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 = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 203
Rule 1122
Rule 1166
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 = 0.01, size = 18, normalized size = 1.00 \[ x+\frac {8}{3} \tan ^{-1}\left (\frac {2}{x}\right )+\frac {1}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 12, normalized size = 0.67 \[ x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.01, size = 12, normalized size = 0.67 \[ x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 13, normalized size = 0.72 \[ x +\frac {\arctan \relax (x )}{3}-\frac {8 \arctan \left (\frac {x}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.14, size = 12, normalized size = 0.67 \[ x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 12, normalized size = 0.67 \[ x-\frac {8\,\mathrm {atan}\left (\frac {x}{2}\right )}{3}+\frac {\mathrm {atan}\relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 14, normalized size = 0.78 \[ x - \frac {8 \operatorname {atan}{\left (\frac {x}{2} \right )}}{3} + \frac {\operatorname {atan}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________