Integrand size = 7, antiderivative size = 10 \[ \int \frac {x}{(1+x)^2} \, dx=\frac {1}{1+x}+\log (1+x) \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {45} \[ \int \frac {x}{(1+x)^2} \, dx=\frac {1}{x+1}+\log (x+1) \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx \\ & = \frac {1}{1+x}+\log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x}{(1+x)^2} \, dx=\frac {1}{1+x}+\log (1+x) \]
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Time = 0.08 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {1}{1+x}+\ln \left (1+x \right )\) | \(11\) |
norman | \(\frac {1}{1+x}+\ln \left (1+x \right )\) | \(11\) |
risch | \(\frac {1}{1+x}+\ln \left (1+x \right )\) | \(11\) |
meijerg | \(-\frac {x}{1+x}+\ln \left (1+x \right )\) | \(14\) |
parallelrisch | \(\frac {\ln \left (1+x \right ) x +1+\ln \left (1+x \right )}{1+x}\) | \(19\) |
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none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {x}{(1+x)^2} \, dx=\frac {{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x}{(1+x)^2} \, dx=\log {\left (x + 1 \right )} + \frac {1}{x + 1} \]
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none
Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x}{(1+x)^2} \, dx=\frac {1}{x + 1} + \log \left (x + 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {x}{(1+x)^2} \, dx=\frac {1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x}{(1+x)^2} \, dx=\ln \left (x+1\right )+\frac {1}{x+1} \]
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