Integrand size = 11, antiderivative size = 35 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=-\frac {1}{16 x}-\frac {3}{16 (2+3 x)}-\frac {3 \log (x)}{16}+\frac {3}{16} \log (2+3 x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=-\frac {1}{16 x}-\frac {3}{16 (3 x+2)}-\frac {3 \log (x)}{16}+\frac {3}{16} \log (3 x+2) \]
[In]
[Out]
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{16 x^2}-\frac {3}{16 x}+\frac {9}{16 (2+3 x)^2}+\frac {9}{16 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{16 x}-\frac {3}{16 (2+3 x)}-\frac {3 \log (x)}{16}+\frac {3}{16} \log (2+3 x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=\frac {1}{16} \left (-\frac {1}{x}-\frac {3}{2+3 x}-3 \log (x)+3 \log (2+3 x)\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {1}{16 x}-\frac {3}{16 \left (2+3 x \right )}-\frac {3 \ln \left (x \right )}{16}+\frac {3 \ln \left (2+3 x \right )}{16}\) | \(28\) |
risch | \(\frac {-\frac {3 x}{8}-\frac {1}{8}}{x \left (2+3 x \right )}-\frac {3 \ln \left (x \right )}{16}+\frac {3 \ln \left (2+3 x \right )}{16}\) | \(31\) |
norman | \(\frac {-\frac {1}{8}+\frac {9 x^{2}}{16}}{x \left (2+3 x \right )}-\frac {3 \ln \left (x \right )}{16}+\frac {3 \ln \left (2+3 x \right )}{16}\) | \(32\) |
meijerg | \(-\frac {1}{16 x}-\frac {3}{32}-\frac {3 \ln \left (x \right )}{16}-\frac {3 \ln \left (3\right )}{16}+\frac {3 \ln \left (2\right )}{16}+\frac {27 x}{64 \left (\frac {9 x}{2}+3\right )}+\frac {3 \ln \left (1+\frac {3 x}{2}\right )}{16}\) | \(38\) |
parallelrisch | \(-\frac {9 \ln \left (x \right ) x^{2}-9 \ln \left (\frac {2}{3}+x \right ) x^{2}+2+6 \ln \left (x \right ) x -6 \ln \left (\frac {2}{3}+x \right ) x -9 x^{2}}{16 x \left (2+3 x \right )}\) | \(48\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )} \log \left (3 \, x + 2\right ) - 3 \, {\left (3 \, x^{2} + 2 \, x\right )} \log \left (x\right ) - 6 \, x - 2}{16 \, {\left (3 \, x^{2} + 2 \, x\right )}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=\frac {- 3 x - 1}{24 x^{2} + 16 x} - \frac {3 \log {\left (x \right )}}{16} + \frac {3 \log {\left (x + \frac {2}{3} \right )}}{16} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=-\frac {3 \, x + 1}{8 \, {\left (3 \, x^{2} + 2 \, x\right )}} + \frac {3}{16} \, \log \left (3 \, x + 2\right ) - \frac {3}{16} \, \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=-\frac {3}{16 \, {\left (3 \, x + 2\right )}} + \frac {3}{32 \, {\left (\frac {2}{3 \, x + 2} - 1\right )}} - \frac {3}{16} \, \log \left ({\left | -\frac {2}{3 \, x + 2} + 1 \right |}\right ) \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 (4+6 x)^2} \, dx=\frac {3\,\ln \left (\frac {6\,x+4}{x}\right )}{16}-\frac {3}{4\,\left (6\,x+4\right )}-\frac {1}{4\,x\,\left (6\,x+4\right )} \]
[In]
[Out]