Optimal. Leaf size=23 \[ \tan ^{-1}\left (2 x+\sqrt {3}\right )-\tan ^{-1}\left (\sqrt {3}-2 x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1161, 618, 204} \[ \tan ^{-1}\left (2 x+\sqrt {3}\right )-\tan ^{-1}\left (\sqrt {3}-2 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+x^2}{1-x^2+x^4} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=-\tan ^{-1}\left (\sqrt {3}-2 x\right )+\tan ^{-1}\left (\sqrt {3}+2 x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 12, normalized size = 0.52 \[ -\tan ^{-1}\left (\frac {x}{x^2-1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 7, normalized size = 0.30 \[ \arctan \left (x^{3}\right ) + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 30, normalized size = 1.30 \[ \frac {1}{4} \, \pi \mathrm {sgn}\relax (x) + \frac {1}{2} \, \arctan \left (\frac {x^{4} - 3 \, x^{2} + 1}{2 \, {\left (x^{3} - x\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 20, normalized size = 0.87 \[ \arctan \left (2 x -\sqrt {3}\right )+\arctan \left (2 x +\sqrt {3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} + 1}{x^{4} - x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.31, size = 7, normalized size = 0.30 \[ \mathrm {atan}\left (x^3\right )+\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.11, size = 7, normalized size = 0.30 \[ \operatorname {atan}{\relax (x )} + \operatorname {atan}{\left (x^{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________