Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {x+x^2}}\right ) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {634, 212} \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]
[In]
[Out]
Rule 212
Rule 634
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right ) \\ & = 2 \text {arctanh}\left (\frac {x}{\sqrt {x+x^2}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(14)=28\).
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.79 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=-\frac {2 \sqrt {x} \sqrt {1+x} \log \left (-\sqrt {x}+\sqrt {1+x}\right )}{\sqrt {x (1+x)}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50
method | result | size |
meijerg | \(2 \,\operatorname {arcsinh}\left (\sqrt {x}\right )\) | \(7\) |
default | \(\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )\) | \(12\) |
pseudoelliptic | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (1+x \right )}}{x}\right )\) | \(15\) |
trager | \(\ln \left (1+2 x +2 \sqrt {x^{2}+x}\right )\) | \(16\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=-\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=\log {\left (2 x + 2 \sqrt {x^{2} + x} + 1 \right )} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=\log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=\frac {1}{4} \, \sqrt {x^{2} + x} {\left (2 \, x + 1\right )} + \frac {1}{8} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {x+x^2}} \, dx=\ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right ) \]
[In]
[Out]