \begin{align*} \frac{2 \,{\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52876, size = 93, normalized size = 0.65 \begin{align*} \frac{2}{21} \,{\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\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 = 1.16036, size = 46, normalized size = 0.32 \begin{align*} \frac{2}{7} \,{\left (x - 1\right )}^{\frac{7}{2}} + 2 \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{16}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{64}{21} i \, \sqrt{2} + 8 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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