\(\int \frac {1}{(3+\sqrt {33}-6 x^2)^{3/2} (-3+\sqrt {33}+6 x^2)^{3/2}} \, dx\) [201]

Optimal result

Integrand size = 33, antiderivative size = 215 \[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {3+\sqrt {33}-6 x^2} \sqrt {-3+\sqrt {33}+6 x^2}}+\frac {1}{528} \sqrt {\frac {1}{6} \left (3+\sqrt {33}\right )} \sqrt {\frac {1}{-3+\sqrt {33}+6 x^2}} \sqrt {-3+\sqrt {33}+6 x^2} E\left (\arctan \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (11-\sqrt {33}\right )\right )+\frac {\sqrt {\frac {1}{-3+\sqrt {33}+6 x^2}} \sqrt {-3+\sqrt {33}+6 x^2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (11-\sqrt {33}\right )\right )}{66 \sqrt {6 \left (3+\sqrt {33}\right )}} \] Output:

1/6*x/(11+33^(1/2))/(3+33^(1/2)-6*x^2)^(1/2)/(-3+33^(1/2)+6*x^2)^(1/2)+1/3 
168*(18+6*33^(1/2))^(1/2)*(1/(-3+33^(1/2)+6*x^2))^(1/2)*(-3+33^(1/2)+6*x^2 
)^(1/2)*EllipticE(6^(1/2)/(-3+33^(1/2))^(1/2)*x/(1+6/(-3+33^(1/2))*x^2)^(1 
/2),1/2*(11-33^(1/2))^(1/2))+1/66*(1/(-3+33^(1/2)+6*x^2))^(1/2)*(-3+33^(1/ 
2)+6*x^2)^(1/2)*InverseJacobiAM(arctan(6^(1/2)/(-3+33^(1/2))^(1/2)*x),1/2* 
(11-33^(1/2))^(1/2))/(18+6*33^(1/2))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {3 \left (3+\sqrt {33}\right )} x \left (7-3 x^2\right )+3 i \left (\sqrt {3}+\sqrt {11}\right ) \sqrt {4+6 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )-i \left (11 \sqrt {3}+3 \sqrt {11}\right ) \sqrt {4+6 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{3168 \sqrt {3+\sqrt {33}} \sqrt {2+3 x^2-3 x^4}} \] Input:

Integrate[1/((3 + Sqrt[33] - 6*x^2)^(3/2)*(-3 + Sqrt[33] + 6*x^2)^(3/2)),x 
]
 

Output:

(2*Sqrt[3*(3 + Sqrt[33])]*x*(7 - 3*x^2) + (3*I)*(Sqrt[3] + Sqrt[11])*Sqrt[ 
4 + 6*x^2 - 6*x^4]*EllipticE[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + S 
qrt[33])/4] - I*(11*Sqrt[3] + 3*Sqrt[11])*Sqrt[4 + 6*x^2 - 6*x^4]*Elliptic 
F[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4])/(3168*Sqrt[3 + 
 Sqrt[33]]*Sqrt[2 + 3*x^2 - 3*x^4])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {316, 27, 402, 27, 399, 321, 327}

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.

