Optimal. Leaf size=31 \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0421315, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2074, 634, 618, 204, 628} \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2074
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1+3 x+3 x^2}{1+2 x+2 x^2+x^3} \, dx &=\int \left (\frac{1}{1+x}+\frac{2 x}{1+x+x^2}\right ) \, dx\\ &=\log (1+x)+2 \int \frac{x}{1+x+x^2} \, dx\\ &=\log (1+x)-\int \frac{1}{1+x+x^2} \, dx+\int \frac{1+2 x}{1+x+x^2} \, dx\\ &=\log (1+x)+\log \left (1+x+x^2\right )+2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{2 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\log (1+x)+\log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0120442, size = 31, normalized size = 1. \[ \log \left (x^2+x+1\right )+\log (x+1)-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 29, normalized size = 0.9 \begin{align*} \ln \left ( 1+x \right ) +\ln \left ({x}^{2}+x+1 \right ) -{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46966, size = 38, normalized size = 1.23 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53569, size = 103, normalized size = 3.32 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.115544, size = 3, normalized size = 0.1 \begin{align*} \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14224, size = 39, normalized size = 1.26 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]