Optimal. Leaf size=25 \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0101712, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {706, 31, 635, 203, 260} \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 706
Rule 31
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(1+x) \left (1+x^2\right )} \, dx &=\frac{1}{2} \int \frac{1}{1+x} \, dx+\frac{1}{2} \int \frac{1-x}{1+x^2} \, dx\\ &=\frac{1}{2} \log (1+x)+\frac{1}{2} \int \frac{1}{1+x^2} \, dx-\frac{1}{2} \int \frac{x}{1+x^2} \, dx\\ &=\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \log (1+x)-\frac{1}{4} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.005603, size = 25, normalized size = 1. \[ -\frac{1}{4} \log \left (x^2+1\right )+\frac{1}{2} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 20, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.62019, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.17692, size = 69, normalized size = 2.76 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.112937, size = 19, normalized size = 0.76 \begin{align*} \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10645, size = 27, normalized size = 1.08 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]