\(\displaystyle \int \frac {1}{\left (-6 x^2+\sqrt {33}+3\right )^{3/2} \left (6 x^2+\sqrt {33}-3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {\int \frac {6 \left (6 x^2+\sqrt {33}+3\right )}{\sqrt {-6 x^2+\sqrt {33}+3} \left (6 x^2+\sqrt {33}-3\right )^{3/2}}dx}{36 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {6 x^2+\sqrt {33}+3}{\sqrt {-6 x^2+\sqrt {33}+3} \left (6 x^2+\sqrt {33}-3\right )^{3/2}}dx}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {x \sqrt {-6 x^2+\sqrt {33}+3}}{\left (11-\sqrt {33}\right ) \sqrt {6 x^2+\sqrt {33}-3}}-\frac {\int -\frac {72 \left (3 x^2+4\right )}{\sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}dx}{36 \left (11-\sqrt {33}\right )}}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {3 x^2+4}{\sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}dx}{11-\sqrt {33}}+\frac {\sqrt {-6 x^2+\sqrt {33}+3} x}{\left (11-\sqrt {33}\right ) \sqrt {6 x^2+\sqrt {33}-3}}}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \left (11-\sqrt {33}\right ) \int \frac {1}{\sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}dx+\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}-3}}{\sqrt {-6 x^2+\sqrt {33}+3}}dx\right )}{11-\sqrt {33}}+\frac {\sqrt {-6 x^2+\sqrt {33}+3} x}{\left (11-\sqrt {33}\right ) \sqrt {6 x^2+\sqrt {33}-3}}}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}-3}}{\sqrt {-6 x^2+\sqrt {33}+3}}dx+\frac {\left (11-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{2 \sqrt {6 \left (\sqrt {33}-3\right )}}\right )}{11-\sqrt {33}}+\frac {\sqrt {-6 x^2+\sqrt {33}+3} x}{\left (11-\sqrt {33}\right ) \sqrt {6 x^2+\sqrt {33}-3}}}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 \left (\frac {\left (11-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{2 \sqrt {6 \left (\sqrt {33}-3\right )}}+\frac {1}{2} \sqrt {\frac {1}{6} \left (\sqrt {33}-3\right )} E\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7-\sqrt {33}\right )\right )\right )}{11-\sqrt {33}}+\frac {\sqrt {-6 x^2+\sqrt {33}+3} x}{\left (11-\sqrt {33}\right ) \sqrt {6 x^2+\sqrt {33}-3}}}{6 \left (11+\sqrt {33}\right )}+\frac {x}{6 \left (11+\sqrt {33}\right ) \sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}\)

Input:

Int[1/((3 + Sqrt[33] - 6*x^2)^(3/2)*(-3 + Sqrt[33] + 6*x^2)^(3/2)),x]
 

Output:

x/(6*(11 + Sqrt[33])*Sqrt[3 + Sqrt[33] - 6*x^2]*Sqrt[-3 + Sqrt[33] + 6*x^2 
]) + ((x*Sqrt[3 + Sqrt[33] - 6*x^2])/((11 - Sqrt[33])*Sqrt[-3 + Sqrt[33] + 
 6*x^2]) + (2*((Sqrt[(-3 + Sqrt[33])/6]*EllipticE[ArcSin[Sqrt[6/(3 + Sqrt[ 
33])]*x], (-7 - Sqrt[33])/4])/2 + ((11 - Sqrt[33])*EllipticF[ArcSin[Sqrt[6 
/(3 + Sqrt[33])]*x], (-7 - Sqrt[33])/4])/(2*Sqrt[6*(-3 + Sqrt[33])])))/(11 
 - Sqrt[33]))/(6*(11 + Sqrt[33]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {x \left (3 x^{2}-7\right )}{264 \sqrt {3+\sqrt {33}-6 x^{2}}\, \sqrt {-3+\sqrt {33}+6 x^{2}}}+\frac {\left (\frac {\sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{66 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}}-\frac {6 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}\, \left (36+12 \sqrt {33}\right )}\right ) \sqrt {\left (3+\sqrt {33}-6 x^{2}\right ) \left (-3+\sqrt {33}+6 x^{2}\right )}}{\sqrt {3+\sqrt {33}-6 x^{2}}\, \sqrt {-3+\sqrt {33}+6 x^{2}}}\)	\(279\)
elliptic	\(-\frac {12 \left (3 x^{4}-3 x^{2}-2\right ) \sqrt {\left (3+\sqrt {33}-6 x^{2}\right ) \left (-3+\sqrt {33}+6 x^{2}\right )}\, \left (\frac {\frac {7}{528} x -\frac {1}{176} x^{3}}{\sqrt {-9 x^{4}+9 x^{2}+6}}+\frac {\sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{66 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}}-\frac {6 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}\, \left (36+12 \sqrt {33}\right )}\right )}{\left (3+\sqrt {33}-6 x^{2}\right )^{\frac {3}{2}} \left (-3+\sqrt {33}+6 x^{2}\right )^{\frac {3}{2}}}\)	\(282\)

Input:

int(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x,method=_RETURNV 
ERBOSE)
 

Output:

