Integrand size = 40, antiderivative size = 63 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-4-4 \sqrt {3} x^2+x^4}}\right ) \] Output:
-1/3*(3+2*3^(1/2))^(1/2)*arctan((1+x+3^(1/2))^2/(9+6*3^(1/2))^(1/2)/(-4-4* 3^(1/2)*x^2+x^4)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(204\) vs. \(2(63)=126\).
Time = 35.51 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.24 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {\sqrt {3+2 \sqrt {3}} \left (1+\sqrt {3}-x\right )^2 \sqrt {-\frac {2 \left (-1+\sqrt {3}\right )-2 \left (-2+\sqrt {3}\right ) x+\left (1+\sqrt {3}\right ) x^2+x^3}{\left (1+\sqrt {3}-x\right )^3}} \arctan \left (\frac {\sqrt {-9+6 \sqrt {3}} \left (1+\sqrt {3}-x\right )^2 \sqrt {-\frac {2 \left (-1+\sqrt {3}\right )-2 \left (-2+\sqrt {3}\right ) x+\left (1+\sqrt {3}\right ) x^2+x^3}{\left (1+\sqrt {3}-x\right )^3}}}{-2-2 \left (-1+\sqrt {3}\right ) x+\left (-2+\sqrt {3}\right ) x^2}\right )}{3 \sqrt {-4-4 \sqrt {3} x^2+x^4}} \] Input:
Integrate[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[-4 - 4*Sqrt[3]*x^2 + x ^4]),x]
Output:
-1/3*(Sqrt[3 + 2*Sqrt[3]]*(1 + Sqrt[3] - x)^2*Sqrt[-((2*(-1 + Sqrt[3]) - 2 *(-2 + Sqrt[3])*x + (1 + Sqrt[3])*x^2 + x^3)/(1 + Sqrt[3] - x)^3)]*ArcTan[ (Sqrt[-9 + 6*Sqrt[3]]*(1 + Sqrt[3] - x)^2*Sqrt[-((2*(-1 + Sqrt[3]) - 2*(-2 + Sqrt[3])*x + (1 + Sqrt[3])*x^2 + x^3)/(1 + Sqrt[3] - x)^3)])/(-2 - 2*(- 1 + Sqrt[3])*x + (-2 + Sqrt[3])*x^2)])/Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4]
Time = 0.49 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2278, 216}
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.
| \(\displaystyle \int \frac {x+\sqrt {3}+1}{\left (x-\sqrt {3}+1\right ) \sqrt {x^4-4 \sqrt {3} x^2-4}} \, dx\) |
| \(\Big \downarrow \) 2278 |
| \(\displaystyle -4 \left (2+\sqrt {3}\right ) \int \frac {1}{\frac {4 \left (x+\sqrt {3}+1\right )^4}{x^4-4 \sqrt {3} x^2-4}+12 \left (3+2 \sqrt {3}\right )}d\frac {\left (x+\sqrt {3}+1\right )^2}{\sqrt {x^4-4 \sqrt {3} x^2-4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (x+\sqrt {3}+1\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {x^4-4 \sqrt {3} x^2-4}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}\) |
Input:
Int[(1 + Sqrt[3] + x)/((1 - Sqrt[3] + x)*Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4]),x ]
Output:
-(((2 + Sqrt[3])*ArcTan[(1 + Sqrt[3] + x)^2/(Sqrt[3*(3 + 2*Sqrt[3])]*Sqrt[ -4 - 4*Sqrt[3]*x^2 + x^4])])/Sqrt[3*(3 + 2*Sqrt[3])])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((A_) + (B_.)*(x_))/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_ .)*(x_)^4]), x_Symbol] :> Simp[(-A^2)*((B*d + A*e)/e) Subst[Int[1/(6*A^3* B*d + 3*A^4*e - a*e*x^2), x], x, (A + B*x)^2/Sqrt[a + b*x^2 + c*x^4]], x] / ; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[B*d - A*e, 0] && EqQ[c^2*d^6 + a*e ^4*(13*c*d^2 + b*e^2), 0] && EqQ[b^2*e^4 - 12*c*d^2*(c*d^2 - b*e^2), 0] && EqQ[4*A*c*d*e + B*(2*c*d^2 - b*e^2), 0]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.02 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.94
| method | result | size |
| elliptic | \(\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ), i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-4-4 \sqrt {3}\, x^{2}+x^{4}}}+2 \sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-8-4 \sqrt {3}\, x^{2}+2 x^{2} \left (\sqrt {3}-1\right )^{2}}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-4}\, \sqrt {-4-4 \sqrt {3}\, x^{2}+x^{4}}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-4}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticPi}\left (\sqrt {-1-\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1-\frac {\sqrt {3}}{2}\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-1-\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1-\frac {\sqrt {3}}{2}}\, \left (\sqrt {3}-1\right ) \sqrt {-4-4 \sqrt {3}\, x^{2}+x^{4}}}\right )\) | \(311\) |
Input:
int((1+3^(1/2)+x)/(1-3^(1/2)+x)/(-4-4*3^(1/2)*x^2+x^4)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/(1/2*I+1/2*I*3^(1/2))*(1-(-1-1/2*3^(1/2))*x^2)^(1/2)*(1-(1-1/2*3^(1/2))* x^2)^(1/2)/(-4-4*3^(1/2)*x^2+x^4)^(1/2)*EllipticF(x*(1/2*I+1/2*I*3^(1/2)), I*(1-4*3^(1/2)*(1-1/2*3^(1/2)))^(1/2))+2*3^(1/2)*(-1/2/((3^(1/2)-1)^4-4*3^ (1/2)*(3^(1/2)-1)^2-4)^(1/2)*arctanh(1/2*(-4*3^(1/2)*(3^(1/2)-1)^2-8-4*3^( 1/2)*x^2+2*x^2*(3^(1/2)-1)^2)/((3^(1/2)-1)^4-4*3^(1/2)*(3^(1/2)-1)^2-4)^(1 /2)/(-4-4*3^(1/2)*x^2+x^4)^(1/2))-1/(-1-1/2*3^(1/2))^(1/2)/(3^(1/2)-1)*(1- (-1-1/2*3^(1/2))*x^2)^(1/2)*(1-(1-1/2*3^(1/2))*x^2)^(1/2)/(-4-4*3^(1/2)*x^ 2+x^4)^(1/2)*EllipticPi((-1-1/2*3^(1/2))^(1/2)*x,1/(-1-1/2*3^(1/2))/(3^(1/ 2)-1)^2,(1-1/2*3^(1/2))^(1/2)/(-1-1/2*3^(1/2))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (45) = 90\).
