Optimal. Leaf size=46 \[ \frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1163, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 1163
Rubi steps
\begin {align*} \int \frac {1-2 x^2}{1+6 x^2+4 x^4} \, dx &=\left (-1-\sqrt {5}\right ) \int \frac {1}{3+\sqrt {5}+4 x^2} \, dx+\left (-1+\sqrt {5}\right ) \int \frac {1}{3-\sqrt {5}+4 x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 84, normalized size = 1.83 \begin {gather*} \frac {-\left (\left (\sqrt {5}-5\right ) \sqrt {3+\sqrt {5}} \tan ^{-1}\left (\frac {2 x}{\sqrt {3-\sqrt {5}}}\right )\right )-\sqrt {3-\sqrt {5}} \left (5+\sqrt {5}\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {5}}}\right )}{4 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^2}{1+6 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 28, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} {\left (x^{3} + x\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 39, normalized size = 0.85 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {4 \, x}{\sqrt {10} + \sqrt {2}}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {4 \, x}{\sqrt {10} - \sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 136, normalized size = 2.96 \begin {gather*} \frac {2 \sqrt {5}\, \arctan \left (\frac {8 x}{2 \sqrt {10}-2 \sqrt {2}}\right )}{2 \sqrt {10}-2 \sqrt {2}}-\frac {2 \arctan \left (\frac {8 x}{2 \sqrt {10}-2 \sqrt {2}}\right )}{2 \sqrt {10}-2 \sqrt {2}}-\frac {2 \sqrt {5}\, \arctan \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{2 \sqrt {10}+2 \sqrt {2}}-\frac {2 \arctan \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{2 \sqrt {10}+2 \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {2 \, x^{2} - 1}{4 \, x^{4} + 6 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.38, size = 30, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (2\,\sqrt {2}\,x^3+2\,\sqrt {2}\,x\right )-\mathrm {atan}\left (\sqrt {2}\,x\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 39, normalized size = 0.85 \begin {gather*} - \frac {\sqrt {2} \left (2 \operatorname {atan}{\left (\sqrt {2} x \right )} - 2 \operatorname {atan}{\left (2 \sqrt {2} x^{3} + 2 \sqrt {2} x \right )}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________