Optimal. Leaf size=35 \[ -\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {2}{3} \tan ^{-1}(x)+\frac {1}{3} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {1876, 209, 634, 618, 204, 628, 203, 295} \[ -\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {2}{3} \tan ^{-1}(x)+\frac {1}{3} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 204
Rule 209
Rule 295
Rule 618
Rule 628
Rule 634
Rule 1876
Rubi steps
\begin {align*} \int \frac {1+x^4}{1+x^6} \, dx &=\int \left (\frac {1}{1+x^6}+\frac {x^4}{1+x^6}\right ) \, dx\\ &=\int \frac {1}{1+x^6} \, dx+\int \frac {x^4}{1+x^6} \, dx\\ &=\frac {1}{3} \int \frac {1-\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{3} \int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{3} \int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx+\frac {1}{3} \int \frac {1+\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx+\frac {2}{3} \int \frac {1}{1+x^2} \, dx\\ &=\frac {2}{3} \tan ^{-1}(x)+2 \left (\frac {1}{12} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx\right )+2 \left (\frac {1}{12} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx\right )\\ &=\frac {2}{3} \tan ^{-1}(x)-2 \left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )\right )-2 \left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\right )\\ &=-\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {2}{3} \tan ^{-1}(x)+\frac {1}{3} \tan ^{-1}\left (\sqrt {3}+2 x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 21, normalized size = 0.60 \[ \frac {2}{3} \tan ^{-1}(x)-\frac {1}{3} \tan ^{-1}\left (\frac {x}{x^2-1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 21, normalized size = 0.60 \[ \frac {1}{3} \tan ^{-1}\left (\frac {x^2-1}{x}\right )+\frac {2}{3} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 9, normalized size = 0.26 \[ \frac {1}{3} \, \arctan \left (x^{3}\right ) + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.88, size = 27, normalized size = 0.77 \[ \frac {1}{3} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {2}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.28, size = 10, normalized size = 0.29
method | result | size |
risch | \(\arctan \relax (x )+\frac {\arctan \left (x^{3}\right )}{3}\) | \(10\) |
default | \(\frac {2 \arctan \relax (x )}{3}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{3}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{3}\) | \(28\) |
meijerg | \(\frac {x^{5} \sqrt {3}\, \ln \left (1-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\left (x^{6}\right )^{\frac {1}{6}}\right )}{3 \left (x^{6}\right )^{\frac {5}{6}}}-\frac {x^{5} \sqrt {3}\, \ln \left (1+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}-\frac {x \sqrt {3}\, \ln \left (1-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\left (x^{6}\right )^{\frac {1}{6}}\right )}{3 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \sqrt {3}\, \ln \left (1+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 27, normalized size = 0.77 \[ \frac {1}{3} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {2}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 9, normalized size = 0.26 \[ \frac {\mathrm {atan}\left (x^3\right )}{3}+\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 8, normalized size = 0.23 \[ \operatorname {atan}{\relax (x )} + \frac {\operatorname {atan}{\left (x^{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________