Integrand size = 90, antiderivative size = 51 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=\frac {-2 x-\sqrt {3} x^2-3 x^3-x^5}{1+3 x^2+\sqrt {3} x^3+3 x^4+x^6} \] Output:
(-2*x-3^(1/2)*x^2-3*x^3-x^5)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=-\frac {x \left (2+\sqrt {3} x+3 x^2+x^4\right )}{1+3 x^2+\sqrt {3} x^3+3 x^4+x^6} \] Input:
Integrate[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 + 6*Sqrt[3]*x^ 5 + 10*x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^ 4 + x^6)^2,x]
Output:
-((x*(2 + Sqrt[3]*x + 3*x^2 + x^4))/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^4 + x^6 ))
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 1.87 (sec) , antiderivative size = 515, normalized size of antiderivative = 10.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2466, 2009}
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.
\(\displaystyle \int \frac {x^{10}+6 x^8+2 \sqrt {3} x^7+10 x^6+6 \sqrt {3} x^5+7 x^4+4 \sqrt {3} x^3-3 x^2-2 \sqrt {3} x-2}{\left (x^6+3 x^4+\sqrt {3} x^3+3 x^2+1\right )^2} \, dx\) |
\(\Big \downarrow \) 2466 |
\(\displaystyle 729 \int \left (-\frac {\sqrt [3]{-1} \left (\sqrt {3} \left (1-(-3)^{2/3}\right )-\left (2+\sqrt [3]{-3}\right ) x\right )}{2187 \sqrt [6]{3} \left (x^2+(-1)^{2/3} \sqrt [6]{3} x+1\right )^2}+\frac {2 i}{2187 \left (2 i x^2-\sqrt [6]{3} \left (i-\sqrt {3}\right ) x+2 i\right )}+\frac {2 i}{2187 \left (2 i x^2-\sqrt [6]{3} \left (i+\sqrt {3}\right ) x+2 i\right )}+\frac {1}{2187 \left (x^2+\sqrt [6]{3} x+1\right )}+\frac {\sqrt {3} \left (1-3^{2/3}\right )-\left (2-\sqrt [3]{3}\right ) x}{2187 \sqrt [6]{3} \left (x^2+\sqrt [6]{3} x+1\right )^2}-\frac {3 \left ((-3)^{2/3}+\sqrt [3]{-1}\right )-\left (3 i+\sqrt {3} \left (1+\sqrt [3]{3}\right )\right ) x}{243\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-\sqrt [3]{-1} \sqrt [6]{3} x+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 729 \left (\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \sqrt [6]{3}-2 x}{\sqrt {4-(-1)^{2/3} \sqrt [3]{3}}}\right )}{2187 \sqrt {\frac {1}{2} \left (8+\sqrt [3]{3}-i 3^{5/6}\right )}}-\frac {2 \arctan \left (\frac {2 x+(-1)^{2/3} \sqrt [6]{3}}{\sqrt {4+\sqrt [3]{-3}}}\right )}{2187 \sqrt {4+\sqrt [3]{-3}}}+\frac {2 i \text {arctanh}\left (\frac {\sqrt [6]{3} \left (-\sqrt {3}+i\right )-4 i x}{\sqrt {2 \left (8+\sqrt [3]{3}-i 3^{5/6}\right )}}\right )}{2187 \sqrt {\frac {1}{2} \left (8+\sqrt [3]{3}-i 3^{5/6}\right )}}+\frac {2 i \text {arctanh}\left (\frac {\sqrt [6]{3} \left (\sqrt {3}+i\right )-4 i x}{\sqrt {2 \left (8+\sqrt [3]{3}+i 3^{5/6}\right )}}\right )}{2187 \sqrt {\frac {1}{2} \left (8+\sqrt [3]{3}+i 3^{5/6}\right )}}+\frac {-\sqrt [6]{3} \left (4-\sqrt [3]{3}\right ) x+3^{2/3}-5 \sqrt [3]{3}+4}{2187 \sqrt [6]{3} \left (4-\sqrt [3]{3}\right ) \left (x^2+\sqrt [6]{3} x+1\right )}-\frac {\sqrt [3]{-1} \left (3+3^{2/3} \left (2+2 