Optimal. Leaf size=108 \[ \frac {\sqrt {\frac {\left (5-\sqrt {17}\right ) x^2+4}{\left (5+\sqrt {17}\right ) x^2+4}} \left (\left (5+\sqrt {17}\right ) x^2+4\right ) F\left (\tan ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {x^4+5 x^2+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1099} \[ \frac {\sqrt {\frac {\left (5-\sqrt {17}\right ) x^2+4}{\left (5+\sqrt {17}\right ) x^2+4}} \left (\left (5+\sqrt {17}\right ) x^2+4\right ) F\left (\tan ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1099
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2+x^4}} \, dx &=\frac {\sqrt {\frac {4+\left (5-\sqrt {17}\right ) x^2}{4+\left (5+\sqrt {17}\right ) x^2}} \left (4+\left (5+\sqrt {17}\right ) x^2\right ) F\left (\tan ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {17}} x\right )|\frac {1}{4} \left (-17+5 \sqrt {17}\right )\right )}{2 \sqrt {5+\sqrt {17}} \sqrt {2+5 x^2+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 103, normalized size = 0.95 \[ -\frac {i \sqrt {2 x^2-\sqrt {17}+5} \sqrt {2 x^2+\sqrt {17}+5} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {17}}} x\right )|\frac {21}{4}+\frac {5 \sqrt {17}}{4}\right )}{\sqrt {2 \left (5-\sqrt {17}\right )} \sqrt {x^4+5 x^2+2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 76, normalized size = 0.70 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {\sqrt {17}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {\sqrt {17}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+\sqrt {17}}\, x}{2}, \frac {5 \sqrt {2}}{4}+\frac {\sqrt {34}}{4}\right )}{\sqrt {-5+\sqrt {17}}\, \sqrt {x^{4}+5 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {x^4+5\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________