Integrand size = 19, antiderivative size = 22 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{x}}}{\sqrt {2}}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1460, 1483, 641, 65, 213} \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{x}+1}}{\sqrt {2}}\right ) \]
[In]
[Out]
Rule 65
Rule 213
Rule 641
Rule 1460
Rule 1483
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+\frac {1}{x}}}{\left (-1+\frac {1}{x^2}\right ) x^2} \, dx \\ & = -\text {Subst}\left (\int \frac {\sqrt {1+x}}{-1+x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {1+x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\right ) \\ & = \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{x}}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1+x}{x}}}{\sqrt {2}}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(17)=34\).
Time = 1.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {\sqrt {\frac {x +1}{x}}\, x \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}+x}}\right )}{2 \sqrt {\left (x +1\right ) x}}\) | \(41\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -4 x \sqrt {-\frac {-x -1}{x}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x -1}\right )}{2}\) | \(48\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} x \sqrt {\frac {x + 1}{x}} + 3 \, x + 1}{x - 1}\right ) \]
[In]
[Out]
\[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=- \int \frac {\sqrt {1 + \frac {1}{x}}}{x^{2} - 1}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\int { -\frac {\sqrt {\frac {1}{x} + 1}}{x^{2} - 1} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (17) = 34\).
Time = 0.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {\sqrt {2} - 1}{\sqrt {2} + 1}\right ) \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + x} + 2 \right |}}\right ) \mathrm {sgn}\left (x\right ) \]
[In]
[Out]
Time = 22.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{x}+1}}{2}\right ) \]
[In]
[Out]