Integrand size = 92, antiderivative size = 28 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (\frac {-5+x \left (-4+x-(4+\log (x (2+x)))^2\right )}{e^3 \log (25)}\right ) \]
[Out]
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6873, 6816} \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (-x^2+20 x+x \log ^2(x (x+2))+8 x \log (x (x+2))+5\right ) \]
[In]
[Out]
Rule 6816
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{(2+x) \left (5+20 x-x^2+8 x \log (x (2+x))+x \log ^2(x (2+x))\right )} \, dx \\ & = \log \left (5+20 x-x^2+8 x \log (x (2+x))+x \log ^2(x (2+x))\right ) \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (5+20 x-x^2+8 x \log (x (2+x))+x \log ^2(x (2+x))\right ) \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21
method | result | size |
norman | \(\ln \left (-\ln \left (x^{2}+2 x \right )^{2} x +x^{2}-8 \ln \left (x^{2}+2 x \right ) x -20 x -5\right )\) | \(34\) |
parallelrisch | \(\ln \left (-\ln \left (x^{2}+2 x \right )^{2} x +x^{2}-8 \ln \left (x^{2}+2 x \right ) x -20 x -5\right )\) | \(34\) |
risch | \(\ln \left (x \right )+\ln \left (\ln \left (x^{2}+2 x \right )^{2}+8 \ln \left (x^{2}+2 x \right )-\frac {x^{2}-20 x -5}{x}\right )\) | \(39\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (x\right ) + \log \left (\frac {x \log \left (x^{2} + 2 \, x\right )^{2} - x^{2} + 8 \, x \log \left (x^{2} + 2 \, x\right ) + 20 \, x + 5}{x}\right ) \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (x^{2} + 2 x \right )}^{2} + 8 \log {\left (x^{2} + 2 x \right )} + \frac {- x^{2} + 20 x + 5}{x} \right )} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (x\right ) + \log \left (\frac {x \log \left (x + 2\right )^{2} + x \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x \log \left (x\right ) + 4 \, x\right )} \log \left (x + 2\right ) + 8 \, x \log \left (x\right ) + 20 \, x + 5}{x}\right ) \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\log \left (-x \log \left (x^{2} + 2 \, x\right )^{2} + x^{2} - 8 \, x \log \left (x^{2} + 2 \, x\right ) - 20 \, x - 5\right ) \]
[In]
[Out]
Time = 8.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {56+32 x-2 x^2+(20+12 x) \log \left (2 x+x^2\right )+(2+x) \log ^2\left (2 x+x^2\right )}{10+45 x+18 x^2-x^3+\left (16 x+8 x^2\right ) \log \left (2 x+x^2\right )+\left (2 x+x^2\right ) \log ^2\left (2 x+x^2\right )} \, dx=\ln \left (8\,\ln \left (x^2+2\,x\right )-x+\frac {5}{x}+{\ln \left (x^2+2\,x\right )}^2+20\right )+\ln \left (x\right ) \]
[In]
[Out]