Integrand size = 41, antiderivative size = 47 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=-\frac {1}{6} \sqrt {3+\sqrt {3}} \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {431} \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=-\frac {1}{6} \sqrt {3+\sqrt {3}} \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \]
[In]
[Out]
Rule 431
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6} \sqrt {3+\sqrt {3}} F\left (\cos ^{-1}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \\ \end{align*}
Time = 1.98 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\frac {\sqrt {3+\sqrt {3}-2 x^2} \sqrt {-3+3 \sqrt {3}-2 \sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{6 \sqrt {\left (-2+\sqrt {3}\right ) \left (3-6 x^2+2 x^4\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(64)=128\).
Time = 2.61 (sec) , antiderivative size = 197, normalized size of antiderivative = 4.19
method | result | size |
default | \(\frac {\sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {2}\, \sqrt {-12 x^{2} \sqrt {3}+18 x^{2}+9 \sqrt {3}-9}\, \sqrt {3}\, \sqrt {-\left (3-3 \sqrt {3}+2 x^{2} \sqrt {3}\right ) \left (\sqrt {3}-1\right )}\, F\left (\frac {x \sqrt {2}\, \sqrt {3}\, \sqrt {\left (2 \sqrt {3}-3\right ) \left (\sqrt {3}-1\right )}}{3 \sqrt {3}-3}, \frac {\sqrt {\left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}}{\sqrt {3}-1}\right ) \left (-3+\sqrt {3}\right )}{54 \left (\sqrt {3}-1\right )^{2} \sqrt {2 \sqrt {3}-3}\, \left (2 x^{4} \sqrt {3}-2 x^{4}-6 x^{2} \sqrt {3}+6 x^{2}+3 \sqrt {3}-3\right )}\) | \(197\) |
elliptic | \(\frac {\sqrt {\left (x^{2} \sqrt {3}-3 x^{2}+3\right ) \left (3-3 \sqrt {3}+2 x^{2} \sqrt {3}\right )}\, \sqrt {6}\, \sqrt {9-\frac {6 \left (2 \sqrt {3}-3\right ) x^{2}}{\sqrt {3}-1}}\, \sqrt {9-\frac {6 \sqrt {3}\, x^{2}}{\sqrt {3}-1}}\, F\left (\frac {x \sqrt {6}\, \sqrt {\frac {2 \sqrt {3}-3}{\sqrt {3}-1}}}{3}, \frac {\sqrt {-9-\frac {6 \left (18 \sqrt {3}-18\right ) \sqrt {3}}{\left (\sqrt {3}-1\right ) \left (6-6 \sqrt {3}\right )}}}{3}\right )}{18 \sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {\frac {2 \sqrt {3}-3}{\sqrt {3}-1}}\, \sqrt {18 x^{2} \sqrt {3}-18 x^{2}+6 x^{4}-6 x^{4} \sqrt {3}+9-9 \sqrt {3}}}\) | \(223\) |
[In]
[Out]
none
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=-\frac {1}{18} \, \sqrt {\sqrt {3} + 3} \sqrt {-9 \, \sqrt {3} + 9} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {3} + 3}\right )\,|\,-\sqrt {3} + 2) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int \frac {1}{\sqrt {- 3 x^{2} + \sqrt {3} x^{2} + 3} \sqrt {2 \sqrt {3} x^{2} - 3 \sqrt {3} + 3}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int \frac {1}{\sqrt {\left (\sqrt {3}-3\right )\,x^2+3}\,\sqrt {2\,\sqrt {3}\,x^2-3\,\sqrt {3}+3}} \,d x \]
[In]
[Out]