Optimal. Leaf size=18 \[ x-\frac{8}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} \tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0140661, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1122, 1166, 203} \[ x-\frac{8}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1122
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4}{4+5 x^2+x^4} \, dx &=x-\int \frac{4+5 x^2}{4+5 x^2+x^4} \, dx\\ &=x+\frac{1}{3} \int \frac{1}{1+x^2} \, dx-\frac{16}{3} \int \frac{1}{4+x^2} \, dx\\ &=x-\frac{8}{3} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} \tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0088147, size = 18, normalized size = 1. \[ x+\frac{8}{3} \tan ^{-1}\left (\frac{2}{x}\right )+\frac{1}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 13, normalized size = 0.7 \begin{align*} x-{\frac{8}{3}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50497, size = 16, normalized size = 0.89 \begin{align*} x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.00811, size = 53, normalized size = 2.94 \begin{align*} x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.128109, size = 14, normalized size = 0.78 \begin{align*} x - \frac{8 \operatorname{atan}{\left (\frac{x}{2} \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13935, size = 16, normalized size = 0.89 \begin{align*} x - \frac{8}{3} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]