Integrand size = 54, antiderivative size = 34 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=-\sqrt [4]{3} \arctan \left (\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {3}}}-\frac {1}{2} 3^{3/4} x^2\right ) \] Output:
3^(1/4)*arctan(-1/6*(-18+12*3^(1/2))^(1/2)+1/2*3^(3/4)*x^2)
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=-\frac {\left (-420+1999 \sqrt {3}\right ) \arctan \left (\frac {1-\sqrt {3}+3 x^2}{2 \sqrt [4]{3}}\right )}{\sqrt [4]{3} \left (-1999+140 \sqrt {3}\right )} \] Input:
Integrate[(12*(-420 + 1999*Sqrt[3])*x)/((-1999 + 140*Sqrt[3])*(-4 - 2*Sqrt [3] - 6*x^2 + 6*Sqrt[3]*x^2 - 9*x^4)),x]
Output:
-(((-420 + 1999*Sqrt[3])*ArcTan[(1 - Sqrt[3] + 3*x^2)/(2*3^(1/4))])/(3^(1/ 4)*(-1999 + 140*Sqrt[3])))
Time = 0.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6, 27, 25, 1432, 1083, 217}
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.
\(\displaystyle \int \frac {12 \left (1999 \sqrt {3}-420\right ) x}{\left (140 \sqrt {3}-1999\right ) \left (-9 x^4+6 \sqrt {3} x^2-6 x^2-2 \sqrt {3}-4\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {12 \left (1999 \sqrt {3}-420\right ) x}{\left (140 \sqrt {3}-1999\right ) \left (-9 x^4+\left (6 \sqrt {3}-6\right ) x^2-2 \sqrt {3}-4\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -12 \sqrt {3} \int -\frac {x}{9 x^4+6 \left (1-\sqrt {3}\right ) x^2+2 \left (2+\sqrt {3}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 12 \sqrt {3} \int \frac {x}{9 x^4+6 \left (1-\sqrt {3}\right ) x^2+2 \left (2+\sqrt {3}\right )}dx\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle 6 \sqrt {3} \int \frac {1}{9 x^4+6 \left (1-\sqrt {3}\right ) x^2+2 \left (2+\sqrt {3}\right )}dx^2\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -12 \sqrt {3} \int \frac {1}{-x^4-144 \sqrt {3}}d\left (18 x^2+6 \left (1-\sqrt {3}\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \sqrt [4]{3} \arctan \left (\frac {18 x^2+6 \left (1-\sqrt {3}\right )}{12 \sqrt [4]{3}}\right )\) |
Input:
Int[(12*(-420 + 1999*Sqrt[3])*x)/((-1999 + 140*Sqrt[3])*(-4 - 2*Sqrt[3] - 6*x^2 + 6*Sqrt[3]*x^2 - 9*x^4)),x]
Output:
3^(1/4)*ArcTan[(6*(1 - Sqrt[3]) + 18*x^2)/(12*3^(1/4))]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18
method | result | size |
default | \(-\frac {\left (-420+1999 \sqrt {3}\right ) 3^{\frac {3}{4}} \arctan \left (\frac {\left (18 x^{2}-6 \sqrt {3}+6\right ) 3^{\frac {3}{4}}}{36}\right )}{3 \left (-1999+140 \sqrt {3}\right )}\) | \(40\) |
Input:
int(12*(-420+1999*3^(1/2))*x/(-1999+140*3^(1/2))/(-4-2*3^(1/2)-6*x^2+6*3^( 1/2)*x^2-9*x^4),x,method=_RETURNVERBOSE)
Output:
-1/3*(-420+1999*3^(1/2))/(-1999+140*3^(1/2))*3^(3/4)*arctan(1/36*(18*x^2-6 *3^(1/2)+6)*3^(3/4))
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=3^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} {\left (3 \, x^{2} + 1\right )} - 3\right )}\right ) \] Input:
integrate(12*(-420+1999*3^(1/2))*x/(-1999+140*3^(1/2))/(-4-2*3^(1/2)-6*x^2 +6*3^(1/2)*x^2-9*x^4),x, algorithm="fricas")