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\right ) \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(-4-4*x^2*3^(1/2)+x^4)^(1/2),x, algo rithm="fricas")
Output:
1/6*sqrt(2*sqrt(3) + 3)*arctan(-(9*x^4 - 30*x^3 + 18*x^2 - 2*sqrt(3)*(2*x^ 4 - 10*x^3 + 3*x^2 - 10*x + 2) + 24)*sqrt(x^4 - 4*sqrt(3)*x^2 - 4)*sqrt(2* sqrt(3) + 3)/(11*x^6 - 42*x^5 + 66*x^4 - 176*x^3 - 132*x^2 - 168*x - 88))
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int \frac {x + 1 + \sqrt {3}}{\left (x - \sqrt {3} + 1\right ) \sqrt {x^{4} - 4 \sqrt {3} x^{2} - 4}}\, dx \] Input:
integrate((1+3**(1/2)+x)/(1-3**(1/2)+x)/(-4-4*x**2*3**(1/2)+x**4)**(1/2),x )
Output:
Integral((x + 1 + sqrt(3))/((x - sqrt(3) + 1)*sqrt(x**4 - 4*sqrt(3)*x**2 - 4)), x)
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}} \,d x } \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(-4-4*x^2*3^(1/2)+x^4)^(1/2),x, algo rithm="maxima")
Output:
integrate((x + sqrt(3) + 1)/(sqrt(x^4 - 4*sqrt(3)*x^2 - 4)*(x - sqrt(3) + 1)), x)
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}} \,d x } \] Input:
integrate((1+3^(1/2)+x)/(1-3^(1/2)+x)/(-4-4*x^2*3^(1/2)+x^4)^(1/2),x, algo rithm="giac")
Output:
integrate((x + sqrt(3) + 1)/(sqrt(x^4 - 4*sqrt(3)*x^2 - 4)*(x - sqrt(3) + 1)), x)
Timed out. \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int \frac {x+\sqrt {3}+1}{\sqrt {x^4-4\,\sqrt {3}\,x^2-4}\,\left (x-\sqrt {3}+1\right )} \,d x \] Input:
int((x + 3^(1/2) + 1)/((x^4 - 4*3^(1/2)*x^2 - 4)^(1/2)*(x - 3^(1/2) + 1)), x)
Output:
int((x + 3^(1/2) + 1)/((x^4 - 4*3^(1/2)*x^2 - 4)^(1/2)*(x - 3^(1/2) + 1)), x)
\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=4 \sqrt {3}\, \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}}{x^{8}-56 x^{4}+16}d x \right )+2 \sqrt {3}\, \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{3}}{x^{8}-56 x^{4}+16}d x \right )+2 \sqrt {3}\, \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{2}}{x^{8}-56 x^{4}+16}d x \right )+8 \sqrt {3}\, \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x}{x^{8}-56 x^{4}+16}d x \right )+8 \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}}{x^{8}-56 x^{4}+16}d x \right )+\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{4}}{x^{8}-56 x^{4}+16}d x +6 \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{2}}{x^{8}-56 x^{4}+16}d x \right )+12 \left (\int \frac {\sqrt {-4 \sqrt {3}\, x^{2}+x^{4}-4}\, x}{x^{8}-56 x^{4}+16}d x \right ) \] Input:
int((1+3^(1/2)+x)/(1-3^(1/2)+x)/(-4-4*x^2*3^(1/2)+x^4)^(1/2),x)
Output:
4*sqrt(3)*int(sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)/(x**8 - 56*x**4 + 16),x) + 2*sqrt(3)*int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x**3)/(x**8 - 56*x**4 + 16),x) + 2*sqrt(3)*int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x**2)/(x**8 - 56*x**4 + 16),x) + 8*sqrt(3)*int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x)/( x**8 - 56*x**4 + 16),x) + 8*int(sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)/(x**8 - 56*x**4 + 16),x) + int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x**4)/(x**8 - 56*x**4 + 16),x) + 6*int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x**2)/(x**8 - 56*x**4 + 16),x) + 12*int((sqrt( - 4*sqrt(3)*x**2 + x**4 - 4)*x)/(x**8 - 56*x**4 + 16),x)