Optimal. Leaf size=39 \[ \frac {-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {638, 618, 204} \[ -\frac {4 x+7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {-3+x}{\left (4+2 x+x^2\right )^2} \, dx &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}-\frac {2}{3} \int \frac {1}{4+2 x+x^2} \, dx\\ &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 x\right )\\ &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}-\frac {2 \tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{3 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 39, normalized size = 1.00 \[ \frac {-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 39, normalized size = 1.00 \[ -\frac {4 \, \sqrt {3} {\left (x^{2} + 2 \, x + 4\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) + 12 \, x + 21}{18 \, {\left (x^{2} + 2 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.93, size = 32, normalized size = 0.82 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {4 \, x + 7}{6 \, {\left (x^{2} + 2 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 35, normalized size = 0.90 \[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x +2\right ) \sqrt {3}}{6}\right )}{9}+\frac {-8 x -14}{12 x^{2}+24 x +48} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.10, size = 32, normalized size = 0.82 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {4 \, x + 7}{6 \, {\left (x^{2} + 2 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 36, normalized size = 0.92 \[ -\frac {\frac {2\,x}{3}+\frac {7}{6}}{x^2+2\,x+4}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 41, normalized size = 1.05 \[ \frac {- 4 x - 7}{6 x^{2} + 12 x + 24} - \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________