i \sqrt {3}\right )\right ) x+5\ 3^{5/6}+2 \sqrt {3}-3 (-1)^{2/3} \sqrt [6]{3}+6 i}{243\ 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-(-1)^{2/3} \sqrt [3]{3}\right ) \left (x^2-\sqrt [3]{-1} \sqrt [6]{3} x+1\right )}-\frac {\sqrt [3]{-1} \left (-\sqrt [6]{3} \left (4 (-1)^{2/3}-\sqrt [3]{3}\right ) x+(-3)^{2/3}+5 \sqrt [3]{-3}+4\right )}{2187 \sqrt [6]{3} \left (4+\sqrt [3]{-3}\right ) \left (x^2+(-1)^{2/3} \sqrt [6]{3} x+1\right )}\right )\) |
Input:
Int[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 + 6*Sqrt[3]*x^5 + 10 *x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^4 + x^ 6)^2,x]
Output:
729*((4 - 5*3^(1/3) + 3^(2/3) - 3^(1/6)*(4 - 3^(1/3))*x)/(2187*3^(1/6)*(4 - 3^(1/3))*(1 + 3^(1/6)*x + x^2)) - (6*I - 3*(-1)^(2/3)*3^(1/6) + 2*Sqrt[3 ] + 5*3^(5/6) + (-1)^(1/3)*(3 + 3^(2/3)*(2 + (2*I)*Sqrt[3]))*x)/(243*3^(2/ 3)*(1 + (-1)^(1/3))^4*(4 - (-1)^(2/3)*3^(1/3))*(1 - (-1)^(1/3)*3^(1/6)*x + x^2)) - ((-1)^(1/3)*(4 + 5*(-3)^(1/3) + (-3)^(2/3) - 3^(1/6)*(4*(-1)^(2/3 ) - 3^(1/3))*x))/(2187*3^(1/6)*(4 + (-3)^(1/3))*(1 + (-1)^(2/3)*3^(1/6)*x + x^2)) + (2*ArcTan[((-1)^(1/3)*3^(1/6) - 2*x)/Sqrt[4 - (-1)^(2/3)*3^(1/3) ]])/(2187*Sqrt[(8 + 3^(1/3) - I*3^(5/6))/2]) - (2*ArcTan[((-1)^(2/3)*3^(1/ 6) + 2*x)/Sqrt[4 + (-3)^(1/3)]])/(2187*Sqrt[4 + (-3)^(1/3)]) + (((2*I)/218 7)*ArcTanh[(3^(1/6)*(I - Sqrt[3]) - (4*I)*x)/Sqrt[2*(8 + 3^(1/3) - I*3^(5/ 6))]])/Sqrt[(8 + 3^(1/3) - I*3^(5/6))/2] + (((2*I)/2187)*ArcTanh[(3^(1/6)* (I + Sqrt[3]) - (4*I)*x)/Sqrt[2*(8 + 3^(1/3) + I*3^(5/6))]])/Sqrt[(8 + 3^( 1/3) + I*3^(5/6))/2])
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Time = 0.70 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {x \left (x^{4}+\sqrt {3}\, x +3 x^{2}+2\right )}{1+3 x^{2}+\sqrt {3}\, x^{3}+3 x^{4}+x^{6}}\) | \(43\) |
orering | \(-\frac {x \left (x^{4}+\sqrt {3}\, x +3 x^{2}+2\right )}{1+3 x^{2}+\sqrt {3}\, x^{3}+3 x^{4}+x^{6}}\) | \(43\) |
default | \(\frac {-2 x -\sqrt {3}\, x^{2}-3 x^{3}-x^{5}}{1+3 x^{2}+\sqrt {3}\, x^{3}+3 x^{4}+x^{6}}\) | \(48\) |
risch | \(\frac {-2 x -\sqrt {3}\, x^{2}-3 x^{3}-x^{5}}{1+3 x^{2}+\sqrt {3}\, x^{3}+3 x^{4}+x^{6}}\) | \(48\) |
parallelrisch | \(\frac {-2 x -\sqrt {3}\, x^{2}-3 x^{3}-x^{5}}{1+3 x^{2}+\sqrt {3}\, x^{3}+3 x^{4}+x^{6}}\) | \(48\) |
norman | \(\frac {-\sqrt {3}\, x^{2}-\sqrt {3}\, x^{4}-2 x -9 x^{3}-13 x^{5}-14 x^{7}-6 x^{9}-x^{11}}{x^{12}+6 x^{10}+15 x^{8}+17 x^{6}+15 x^{4}+6 x^{2}+1}\) | \(79\) |
Input:
int((-2-2*3^(1/2)*x-3*x^2+4*3^(1/2)*x^3+7*x^4+6*3^(1/2)*x^5+10*x^6+2*3^(1/ 2)*x^7+6*x^8+x^10)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)^2,x,method=_RETURNVERBO SE)
Output:
-x*(x^4+3^(1/2)*x+3*x^2+2)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.41 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=-\frac {x^{11} + 6 \, x^{9} + 14 \, x^{7} + 13 \, x^{5} + 9 \, x^{3} + \sqrt {3} {\left (x^{4} + x^{2}\right )} + 2 \, x}{x^{12} + 6 \, x^{10} + 15 \, x^{8} + 17 \, x^{6} + 15 \, x^{4} + 6 \, x^{2} + 1} \] Input:
