Optimal. Leaf size=29 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-9 x^2-12 x-4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {608, 31} \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-9 x^2-12 x-4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 608
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx &=-\left (-\frac {(-6-9 x) \int \frac {1}{-6-9 x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right )\\ &=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-4-12 x-9 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 28, normalized size = 0.97 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.86, size = 6, normalized size = 0.21 \[ -\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.34, size = 23, normalized size = 0.79 \[ \frac {i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm {sgn}\left (-3 \, x - 2\right )\right )}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 25, normalized size = 0.86 \[ \frac {\left (3 x +2\right ) \ln \left (3 x +2\right )}{3 \sqrt {-\left (3 x +2\right )^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 2.78, size = 6, normalized size = 0.21 \[ \frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 15, normalized size = 0.52 \[ -\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (3 x + 2\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________