Optimal. Leaf size=18 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0075449, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 43} \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{1+x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0026922, size = 18, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 15, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{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 = 0.927049, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} - \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 = 1.81593, size = 38, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.069481, size = 12, normalized size = 0.67 \begin{align*} \frac{x^{2}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05878, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]