Integrand size = 15, antiderivative size = 126 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6311, 6316, 98, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{3 \sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}} \]
[In]
[Out]
Rule 95
Rule 96
Rule 98
Rule 212
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \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}} \text {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}} \text {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 ) \text {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 ) \text {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}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}} (4+x)-3 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{3 \sqrt {1-x}} \]
[In]
[Out]
Time = 0.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {2 \sqrt {1-x}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-\sqrt {-1-x}\, x -4 \sqrt {-1-x}\right )}{3 \sqrt {\frac {x -1}{1+x}}\, \sqrt {-1-x}}\) | \(66\) |
risch | \(\frac {2 \left (x +4\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{3 \sqrt {-1-x}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(111\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x^{2} + 5 \, x + 4\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (x - 1\right )}} \]
[In]
[Out]
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \sqrt {1 - x}}\, dx \]
[In]
[Out]
\[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int { \frac {x}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\text {Exception raised: NotImplementedError} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx=\int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \]
[In]
[Out]