Integrand size = 22, antiderivative size = 29 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=-4-e^4+\frac {1-x}{x}-x+\log (x)-\log (2+2 x) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 1634} \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=-x+\frac {1}{x}+\log (x)-\log (x+1) \]
[In]
[Out]
Rule 1607
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1-x^2-x^3}{x^2 (1+x)} \, dx \\ & = \int \left (-1+\frac {1}{-1-x}-\frac {1}{x^2}+\frac {1}{x}\right ) \, dx \\ & = \frac {1}{x}-x+\log (x)-\log (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=\frac {1}{x}-x+\log (x)-\log (1+x) \]
[In]
[Out]
Time = 1.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.55
method | result | size |
default | \(-x +\ln \left (x \right )+\frac {1}{x}-\ln \left (1+x \right )\) | \(16\) |
meijerg | \(-x +\ln \left (x \right )+\frac {1}{x}-\ln \left (1+x \right )\) | \(16\) |
risch | \(-x +\ln \left (x \right )+\frac {1}{x}-\ln \left (1+x \right )\) | \(16\) |
parallelrisch | \(\frac {x \ln \left (x \right )-\ln \left (1+x \right ) x -x^{2}+1}{x}\) | \(23\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=-\frac {x^{2} + x \log \left (x + 1\right ) - x \log \left (x\right ) - 1}{x} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.41 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=- x + \log {\left (x \right )} - \log {\left (x + 1 \right )} + \frac {1}{x} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=-x + \frac {1}{x} - \log \left (x + 1\right ) + \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=-x + \frac {1}{x} - \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {-1-x^2-x^3}{x^2+x^3} \, dx=\frac {1}{x}-2\,\mathrm {atanh}\left (2\,x+1\right )-x \]
[In]
[Out]