Optimal. Leaf size=130 \[ -\frac{\sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{2 (1-x)^{3/2}}+\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \sqrt{\frac{1}{x}+1} x^2}{2 (1-x)^{3/2}}-\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13163, antiderivative size = 130, 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} \[ -\frac{\sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{2 (1-x)^{3/2}}+\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \sqrt{\frac{1}{x}+1} x^2}{2 (1-x)^{3/2}}-\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}} \]
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}{(1-x)^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{\coth ^{-1}(x)}}{\left (1-\frac{1}{x}\right )^{3/2} \sqrt{x}} \, dx}{(1-x)^{3/2}}\\ &=-\frac{\left (1-\frac{1}{x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x)^2 x^{3/2}} \, dx,x,\frac{1}{x}\right )}{(1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac{\left (5 \left (1-\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x) x^{3/2}} \, dx,x,\frac{1}{x}\right )}{4 (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \sqrt{1+\frac{1}{x}} x^2}{2 (1-x)^{3/2}}-\frac{\sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac{\left (5 \left (1-\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x} \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )}{2 (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \sqrt{1+\frac{1}{x}} x^2}{2 (1-x)^{3/2}}-\frac{\sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac{\left (5 \left (1-\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{(1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \sqrt{1+\frac{1}{x}} x^2}{2 (1-x)^{3/2}}-\frac{\sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac{5 \left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0750892, size = 75, normalized size = 0.58 \[ -\frac{\sqrt{\frac{x-1}{x}} x \left (2 \sqrt{\frac{1}{x}+1} (3-2 x)+5 \sqrt{2} (x-1) \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )\right )}{2 (1-x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.115, size = 90, normalized size = 0.7 \begin{align*}{\frac{1}{-2+2\,x}\sqrt{1-x} \left ( -5\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{-1-x}\sqrt{2} \right ) x+4\,\sqrt{-1-x}x+5\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{-1-x}\sqrt{2} \right ) -6\,\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}{{\left (-x + 1\right )}^{\frac{3}{2}} \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 = 1.46734, size = 221, normalized size = 1.7 \begin{align*} -\frac{5 \, \sqrt{2}{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) - 2 \,{\left (2 \, x^{2} - x - 3\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14619, size = 73, normalized size = 0.56 \begin{align*} \frac{{\left (5 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x - 1}\right ) - 4 \, \sqrt{-x - 1} + \frac{2 \, \sqrt{-x - 1}}{x - 1}\right )} \mathrm{sgn}\left (x\right )}{2 \, \mathrm{sgn}\left (-x - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]