Integrand size = 11, antiderivative size = 22 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 196, 45} \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \sqrt {x+1}-2 \log \left (\sqrt {x+1}+1\right ) \]
[In]
[Out]
Rule 45
Rule 196
Rule 253
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+\sqrt {x}} \, dx,x,1+x\right ) \\ & = 2 \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \sqrt {1+x}-2 \log \left (1+\sqrt {1+x}\right ) \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-2 \ln \left (1+\sqrt {1+x}\right )+2 \sqrt {1+x}\) | \(19\) |
trager | \(2 \sqrt {1+x}-\ln \left (2 \sqrt {1+x}+2+x \right )\) | \(22\) |
default | \(2 \sqrt {1+x}+\ln \left (-1+\sqrt {1+x}\right )-\ln \left (1+\sqrt {1+x}\right )-\ln \left (x \right )\) | \(31\) |
meijerg | \(\frac {-4 \sqrt {\pi }+4 \sqrt {\pi }\, \sqrt {1+x}-4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+x}}{2}\right )}{2 \sqrt {\pi }}\) | \(37\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} + 1 \right )} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{1+\sqrt {1+x}} \, dx=2\,\sqrt {x+1}-2\,\ln \left (\sqrt {x+1}+1\right ) \]
[In]
[Out]