Output:
3^(1/4)*arctan(1/6*3^(1/4)*(sqrt(3)*(3*x^2 + 1) - 3))
Exception generated. \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(12*(-420+1999*3**(1/2))*x/(-1999+140*3**(1/2))/(-4-2*3**(1/2)-6* x**2+6*3**(1/2)*x**2-9*x**4),x)
Output:
Exception raised: PolynomialError >> 1/(-173135830797577185636502362255200 6976*_t**2 + 987143636854743804260619971714841888*sqrt(3)*_t**2 - 45428740 085141256258642586775496*_t + 27842854251208551842166254171016*sqrt(3)*_t - 3639190287298
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=-\frac {3^{\frac {3}{4}} {\left (1999 \, \sqrt {3} - 420\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} {\left (3 \, x^{2} - \sqrt {3} + 1\right )}\right )}{3 \, {\left (140 \, \sqrt {3} - 1999\right )}} \] Input:
integrate(12*(-420+1999*3^(1/2))*x/(-1999+140*3^(1/2))/(-4-2*3^(1/2)-6*x^2 +6*3^(1/2)*x^2-9*x^4),x, algorithm="maxima")
Output:
-1/3*3^(3/4)*(1999*sqrt(3) - 420)*arctan(1/6*3^(3/4)*(3*x^2 - sqrt(3) + 1) )/(140*sqrt(3) - 1999)
Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=-\frac {3^{\frac {3}{4}} {\left (1999 \, \sqrt {3} - 420\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} {\left (3 \, x^{2} - \sqrt {3} + 1\right )}\right )}{3 \, {\left (140 \, \sqrt {3} - 1999\right )}} \] Input:
integrate(12*(-420+1999*3^(1/2))*x/(-1999+140*3^(1/2))/(-4-2*3^(1/2)-6*x^2 +6*3^(1/2)*x^2-9*x^4),x, algorithm="giac")
Output:
-1/3*3^(3/4)*(1999*sqrt(3) - 420)*arctan(1/6*3^(3/4)*(3*x^2 - sqrt(3) + 1) )/(140*sqrt(3) - 1999)
Time = 9.58 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=-\frac {2\,\sqrt {3}\,\mathrm {atanh}\left (\frac {32\,x^2}{\sqrt {-4\,\sqrt {3}}\,\left (\frac {32\,\sqrt {3}}{9}+\frac {8\,\sqrt {3}\,x^2}{3}-8\,x^2+\frac {16}{3}\right )}\right )}{\sqrt {-4\,\sqrt {3}}} \] Input:
int(-(12*x*(1999*3^(1/2) - 420))/((140*3^(1/2) - 1999)*(2*3^(1/2) - 6*3^(1 /2)*x^2 + 6*x^2 + 9*x^4 + 4)),x)
Output:
-(2*3^(1/2)*atanh((32*x^2)/((-4*3^(1/2))^(1/2)*((32*3^(1/2))/9 + (8*3^(1/2 )*x^2)/3 - 8*x^2 + 16/3))))/(-4*3^(1/2))^(1/2)
\[ \int \frac {12 \left (-420+1999 \sqrt {3}\right ) x}{\left (-1999+140 \sqrt {3}\right ) \left (-4-2 \sqrt {3}-6 x^2+6 \sqrt {3} x^2-9 x^4\right )} \, dx=108 \sqrt {3}\, \left (\int \frac {x^{5}}{81 x^{8}+108 x^{6}+120 x^{2}+4}d x \right )+72 \sqrt {3}\, \left (\int \frac {x^{3}}{81 x^{8}+108 x^{6}+120 x^{2}+4}d x \right )+48 \sqrt {3}\, \left (\int \frac {x}{81 x^{8}+108 x^{6}+120 x^{2}+4}d x \right )+216 \left (\int \frac {x^{3}}{81 x^{8}+108 x^{6}+120 x^{2}+4}d x \right )-72 \left (\int \frac {x}{81 x^{8}+108 x^{6}+120 x^{2}+4}d x \right ) \] Input:
int(12*(-420+1999*3^(1/2))*x/(-1999+140*3^(1/2))/(-4-2*3^(1/2)-6*x^2+6*3^( 1/2)*x^2-9*x^4),x)
Output:
12*(9*sqrt(3)*int(x**5/(81*x**8 + 108*x**6 + 120*x**2 + 4),x) + 6*sqrt(3)* int(x**3/(81*x**8 + 108*x**6 + 120*x**2 + 4),x) + 4*sqrt(3)*int(x/(81*x**8 + 108*x**6 + 120*x**2 + 4),x) + 18*int(x**3/(81*x**8 + 108*x**6 + 120*x** 2 + 4),x) - 6*int(x/(81*x**8 + 108*x**6 + 120*x**2 + 4),x))