\(\int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx\) [98]

Optimal result

Integrand size = 38, antiderivative size = 111 \[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=-2^{3/4} \sqrt {-1+\sqrt {1+2 \sqrt {3}}} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (-1+\sqrt {1+2 \sqrt {3}}\right )}}\right )-2^{3/4} \sqrt {1+\sqrt {1+2 \sqrt {3}}} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (1+\sqrt {1+2 \sqrt {3}}\right )}}\right ) \] Output:

-2^(3/4)*(-1+(1+2*3^(1/2))^(1/2))^(1/2)*arctan(2^(1/4)*x/(-3+3*(1+2*3^(1/2 
))^(1/2))^(1/2))-2^(3/4)*(1+(1+2*3^(1/2))^(1/2))^(1/2)*arctanh(2^(1/4)*x/( 
3+3*(1+2*3^(1/2))^(1/2))^(1/2))
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.56 \[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=-\frac {2^{3/4} \left (\frac {\left (6+\sqrt {3}-\sqrt {3+6 \sqrt {3}}\right ) \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (-1+\sqrt {1+2 \sqrt {3}}\right )}}\right )}{\sqrt {\left (1+2 \sqrt {3}\right ) \left (-1+\sqrt {1+2 \sqrt {3}}\right )}}+\frac {\left (6+\sqrt {3}+\sqrt {3+6 \sqrt {3}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (1+\sqrt {1+2 \sqrt {3}}\right )}}\right )}{\sqrt {\left (1+2 \sqrt {3}\right ) \left (1+\sqrt {1+2 \sqrt {3}}\right )}}\right )}{\sqrt {3}} \] Input:

Integrate[(-36 - 2*Sqrt[6]*x^2)/(9*Sqrt[3] + 3*Sqrt[2]*x^2 - x^4),x]
 

Output:

-((2^(3/4)*(((6 + Sqrt[3] - Sqrt[3 + 6*Sqrt[3]])*ArcTan[(2^(1/4)*x)/Sqrt[3 
*(-1 + Sqrt[1 + 2*Sqrt[3]])]])/Sqrt[(1 + 2*Sqrt[3])*(-1 + Sqrt[1 + 2*Sqrt[ 
3]])] + ((6 + Sqrt[3] + Sqrt[3 + 6*Sqrt[3]])*ArcTanh[(2^(1/4)*x)/Sqrt[3*(1 
 + Sqrt[1 + 2*Sqrt[3]])]])/Sqrt[(1 + 2*Sqrt[3])*(1 + Sqrt[1 + 2*Sqrt[3]])] 
))/Sqrt[3])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1480, 217, 219}

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.

\(\displaystyle \int \frac {-2 \sqrt {6} x^2-36}{-x^4+3 \sqrt {2} x^2+9 \sqrt {3}} \, dx\)

\(\Big \downarrow \) 1480

\(\displaystyle -\sqrt {6} \left (1-\sqrt {1+2 \sqrt {3}}\right ) \int \frac {1}{\frac {3 \left (1-\sqrt {1+2 \sqrt {3}}\right )}{\sqrt {2}}-x^2}dx-\sqrt {6} \left (1+\sqrt {1+2 \sqrt {3}}\right ) \int \frac {1}{\frac {3 \left (1+\sqrt {1+2 \sqrt {3}}\right )}{\sqrt {2}}-x^2}dx\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2^{3/4} \left (1-\sqrt {1+2 \sqrt {3}}\right ) \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (\sqrt {1+2 \sqrt {3}}-1\right )}}\right )}{\sqrt {\sqrt {1+2 \sqrt {3}}-1}}-\sqrt {6} \left (1+\sqrt {1+2 \sqrt {3}}\right ) \int \frac {1}{\frac {3 \left (1+\sqrt {1+2 \sqrt {3}}\right )}{\sqrt {2}}-x^2}dx\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2^{3/4} \left (1-\sqrt {1+2 \sqrt {3}}\right ) \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (\sqrt {1+2 \sqrt {3}}-1\right )}}\right )}{\sqrt {\sqrt {1+2 \sqrt {3}}-1}}-2^{3/4} \sqrt {1+\sqrt {1+2 \sqrt {3}}} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {3 \left (1+\sqrt {1+2 \sqrt {3}}\right )}}\right )\)

Input:

Int[(-36 - 2*Sqrt[6]*x^2)/(9*Sqrt[3] + 3*Sqrt[2]*x^2 - x^4),x]
 

Output:

