Integrand size = 18, antiderivative size = 27 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {(2+3 x) \text {arctanh}(1+3 x)}{\sqrt {-4-12 x-9 x^2}} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {660, 36, 31, 29} \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {(3 x+2) \log (x)}{2 \sqrt {-9 x^2-12 x-4}}-\frac {(3 x+2) \log (3 x+2)}{2 \sqrt {-9 x^2-12 x-4}} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 660
Rubi steps \begin{align*} \text {integral}& = -\left (-\frac {(-6-9 x) \int \frac {1}{(-6-9 x) x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right ) \\ & = -\frac {(3 (-6-9 x)) \int \frac {1}{-6-9 x} \, dx}{2 \sqrt {-4-12 x-9 x^2}}+-\frac {(-6-9 x) \int \frac {1}{x} \, dx}{6 \sqrt {-4-12 x-9 x^2}} \\ & = \frac {(2+3 x) \log (x)}{2 \sqrt {-4-12 x-9 x^2}}-\frac {(2+3 x) \log (2+3 x)}{2 \sqrt {-4-12 x-9 x^2}} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {(2+3 x) (\log (x)-\log (2+3 x))}{2 \sqrt {-(2+3 x)^2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
meijerg | \(-\frac {i \left (\ln \left (x \right )+\ln \left (3\right )-\ln \left (2\right )-\ln \left (1+\frac {3 x}{2}\right )\right )}{2}\) | \(21\) |
default | \(\frac {\left (2+3 x \right ) \left (\ln \left (x \right )-\ln \left (2+3 x \right )\right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(30\) |
risch | \(\frac {\left (2+3 x \right ) \ln \left (x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}-\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(46\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {1}{2} i \, \log \left (x + \frac {2}{3}\right ) - \frac {1}{2} i \, \log \left (x\right ) \]
[In]
[Out]
\[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\int \frac {1}{x \sqrt {- \left (3 x + 2\right )^{2}}}\, dx \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{2} i \, \left (-1\right )^{12 \, x + 8} \log \left (\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {i \, \log \left ({\left | 3 \, x + 2 \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x - 2\right )} + \frac {i \, \log \left ({\left | x \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]
[In]
[Out]
Time = 10.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {\ln \left (\frac {6\,x+4-\sqrt {-{\left (3\,x+2\right )}^2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]
[In]
[Out]