Optimal. Leaf size=39 \[ \tan ^{-1}\left (\frac {1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt {3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2095} \[ \tan ^{-1}\left (\frac {1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt {3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2095
Rubi steps
\begin {align*} \int \frac {6-3 x^2+x^4}{4+5 x^2-5 x^4+x^6} \, dx &=-\tan ^{-1}\left (\sqrt {3}-2 x\right )+\tan ^{-1}\left (\sqrt {3}+2 x\right )+\tan ^{-1}\left (\frac {1}{2} x \left (1-3 x^2+x^4\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 41, normalized size = 1.05 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x \left (x^2-3\right )}{x^2-2}\right )-\frac {1}{2} \tan ^{-1}\left (\frac {x \left (x^2-3\right )}{2-x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 22, normalized size = 0.56 \[ \arctan \left (\frac {1}{2} \, x^{5} - \frac {3}{2} \, x^{3} + \frac {1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.97, size = 22, normalized size = 0.56 \[ \arctan \left (\frac {1}{2} \, x^{5} - \frac {3}{2} \, x^{3} + \frac {1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 23, normalized size = 0.59 \[ \arctan \relax (x )+\arctan \left (x^{3}\right )+\arctan \left (\frac {1}{2} x^{5}-\frac {3}{2} x^{3}+\frac {1}{2} x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} - 3 \, x^{2} + 6}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.27, size = 22, normalized size = 0.56 \[ \mathrm {atan}\left (\frac {x^5}{2}-\frac {3\,x^3}{2}+\frac {x}{2}\right )+\mathrm {atan}\left (x^3\right )+\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 24, normalized size = 0.62 \[ \operatorname {atan}{\relax (x )} + \operatorname {atan}{\left (x^{3} \right )} + \operatorname {atan}{\left (\frac {x^{5}}{2} - \frac {3 x^{3}}{2} + \frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________