(2^(3/4)*(1 - Sqrt[1 + 2*Sqrt[3]])*ArcTan[(2^(1/4)*x)/Sqrt[3*(-1 + Sqrt[1 
+ 2*Sqrt[3]])]])/Sqrt[-1 + Sqrt[1 + 2*Sqrt[3]]] - 2^(3/4)*Sqrt[1 + Sqrt[1 
+ 2*Sqrt[3]]]*ArcTanh[(2^(1/4)*x)/Sqrt[3*(1 + Sqrt[1 + 2*Sqrt[3]])]]
 

Defintions of rubi rules used

rule 217

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])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68

method	result	size
risch	\(\sqrt {3}\, \sqrt {2}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3, \operatorname {index} =1\right )-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2, \operatorname {index} =1\right ) \textit {\_Z}^{2}+\textit {\_Z}^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-3 \sqrt {3}\, \sqrt {2}\right ) \ln \left (x -\textit {\_R} \right )}{3 \sqrt {2}\, \textit {\_R} -2 \textit {\_R}^{3}}\right )\)	\(76\)
default	\(-\frac {2 \left (\sqrt {2}\, \sqrt {6}-\sqrt {6}\, \sqrt {2+4 \sqrt {3}}+12\right ) \arctan \left (\frac {2 x}{\sqrt {-6 \sqrt {2}+6 \sqrt {2+4 \sqrt {3}}}}\right )}{\sqrt {2+4 \sqrt {3}}\, \sqrt {-6 \sqrt {2}+6 \sqrt {2+4 \sqrt {3}}}}+\frac {2 \left (-\sqrt {2}\, \sqrt {6}-\sqrt {6}\, \sqrt {2+4 \sqrt {3}}-12\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {6 \sqrt {2}+6 \sqrt {2+4 \sqrt {3}}}}\right )}{\sqrt {2+4 \sqrt {3}}\, \sqrt {6 \sqrt {2}+6 \sqrt {2+4 \sqrt {3}}}}\)	\(155\)

Input:

int((-36-2*6^(1/2)*x^2)/(9*3^(1/2)+3*2^(1/2)*x^2-x^4),x,method=_RETURNVERB 
OSE)
 

Output:

3^(1/2)*2^(1/2)*sum((-_R^2-3*3^(1/2)*2^(1/2))/(3*2^(1/2)*_R-2*_R^3)*ln(x-_ 
R),_R=RootOf(-9*RootOf(_Z^2-3,index=1)-3*RootOf(_Z^2-2,index=1)*_Z^2+_Z^4) 
)
 

Fricas [F]

\[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=\int { \frac {2 \, {\left (\sqrt {6} x^{2} + 18\right )}}{x^{4} - 3 \, \sqrt {2} x^{2} - 9 \, \sqrt {3}} \,d x } \] Input:

integrate((-36-2*6^(1/2)*x^2)/(9*3^(1/2)+3*2^(1/2)*x^2-x^4),x, algorithm=" 
fricas")
 

Output:

0
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((-36-2*6**(1/2)*x**2)/(9*3**(1/2)+3*2**(1/2)*x**2-x**4),x)
 

Output:

Exception raised: PolynomialError >> 1/(2*_t**4 - 2*sqrt(2)*_t**2 + 1) con 
tains an element of the set of generators.
 

Maxima [F]

\[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=\int { \frac {2 \, {\left (\sqrt {6} x^{2} + 18\right )}}{x^{4} - 3 \, \sqrt {2} x^{2} - 9 \, \sqrt {3}} \,d x } \] Input:

integrate((-36-2*6^(1/2)*x^2)/(9*3^(1/2)+3*2^(1/2)*x^2-x^4),x, algorithm=" 
maxima")
 

Output:

2*integrate((sqrt(6)*x^2 + 18)/(x^4 - 3*sqrt(2)*x^2 - 9*sqrt(3)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-36-2*6^(1/2)*x^2)/(9*3^(1/2)+3*2^(1/2)*x^2-x^4),x, algorithm=" 
giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:  ***   Warning: increasing stack s 
ize to 4096000.  ***   Warning: increasing stack size to 8192000.  ***   W 
arning: i
 

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 2068, normalized size of antiderivative = 18.63 \[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=\text {Too large to display} \] Input:

int(-(2*6^(1/2)*x^2 + 36)/(9*3^(1/2) + 3*2^(1/2)*x^2 - x^4),x)
 

Output:

- atan(((x*(864*2^(1/2)*6^(1/2) + 864*3^(1/2) + 6048) + ((48*3^(1/2)*6^(1/ 
2) - 132*2^(1/2)*3^(1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/ 
2) + 36*((4*3^(1/2) + 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2)*(5184*3^(1 
/2) - x*(864*2^(1/2)*3^(1/2) + 432*2^(1/2))*((48*3^(1/2)*6^(1/2) - 132*2^( 
1/2)*3^(1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/2) + 36*((4* 
3^(1/2) + 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2) + 2592))*((48*3^(1/2)* 
6^(1/2) - 132*2^(1/2)*3^(1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3 
)^(1/2) + 36*((4*3^(1/2) + 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2)*1i + 
(x*(864*2^(1/2)*6^(1/2) + 864*3^(1/2) + 6048) - ((48*3^(1/2)*6^(1/2) - 132 
*2^(1/2)*3^(1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/2) + 36* 
((4*3^(1/2) + 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2)*(5184*3^(1/2) + x* 
(864*2^(1/2)*3^(1/2) + 432*2^(1/2))*((48*3^(1/2)*6^(1/2) - 132*2^(1/2)*3^( 
1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/2) + 36*((4*3^(1/2) 
+ 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2) + 2592))*((48*3^(1/2)*6^(1/2) 
- 132*2^(1/2)*3^(1/2) + 288*6^(1/2) + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/2) 
+ 36*((4*3^(1/2) + 2)^3)^(1/2))/(6*(52*3^(1/2) + 48)))^(1/2)*1i)/(5184*2^( 
1/2) - 864*3^(1/2)*6^(1/2) + 5184*6^(1/2) - (x*(864*2^(1/2)*6^(1/2) + 864* 
3^(1/2) + 6048) + ((48*3^(1/2)*6^(1/2) - 132*2^(1/2)*3^(1/2) + 288*6^(1/2) 
 + 6*3^(1/2)*((4*3^(1/2) + 2)^3)^(1/2) + 36*((4*3^(1/2) + 2)^3)^(1/2))/(6* 
(52*3^(1/2) + 48)))^(1/2)*(5184*3^(1/2) - x*(864*2^(1/2)*3^(1/2) + 432*...
 

Reduce [F]

\[ \int \frac {-36-2 \sqrt {6} x^2}{9 \sqrt {3}+3 \sqrt {2} x^2-x^4} \, dx=2 \sqrt {6}\, \left (\int \frac {x^{14}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )-36 \sqrt {6}\, \left (\int \frac {x^{10}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+1458 \sqrt {6}\, \left (\int \frac {x^{6}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+12 \sqrt {3}\, \left (\int \frac {x^{12}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+108 \sqrt {3}\, \left (\int \frac {x^{8}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+8748 \sqrt {3}\, \left (\int \frac {x^{4}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )-78732 \sqrt {3}\, \left (\int \frac {1}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+162 \sqrt {2}\, \left (\int \frac {x^{10}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )-972 \sqrt {2}\, \left (\int \frac {x^{6}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+13122 \sqrt {2}\, \left (\int \frac {x^{2}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )+36 \left (\int \frac {x^{12}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right )-8748 \left (\int \frac {x^{4}}{x^{16}-36 x^{12}-162 x^{8}-8748 x^{4}+59049}d x \right ) \] Input:

int((-36-2*6^(1/2)*x^2)/(9*3^(1/2)+3*2^(1/2)*x^2-x^4),x)
 

Output:

2*(sqrt(6)*int(x**14/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 59049),x) 
- 18*sqrt(6)*int(x**10/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 59049),x 
) + 729*sqrt(6)*int(x**6/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 59049) 
,x) + 6*sqrt(3)*int(x**12/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 59049 
),x) + 54*sqrt(3)*int(x**8/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 5904 
9),x) + 4374*sqrt(3)*int(x**4/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 5 
9049),x) - 39366*sqrt(3)*int(1/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 
59049),x) + 81*sqrt(2)*int(x**10/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 
+ 59049),x) - 486*sqrt(2)*int(x**6/(x**16 - 36*x**12 - 162*x**8 - 8748*x** 
4 + 59049),x) + 6561*sqrt(2)*int(x**2/(x**16 - 36*x**12 - 162*x**8 - 8748* 
x**4 + 59049),x) + 18*int(x**12/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 
 59049),x) - 4374*int(x**4/(x**16 - 36*x**12 - 162*x**8 - 8748*x**4 + 5904 
9),x))