-1/264*x*(3*x^2-7)/(3+33^(1/2)-6*x^2)^(1/2)/(-3+33^(1/2)+6*x^2)^(1/2)+(1/6 
6/(-3+33^(1/2))^(1/2)*(1-(-3/4+1/4*33^(1/2))*x^2)^(1/2)*(1-(-3/4-1/4*33^(1 
/2))*x^2)^(1/2)/(-9*x^4+9*x^2+6)^(1/2)*EllipticF(1/2*(-3+33^(1/2))^(1/2)*x 
,1/4*I*6^(1/2)+1/4*I*22^(1/2))-6/11/(-3+33^(1/2))^(1/2)*(1-(-3/4+1/4*33^(1 
/2))*x^2)^(1/2)*(1-(-3/4-1/4*33^(1/2))*x^2)^(1/2)/(-9*x^4+9*x^2+6)^(1/2)/( 
36+12*33^(1/2))*(EllipticF(1/2*(-3+33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*2 
2^(1/2))-EllipticE(1/2*(-3+33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2)) 
))*((3+33^(1/2)-6*x^2)*(-3+33^(1/2)+6*x^2))^(1/2)/(3+33^(1/2)-6*x^2)^(1/2) 
/(-3+33^(1/2)+6*x^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {33} \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} - 3 \, \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + {\left (\sqrt {33} \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} + 21 \, \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + 12 \, {\left (3 \, x^{3} - 7 \, x\right )} \sqrt {6 \, x^{2} + \sqrt {33} - 3} \sqrt {-6 \, x^{2} + \sqrt {33} + 3}}{38016 \, {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}} \] Input:

integrate(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x, algorith 
m="fricas")
 

Output:

1/38016*(3*(sqrt(33)*sqrt(6)*(3*x^4 - 3*x^2 - 2) - 3*sqrt(6)*(3*x^4 - 3*x^ 
2 - 2))*sqrt(sqrt(33) - 3)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(33) - 3)), -1 
/4*sqrt(33) - 7/4) + (sqrt(33)*sqrt(6)*(3*x^4 - 3*x^2 - 2) + 21*sqrt(6)*(3 
*x^4 - 3*x^2 - 2))*sqrt(sqrt(33) - 3)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(33 
) - 3)), -1/4*sqrt(33) - 7/4) + 12*(3*x^3 - 7*x)*sqrt(6*x^2 + sqrt(33) - 3 
)*sqrt(-6*x^2 + sqrt(33) + 3))/(3*x^4 - 3*x^2 - 2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(3+33**(1/2)-6*x**2)**(3/2)/(-3+33**(1/2)+6*x**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (6 \, x^{2} + \sqrt {33} - 3\right )}^{\frac {3}{2}} {\left (-6 \, x^{2} + \sqrt {33} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x, algorith 
m="maxima")
 

Output:

integrate(1/((6*x^2 + sqrt(33) - 3)^(3/2)*(-6*x^2 + sqrt(33) + 3)^(3/2)), 
x)
 

Giac [F]

\[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (6 \, x^{2} + \sqrt {33} - 3\right )}^{\frac {3}{2}} {\left (-6 \, x^{2} + \sqrt {33} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x, algorith 
m="giac")
 

Output:

integrate(1/((6*x^2 + sqrt(33) - 3)^(3/2)*(-6*x^2 + sqrt(33) + 3)^(3/2)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (-6\,x^2+\sqrt {33}+3\right )}^{3/2}\,{\left (6\,x^2+\sqrt {33}-3\right )}^{3/2}} \,d x \] Input:

int(1/((33^(1/2) - 6*x^2 + 3)^(3/2)*(33^(1/2) + 6*x^2 - 3)^(3/2)),x)
 

Output:

int(1/((33^(1/2) - 6*x^2 + 3)^(3/2)*(33^(1/2) + 6*x^2 - 3)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{\left (3+\sqrt {33}-6 x^2\right )^{3/2} \left (-3+\sqrt {33}+6 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (3+\sqrt {33}-6 x^{2}\right )^{\frac {3}{2}} \left (-3+\sqrt {33}+6 x^{2}\right )^{\frac {3}{2}}}d x \] Input:

int(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x)
 

Output:

int(1/(3+33^(1/2)-6*x^2)^(3/2)/(-3+33^(1/2)+6*x^2)^(3/2),x)