Integrand size = 41, antiderivative size = 47 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=-\frac {1}{6} \sqrt {3+\sqrt {3}} \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \] Output:
-1/6*(3+3^(1/2))^(1/2)*InverseJacobiAM(arccos(1/3*(9-3*3^(1/2))^(1/2)*x),1 /2*(2+2*3^(1/2))^(1/2))
Time = 1.83 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.94 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\frac {\sqrt {3+\sqrt {3}-2 x^2} \sqrt {-3+3 \sqrt {3}-2 \sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{6 \sqrt {\left (-2+\sqrt {3}\right ) \left (3-6 x^2+2 x^4\right )}} \] Input:
Integrate[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x ^2]),x]
Output:
(Sqrt[3 + Sqrt[3] - 2*x^2]*Sqrt[-3 + 3*Sqrt[3] - 2*Sqrt[3]*x^2]*EllipticF[ ArcSin[Sqrt[1 - 1/Sqrt[3]]*x], 2 + Sqrt[3]])/(6*Sqrt[(-2 + Sqrt[3])*(3 - 6 *x^2 + 2*x^4)])
Time = 0.19 (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 = {322}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {2 \sqrt {3} x^2-3 \sqrt {3}+3} \sqrt {\left (\sqrt {3}-3\right ) x^2+3}} \, dx\) |
\(\Big \downarrow \) 322 |
\(\displaystyle -\frac {1}{6} \sqrt {3+\sqrt {3}} \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )\) |
Input:
Int[1/(Sqrt[3 - 3*Sqrt[3] + 2*Sqrt[3]*x^2]*Sqrt[3 + (-3 + Sqrt[3])*x^2]),x ]
Output:
-1/6*(Sqrt[3 + Sqrt[3]]*EllipticF[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sq rt[3])/2])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a - b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/ c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]
Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(34)=68\).
Time = 1.38 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.04
method | result | size |
default | \(\frac {\sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {2}\, \sqrt {-12 x^{2} \sqrt {3}+18 x^{2}+9 \sqrt {3}-9}\, \sqrt {3}\, \sqrt {-3+3 \sqrt {3}-2 x^{2} \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {2}\, \sqrt {\left (-3+2 \sqrt {3}\right ) \left (\sqrt {3}-1\right )}}{3 \sqrt {3}-3}, \frac {\sqrt {\left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}}{\sqrt {3}-1}\right ) \left (-3+\sqrt {3}\right )}{54 \left (\sqrt {3}-1\right )^{\frac {3}{2}} \sqrt {-3+2 \sqrt {3}}\, \left (2 x^{4} \sqrt {3}-2 x^{4}-6 x^{2} \sqrt {3}+6 x^{2}+3 \sqrt {3}-3\right )}\) | \(190\) |
elliptic | \(\frac {\sqrt {\left (3-3 \sqrt {3}+2 x^{2} \sqrt {3}\right ) \left (x^{2} \sqrt {3}-3 x^{2}+3\right )}\, \sqrt {6}\, \sqrt {9-\frac {6 \left (-3+2 \sqrt {3}\right ) x^{2}}{\sqrt {3}-1}}\, \sqrt {9-\frac {6 \sqrt {3}\, x^{2}}{\sqrt {3}-1}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6}\, \sqrt {\frac {-3+2 \sqrt {3}}{\sqrt {3}-1}}}{3}, \frac {\sqrt {-9-\frac {6 \left (18 \sqrt {3}-18\right ) \sqrt {3}}{\left (\sqrt {3}-1\right ) \left (6-6 \sqrt {3}\right )}}}{3}\right )}{18 \sqrt {x^{2} \sqrt {3}-3 x^{2}+3}\, \sqrt {3-3 \sqrt {3}+2 x^{2} \sqrt {3}}\, \sqrt {\frac {-3+2 \sqrt {3}}{\sqrt {3}-1}}\, \sqrt {18 x^{2} \sqrt {3}-18 x^{2}+9-9 \sqrt {3}+6 x^{4}-6 x^{4} \sqrt {3}}}\) | \(223\) |
