Optimal. Leaf size=19 \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0132435, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1810, 627, 31} \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1810
Rule 627
Rule 31
Rubi steps
\begin{align*} \int \frac{-1+x^5}{-1+x^2} \, dx &=\int \left (x+x^3-\frac{1-x}{-1+x^2}\right ) \, dx\\ &=\frac{x^2}{2}+\frac{x^4}{4}-\int \frac{1-x}{-1+x^2} \, dx\\ &=\frac{x^2}{2}+\frac{x^4}{4}-\int \frac{1}{-1-x} \, dx\\ &=\frac{x^2}{2}+\frac{x^4}{4}+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0041398, size = 19, normalized size = 1. \[ \frac{x^4}{4}+\frac{x^2}{2}+\log (x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 16, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{{x}^{4}}{4}}+\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.962015, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.4384, size = 43, normalized size = 2.26 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.075318, size = 14, normalized size = 0.74 \begin{align*} \frac{x^{4}}{4} + \frac{x^{2}}{2} + \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17694, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, x^{2} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]