\(\int \frac {1}{(24+36 x^2-36 x^4)^{3/2}} \, dx\) [200]

Optimal result

Integrand size = 16, antiderivative size = 136 \[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {2+3 x^2-3 x^4}}+\frac {1}{528} \sqrt {\frac {1}{6} \left (-3+\sqrt {33}\right )} E\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7-\sqrt {33}\right )\right )+\frac {1}{144} \sqrt {\frac {1}{22} \left (-3+\sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right ) \] Output:

1/1584*x*(-3*x^2+7)*3^(1/2)/(-3*x^4+3*x^2+2)^(1/2)+1/3168*(-18+6*33^(1/2)) 
^(1/2)*EllipticE(6^(1/2)/(3+33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2) 
)+1/3168*(-66+22*33^(1/2))^(1/2)*EllipticF(6^(1/2)/(3+33^(1/2))^(1/2)*x,1/ 
4*I*6^(1/2)+1/4*I*22^(1/2))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.83 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\frac {2 \sqrt {3 \left (3+\sqrt {33}\right )} x \left (7-3 x^2\right )+3 i \left (\sqrt {3}+\sqrt {11}\right ) \sqrt {4+6 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )-i \left (11 \sqrt {3}+3 \sqrt {11}\right ) \sqrt {4+6 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{3168 \sqrt {3+\sqrt {33}} \sqrt {2+3 x^2-3 x^4}} \] Input:

Integrate[(24 + 36*x^2 - 36*x^4)^(-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 = 0.57 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1494, 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.

\(\displaystyle \int \frac {1}{\left (-36 x^4+36 x^2+24\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}-\frac {\int -\frac {72 \sqrt {3} \left (3 x^2+4\right )}{\sqrt {-3 x^4+3 x^2+2}}dx}{114048}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 x^2+4}{\sqrt {-3 x^4+3 x^2+2}}dx}{528 \sqrt {3}}+\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {1}{264} \int \frac {3 x^2+4}{\sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}dx+\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{264} \left (\frac {1}{2} \left (11-\sqrt {33}\right ) \int \frac {1}{\sqrt {-6 x^2+\sqrt {33}+3} \sqrt {6 x^2+\sqrt {33}-3}}dx+\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}-3}}{\sqrt {-6 x^2+\sqrt {33}+3}}dx\right )+\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{264} \left (\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}-3}}{\sqrt {-6 x^2+\sqrt {33}+3}}dx+\frac {\left (11-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{2 \sqrt {6 \left (\sqrt {33}-3\right )}}\right )+\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{264} \left (\frac {\left (11-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{2 \sqrt {6 \left (\sqrt {33}-3\right )}}+\frac {1}{2} \sqrt {\frac {1}{6} \left (\sqrt {33}-3\right )} E\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7-\sqrt {33}\right )\right )\right )+\frac {x \left (7-3 x^2\right )}{528 \sqrt {3} \sqrt {-3 x^4+3 x^2+2}}\)

Input:

Int[(24 + 36*x^2 - 36*x^4)^(-3/2),x]
 

Output:

(x*(7 - 3*x^2))/(528*Sqrt[3]*Sqrt[2 + 3*x^2 - 3*x^4]) + ((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])]))/264
 

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 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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 1494

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c]   Int[(d + e*x^2)/(Sqr 
t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e 
}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (102 ) = 204\).

Time = 2.92 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {x \left (3 x^{2}-7\right )}{528 \sqrt {-9 x^{4}+9 x^{2}+6}}+\frac {\sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{66 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}}-\frac {3 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{22 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}\, \left (9+3 \sqrt {33}\right )}\)	\(220\)
default	\(\frac {\frac {7}{528} x -\frac {1}{176} x^{3}}{\sqrt {-9 x^{4}+9 x^{2}+6}}+\frac {\sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{66 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}}-\frac {3 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{22 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}\, \left (9+3 \sqrt {33}\right )}\)	\(221\)
elliptic	\(\frac {\frac {7}{528} x -\frac {1}{176} x^{3}}{\sqrt {-9 x^{4}+9 x^{2}+6}}+\frac {\sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{66 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}}-\frac {6 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {-3+\sqrt {33}}\, \sqrt {-9 x^{4}+9 x^{2}+6}\, \left (36+12 \sqrt {33}\right )}\)	\(221\)

Input:

int(1/(-36*x^4+36*x^2+24)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/528*x*(3*x^2-7)/(-9*x^4+9*x^2+6)^(1/2)+1/66/(-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) 
)-3/22/(-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)/(9+3*33^(1/2))*(EllipticF(1/2* 
(-3+33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2))-EllipticE(1/2*(-3+33^( 
1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {33} \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} - 3 \, \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + {\left (\sqrt {33} \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} + 21 \, \sqrt {6} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + 12 \, \sqrt {-36 \, x^{4} + 36 \, x^{2} + 24} {\left (3 \, x^{3} - 7 \, x\right )}}{38016 \, {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}} \] Input:

integrate(1/(-36*x^4+36*x^2+24)^(3/2),x, algorithm="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*sqrt(-36*x^4 + 36*x^2 + 24)*(3*x^3 - 7* 
x))/(3*x^4 - 3*x^2 - 2)
 

Sympy [F]

\[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\frac {\sqrt {3} \int \frac {1}{- 3 x^{4} \sqrt {- 3 x^{4} + 3 x^{2} + 2} + 3 x^{2} \sqrt {- 3 x^{4} + 3 x^{2} + 2} + 2 \sqrt {- 3 x^{4} + 3 x^{2} + 2}}\, dx}{72} \] Input:

integrate(1/(-36*x**4+36*x**2+24)**(3/2),x)
 

Output:

sqrt(3)*Integral(1/(-3*x**4*sqrt(-3*x**4 + 3*x**2 + 2) + 3*x**2*sqrt(-3*x* 
*4 + 3*x**2 + 2) + 2*sqrt(-3*x**4 + 3*x**2 + 2)), x)/72
 

Maxima [F]

\[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-36 \, x^{4} + 36 \, x^{2} + 24\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-36*x^4+36*x^2+24)^(3/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((-36*x^4 + 36*x^2 + 24)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-36 \, x^{4} + 36 \, x^{2} + 24\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-36*x^4+36*x^2+24)^(3/2),x, algorithm="giac")
 

Output:

integrate((-36*x^4 + 36*x^2 + 24)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-36\,x^4+36\,x^2+24\right )}^{3/2}} \,d x \] Input:

int(1/(36*x^2 - 36*x^4 + 24)^(3/2),x)
 

Output:

int(1/(36*x^2 - 36*x^4 + 24)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (24+36 x^2-36 x^4\right )^{3/2}} \, dx=\frac {\sqrt {3}\, \left (\int \frac {\sqrt {-3 x^{4}+3 x^{2}+2}}{9 x^{8}-18 x^{6}-3 x^{4}+12 x^{2}+4}d x \right )}{72} \] Input:

int(1/(-36*x^4+36*x^2+24)^(3/2),x)
 

Output:

(sqrt(3)*int(sqrt( - 3*x**4 + 3*x**2 + 2)/(9*x**8 - 18*x**6 - 3*x**4 + 12* 
x**2 + 4),x))/72