Input:
int(1/(3-3*3^(1/2)+2*x^2*3^(1/2))^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x,metho d=_RETURNVERBOSE)
Output:
1/54*(3-3*3^(1/2)+2*x^2*3^(1/2))^(1/2)*(x^2*3^(1/2)-3*x^2+3)^(1/2)*2^(1/2) *(-12*x^2*3^(1/2)+18*x^2+9*3^(1/2)-9)^(1/2)/(3^(1/2)-1)^(3/2)*3^(1/2)*(-3+ 3*3^(1/2)-2*x^2*3^(1/2))^(1/2)*EllipticF(1/3*x*3^(1/2)*2^(1/2)/(3^(1/2)-1) *((-3+2*3^(1/2))*(3^(1/2)-1))^(1/2),1/(3^(1/2)-1)*((3^(1/2)-1)*(1+3^(1/2)) )^(1/2))*(-3+3^(1/2))/(-3+2*3^(1/2))^(1/2)/(2*x^4*3^(1/2)-2*x^4-6*x^2*3^(1 /2)+6*x^2+3*3^(1/2)-3)
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=-\frac {1}{18} \, \sqrt {3} \sqrt {\frac {1}{3} \, \sqrt {3} + 1} \sqrt {-9 \, \sqrt {3} + 9} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} + 1}\right )\,|\,-\sqrt {3} + 2) \] Input:
integrate(1/(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x , algorithm="fricas")
Output:
-1/18*sqrt(3)*sqrt(1/3*sqrt(3) + 1)*sqrt(-9*sqrt(3) + 9)*elliptic_f(arcsin (x*sqrt(1/3*sqrt(3) + 1)), -sqrt(3) + 2)
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int \frac {1}{\sqrt {- 3 x^{2} + \sqrt {3} x^{2} + 3} \sqrt {2 \sqrt {3} x^{2} - 3 \sqrt {3} + 3}}\, dx \] Input:
integrate(1/(3-3*3**(1/2)+2*3**(1/2)*x**2)**(1/2)/(3+(-3+3**(1/2))*x**2)** (1/2),x)
Output:
Integral(1/(sqrt(-3*x**2 + sqrt(3)*x**2 + 3)*sqrt(2*sqrt(3)*x**2 - 3*sqrt( 3) + 3)), x)
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}} \,d x } \] Input:
integrate(1/(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x , algorithm="maxima")
Output:
integrate(1/(sqrt(x^2*(sqrt(3) - 3) + 3)*sqrt(2*sqrt(3)*x^2 - 3*sqrt(3) + 3)), x)
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}} \,d x } \] Input:
integrate(1/(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x , algorithm="giac")
Output:
integrate(1/(sqrt(x^2*(sqrt(3) - 3) + 3)*sqrt(2*sqrt(3)*x^2 - 3*sqrt(3) + 3)), x)
Timed out. \[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int \frac {1}{\sqrt {\left (\sqrt {3}-3\right )\,x^2+3}\,\sqrt {2\,\sqrt {3}\,x^2-3\,\sqrt {3}+3}} \,d x \] Input:
int(1/((x^2*(3^(1/2) - 3) + 3)^(1/2)*(2*3^(1/2)*x^2 - 3*3^(1/2) + 3)^(1/2) ),x)
Output:
int(1/((x^2*(3^(1/2) - 3) + 3)^(1/2)*(2*3^(1/2)*x^2 - 3*3^(1/2) + 3)^(1/2) ), x)
\[ \int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3} x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2}} \, dx=\int \frac {1}{\sqrt {3-3 \sqrt {3}+2 \sqrt {3}\, x^{2}}\, \sqrt {3+\left (-3+\sqrt {3}\right ) x^{2}}}d x \] Input:
int(1/(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x)
Output:
int(1/(3-3*3^(1/2)+2*3^(1/2)*x^2)^(1/2)/(3+(-3+3^(1/2))*x^2)^(1/2),x)