\(\int \frac {1}{(2-3 x^2+3 x^4)^{3/2}} \, dx\) [241]

Optimal result

Integrand size = 16, antiderivative size = 257 \[ \int \frac {1}{\left (2-3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {x \left (1+3 x^2\right )}{10 \sqrt {2-3 x^2+3 x^4}}-\frac {3 x \sqrt {2-3 x^2+3 x^4}}{10 \left (\sqrt {6}+3 x^2\right )}+\frac {\sqrt [4]{3} \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-3 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{8} \left (4+\sqrt {6}\right )\right )}{5\ 2^{3/4} \sqrt {2-3 x^2+3 x^4}}-\frac {\left (3-2 \sqrt {6}\right ) \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-3 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{8} \left (4+\sqrt {6}\right )\right )}{10\ 6^{3/4} \sqrt {2-3 x^2+3 x^4}} \] Output:

1/10*x*(3*x^2+1)/(3*x^4-3*x^2+2)^(1/2)-3*x*(3*x^4-3*x^2+2)^(1/2)/(10*6^(1/ 
2)+30*x^2)+1/10*3^(1/4)*(2+6^(1/2)*x^2)*((3*x^4-3*x^2+2)/(2+6^(1/2)*x^2)^2 
)^(1/2)*EllipticE(sin(2*arctan(1/2*3^(1/4)*2^(3/4)*x)),1/4*(8+2*6^(1/2))^( 
1/2))*2^(1/4)/(3*x^4-3*x^2+2)^(1/2)-1/60*(3-2*6^(1/2))*(2+6^(1/2)*x^2)*((3 
*x^4-3*x^2+2)/(2+6^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(1/2*3^(1/4 
)*2^(3/4)*x),1/4*(8+2*6^(1/2))^(1/2))*6^(1/4)/(3*x^4-3*x^2+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.15 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\left (2-3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {-\frac {i}{3 i+\sqrt {15}}} x \left (1+3 x^2\right )+3 \sqrt {2} \left (-i \sqrt {3}+\sqrt {5}\right ) \sqrt {\frac {3 i+\sqrt {15}-6 i x^2}{3 i+\sqrt {15}}} \sqrt {\frac {-3 i+\sqrt {15}+6 i x^2}{-3 i+\sqrt {15}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{3 i+\sqrt {15}}} x\right )|\frac {3 i+\sqrt {15}}{3 i-\sqrt {15}}\right )-i \sqrt {2} \left (5 \sqrt {3}-3 i \sqrt {5}\right ) \sqrt {\frac {3 i+\sqrt {15}-6 i x^2}{3 i+\sqrt {15}}} \sqrt {\frac {-3 i+\sqrt {15}+6 i x^2}{-3 i+\sqrt {15}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{3 i+\sqrt {15}}} x\right ),\frac {3 i+\sqrt {15}}{3 i-\sqrt {15}}\right )}{120 \sqrt {-\frac {i}{3 i+\sqrt {15}}} \sqrt {2-3 x^2+3 x^4}} \] Input:

Integrate[(2 - 3*x^2 + 3*x^4)^(-3/2),x]
 

Output:

(12*Sqrt[(-I)/(3*I + Sqrt[15])]*x*(1 + 3*x^2) + 3*Sqrt[2]*((-I)*Sqrt[3] + 
Sqrt[5])*Sqrt[(3*I + Sqrt[15] - (6*I)*x^2)/(3*I + Sqrt[15])]*Sqrt[(-3*I + 
Sqrt[15] + (6*I)*x^2)/(-3*I + Sqrt[15])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/( 
3*I + Sqrt[15])]*x], (3*I + Sqrt[15])/(3*I - Sqrt[15])] - I*Sqrt[2]*(5*Sqr 
t[3] - (3*I)*Sqrt[5])*Sqrt[(3*I + Sqrt[15] - (6*I)*x^2)/(3*I + Sqrt[15])]* 
Sqrt[(-3*I + Sqrt[15] + (6*I)*x^2)/(-3*I + Sqrt[15])]*EllipticF[I*ArcSinh[ 
Sqrt[(-6*I)/(3*I + Sqrt[15])]*x], (3*I + Sqrt[15])/(3*I - Sqrt[15])])/(120 
*Sqrt[(-I)/(3*I + Sqrt[15])]*Sqrt[2 - 3*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{30} \int \frac {3 \left (4-3 x^2\right )}{\sqrt {3 x^4-3 x^2+2}}dx+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \int \frac {4-3 x^2}{\sqrt {3 x^4-3 x^2+2}}dx+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {1}{10} \left (\left (4-\sqrt {6}\right ) \int \frac {1}{\sqrt {3 x^4-3 x^2+2}}dx+\sqrt {6} \int \frac {2-\sqrt {6} x^2}{2 \sqrt {3 x^4-3 x^2+2}}dx\right )+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\left (4-\sqrt {6}\right ) \int \frac {1}{\sqrt {3 x^4-3 x^2+2}}dx+\sqrt {\frac {3}{2}} \int \frac {2-\sqrt {6} x^2}{\sqrt {3 x^4-3 x^2+2}}dx\right )+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{10} \left (\sqrt {\frac {3}{2}} \int \frac {2-\sqrt {6} x^2}{\sqrt {3 x^4-3 x^2+2}}dx+\frac {\left (4-\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-3 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{8} \left (4+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-3 x^2+2}}\right )+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{10} \left (\frac {\left (4-\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-3 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{8} \left (4+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-3 x^2+2}}+\sqrt {\frac {3}{2}} \left (\frac {2^{3/4} \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-3 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{8} \left (4+\sqrt {6}\right )\right )}{\sqrt [4]{3} \sqrt {3 x^4-3 x^2+2}}-\frac {2 x \sqrt {3 x^4-3 x^2+2}}{\sqrt {6} x^2+2}\right )\right )+\frac {x \left (3 x^2+1\right )}{10 \sqrt {3 x^4-3 x^2+2}}\)

