Optimal. Leaf size=27 \[ -\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{-9 x^2-12 x-4}} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 0.0137119, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {646, 36, 31, 29} \[ \frac{(3 x+2) \log (x)}{2 \sqrt{-9 x^2-12 x-4}}-\frac{(3 x+2) \log (3 x+2)}{2 \sqrt{-9 x^2-12 x-4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 36
Rule 31
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-4-12 x-9 x^2}} \, dx &=-\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*}
Mathematica [A] time = 0.0071819, size = 33, normalized size = 1.22 \[ \frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{-(3 x+2)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.103, size = 30, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( 2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.51444, size = 32, normalized size = 1.19 \begin{align*} -\frac{1}{2} i \, \left (-1\right )^{12 \, x + 8} \log \left (\frac{12 \, x}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.00307, size = 47, normalized size = 1.74 \begin{align*} \frac{1}{2} i \, \log \left (x + \frac{2}{3}\right ) - \frac{1}{2} i \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- \left (3 x + 2\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]