Integrand size = 11, antiderivative size = 11 \[ \int \frac {1}{(1+x) (2+x)} \, dx=\log (1+x)-\log (2+x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {36, 31} \[ \int \frac {1}{(1+x) (2+x)} \, dx=\log (x+1)-\log (x+2) \]
[In]
[Out]
Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{1+x} \, dx-\int \frac {1}{2+x} \, dx \\ & = \log (1+x)-\log (2+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+x) (2+x)} \, dx=\log (1+x)-\log (2+x) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
default | \(\ln \left (1+x \right )-\ln \left (2+x \right )\) | \(12\) |
norman | \(\ln \left (1+x \right )-\ln \left (2+x \right )\) | \(12\) |
risch | \(\ln \left (1+x \right )-\ln \left (2+x \right )\) | \(12\) |
parallelrisch | \(\ln \left (1+x \right )-\ln \left (2+x \right )\) | \(12\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+x) (2+x)} \, dx=-\log \left (x + 2\right ) + \log \left (x + 1\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1+x) (2+x)} \, dx=\log {\left (x + 1 \right )} - \log {\left (x + 2 \right )} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+x) (2+x)} \, dx=-\log \left (x + 2\right ) + \log \left (x + 1\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(1+x) (2+x)} \, dx=-\log \left ({\left | x + 2 \right |}\right ) + \log \left ({\left | x + 1 \right |}\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1+x) (2+x)} \, dx=\ln \left (1-\frac {1}{x+2}\right ) \]
[In]
[Out]