integrate((-2-2*3^(1/2)*x-3*x^2+4*3^(1/2)*x^3+7*x^4+6*3^(1/2)*x^5+10*x^6+2 *3^(1/2)*x^7+6*x^8+x^10)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)^2,x, algorithm="f ricas")
Output:
-(x^11 + 6*x^9 + 14*x^7 + 13*x^5 + 9*x^3 + sqrt(3)*(x^4 + x^2) + 2*x)/(x^1 2 + 6*x^10 + 15*x^8 + 17*x^6 + 15*x^4 + 6*x^2 + 1)
Time = 0.38 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=\frac {- x^{5} - 3 x^{3} - \sqrt {3} x^{2} - 2 x}{x^{6} + 3 x^{4} + \sqrt {3} x^{3} + 3 x^{2} + 1} \] Input:
integrate((-2-2*3**(1/2)*x-3*x**2+4*3**(1/2)*x**3+7*x**4+6*3**(1/2)*x**5+1 0*x**6+2*3**(1/2)*x**7+6*x**8+x**10)/(1+3*x**2+3**(1/2)*x**3+3*x**4+x**6)* *2,x)
Output:
(-x**5 - 3*x**3 - sqrt(3)*x**2 - 2*x)/(x**6 + 3*x**4 + sqrt(3)*x**3 + 3*x* *2 + 1)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=-\frac {\sqrt {3} x^{5} + 3 \, \sqrt {3} x^{3} + 3 \, x^{2} + 2 \, \sqrt {3} x}{\sqrt {3} x^{6} + 3 \, \sqrt {3} x^{4} + 3 \, x^{3} + 3 \, \sqrt {3} x^{2} + \sqrt {3}} \] Input:
integrate((-2-2*3^(1/2)*x-3*x^2+4*3^(1/2)*x^3+7*x^4+6*3^(1/2)*x^5+10*x^6+2 *3^(1/2)*x^7+6*x^8+x^10)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)^2,x, algorithm="m axima")
Output:
-(sqrt(3)*x^5 + 3*sqrt(3)*x^3 + 3*x^2 + 2*sqrt(3)*x)/(sqrt(3)*x^6 + 3*sqrt (3)*x^4 + 3*x^3 + 3*sqrt(3)*x^2 + sqrt(3))
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=-\frac {x^{5} + 3 \, x^{3} + \sqrt {3} x^{2} + 2 \, x}{x^{6} + 3 \, x^{4} + \sqrt {3} x^{3} + 3 \, x^{2} + 1} \] Input:
integrate((-2-2*3^(1/2)*x-3*x^2+4*3^(1/2)*x^3+7*x^4+6*3^(1/2)*x^5+10*x^6+2 *3^(1/2)*x^7+6*x^8+x^10)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)^2,x, algorithm="g iac")
Output:
-(x^5 + 3*x^3 + sqrt(3)*x^2 + 2*x)/(x^6 + 3*x^4 + sqrt(3)*x^3 + 3*x^2 + 1)
Time = 0.65 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=-\frac {x\,\left (x^4+3\,x^2+\sqrt {3}\,x+2\right )}{x^6+3\,x^4+\sqrt {3}\,x^3+3\,x^2+1} \] Input:
int((4*3^(1/2)*x^3 - 2*3^(1/2)*x + 6*3^(1/2)*x^5 + 2*3^(1/2)*x^7 - 3*x^2 + 7*x^4 + 10*x^6 + 6*x^8 + x^10 - 2)/(3^(1/2)*x^3 + 3*x^2 + 3*x^4 + x^6 + 1 )^2,x)
Output:
-(x*(3^(1/2)*x + 3*x^2 + x^4 + 2))/(3^(1/2)*x^3 + 3*x^2 + 3*x^4 + x^6 + 1)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.43 \[ \int \frac {-2-2 \sqrt {3} x-3 x^2+4 \sqrt {3} x^3+7 x^4+6 \sqrt {3} x^5+10 x^6+2 \sqrt {3} x^7+6 x^8+x^{10}}{\left (1+3 x^2+\sqrt {3} x^3+3 x^4+x^6\right )^2} \, dx=\frac {x \left (-\sqrt {3}\, x^{3}-\sqrt {3}\, x -x^{10}-6 x^{8}-14 x^{6}-13 x^{4}-9 x^{2}-2\right )}{x^{12}+6 x^{10}+15 x^{8}+17 x^{6}+15 x^{4}+6 x^{2}+1} \] Input:
int((-2-2*3^(1/2)*x-3*x^2+4*3^(1/2)*x^3+7*x^4+6*3^(1/2)*x^5+10*x^6+2*3^(1/ 2)*x^7+6*x^8+x^10)/(1+3*x^2+3^(1/2)*x^3+3*x^4+x^6)^2,x)
Output:
(x*( - sqrt(3)*x**3 - sqrt(3)*x - x**10 - 6*x**8 - 14*x**6 - 13*x**4 - 9*x **2 - 2))/(x**12 + 6*x**10 + 15*x**8 + 17*x**6 + 15*x**4 + 6*x**2 + 1)