Optimal. Leaf size=126 \[ \frac{2 \sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{3 \sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.114503, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6176, 6181, 96, 94, 93, 206} \[ \frac{2 \sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{3 \sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6176
Rule 6181
Rule 96
Rule 94
Rule 93
Rule 206
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{\sqrt{1+x}}{(1-x) x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\sqrt{1-\frac{1}{x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x) x^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\left (2 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x} \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\left (4 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0467435, size = 69, normalized size = 0.55 \[ \frac{2 \sqrt{\frac{x-1}{x}} x \left (\sqrt{\frac{1}{x}+1} (x+4)-3 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )\right )}{3 \sqrt{1-x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.118, size = 66, normalized size = 0.5 \begin{align*}{\frac{2}{3}\sqrt{1-x} \left ( 3\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{-1-x}\sqrt{2} \right ) -\sqrt{-1-x}x-4\,\sqrt{-1-x} \right ){\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57471, size = 196, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{2}{\left (x - 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) -{\left (x^{2} + 5 \, x + 4\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}\right )}}{3 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\frac{x - 1}{x + 1}} \sqrt{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]