Optimal. Leaf size=29 \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0247806, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1804, 635, 203, 260} \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1804
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{x \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx &=-\frac{x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{-1-2 x}{1+x^2} \, dx\\ &=-\frac{x}{2 \left (1+x^2\right )}+\frac{1}{2} \int \frac{1}{1+x^2} \, dx+\int \frac{x}{1+x^2} \, dx\\ &=-\frac{x}{2 \left (1+x^2\right )}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0084116, size = 23, normalized size = 0.79 \[ \frac{1}{2} \left (-\frac{x}{x^2+1}+\log \left (x^2+1\right )+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 24, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46719, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.989002, size = 89, normalized size = 3.07 \begin{align*} \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) +{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.114108, size = 20, normalized size = 0.69 \begin{align*} - \frac{x}{2 x^{2} + 2} + \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1516, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]