Optimal. Leaf size=23 \[ \frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1366, 632, 210}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 1366
Rubi steps
\begin {align*} \int \frac {x^3}{1+x^4+x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^4\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 19, normalized size = 0.83
method | result | size |
default | \(\frac {\arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}\) | \(19\) |
risch | \(\frac {\arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.05, size = 26, normalized size = 1.13 \begin {gather*} \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.02, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.30, size = 17, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,x^4}{3}+\frac {1}{3}\right )\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________