Input:

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

Output:

(x*(1 + 3*x^2))/(10*Sqrt[2 - 3*x^2 + 3*x^4]) + (Sqrt[3/2]*((-2*x*Sqrt[2 - 
3*x^2 + 3*x^4])/(2 + Sqrt[6]*x^2) + (2^(3/4)*(2 + Sqrt[6]*x^2)*Sqrt[(2 - 3 
*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(3/2)^(1/4)*x], (4 + 
 Sqrt[6])/8])/(3^(1/4)*Sqrt[2 - 3*x^2 + 3*x^4])) + ((4 - Sqrt[6])*(2 + Sqr 
t[6]*x^2)*Sqrt[(2 - 3*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan 
[(3/2)^(1/4)*x], (4 + Sqrt[6])/8])/(2*6^(1/4)*Sqrt[2 - 3*x^2 + 3*x^4]))/10
 

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 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 1416

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

rule 1509

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

rule 1511

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.67 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {x \left (3 x^{2}+1\right )}{10 \sqrt {3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}\, \left (-3+i \sqrt {15}\right )}\)	\(237\)
default	\(-\frac {6 \left (-\frac {1}{60} x -\frac {1}{20} x^{3}\right )}{\sqrt {3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}\, \left (-3+i \sqrt {15}\right )}\)	\(238\)
elliptic	\(-\frac {6 \left (-\frac {1}{60} x -\frac {1}{20} x^{3}\right )}{\sqrt {3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {i \sqrt {15}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {3+i \sqrt {15}}}{2}, \frac {\sqrt {-1-i \sqrt {15}}}{2}\right )\right )}{5 \sqrt {3+i \sqrt {15}}\, \sqrt {3 x^{4}-3 x^{2}+2}\, \left (-3+i \sqrt {15}\right )}\)	\(238\)

Input:

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

Output:

1/10*x*(3*x^2+1)/(3*x^4-3*x^2+2)^(1/2)+4/5/(3+I*15^(1/2))^(1/2)*(1-(3/4+1/ 
4*I*15^(1/2))*x^2)^(1/2)*(1-(3/4-1/4*I*15^(1/2))*x^2)^(1/2)/(3*x^4-3*x^2+2 
)^(1/2)*EllipticF(1/2*x*(3+I*15^(1/2))^(1/2),1/2*(-1-I*15^(1/2))^(1/2))+12 
/5/(3+I*15^(1/2))^(1/2)*(1-(3/4+1/4*I*15^(1/2))*x^2)^(1/2)*(1-(3/4-1/4*I*1 
5^(1/2))*x^2)^(1/2)/(3*x^4-3*x^2+2)^(1/2)/(-3+I*15^(1/2))*(EllipticF(1/2*x 
*(3+I*15^(1/2))^(1/2),1/2*(-1-I*15^(1/2))^(1/2))-EllipticE(1/2*x*(3+I*15^( 
1/2))^(1/2),1/2*(-1-I*15^(1/2))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (2-3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3 \, \sqrt {2} {\left (9 \, x^{4} - 9 \, x^{2} + \sqrt {-15} {\left (3 \, x^{4} - 3 \, x^{2} + 2\right )} + 6\right )} \sqrt {\sqrt {-15} + 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-15} + 3}\right )\,|\,-\frac {1}{4} \, \sqrt {-15} - \frac {1}{4}) + \sqrt {2} {\left (9 \, x^{4} - 9 \, x^{2} - 7 \, \sqrt {-15} {\left (3 \, x^{4} - 3 \, x^{2} + 2\right )} + 6\right )} \sqrt {\sqrt {-15} + 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-15} + 3}\right )\,|\,-\frac {1}{4} \, \sqrt {-15} - \frac {1}{4}) + 24 \, \sqrt {3 \, x^{4} - 3 \, x^{2} + 2} {\left (3 \, x^{3} + x\right )}}{240 \, {\left (3 \, x^{4} - 3 \, x^{2} + 2\right )}} \] Input:

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

Output:

1/240*(3*sqrt(2)*(9*x^4 - 9*x^2 + sqrt(-15)*(3*x^4 - 3*x^2 + 2) + 6)*sqrt( 
sqrt(-15) + 3)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(-15) + 3)), -1/4*sqrt(-15 
) - 1/4) + sqrt(2)*(9*x^4 - 9*x^2 - 7*sqrt(-15)*(3*x^4 - 3*x^2 + 2) + 6)*s 
qrt(sqrt(-15) + 3)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(-15) + 3)), -1/4*sqrt 
(-15) - 1/4) + 24*sqrt(3*x^4 - 3*x^2 + 2)*(3*x^3 + x))/(3*x^4 - 3*x^2 + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (2-3 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}-3 x^{2}+2}}{9 x^{8}-18 x^{6}+21 x^{4}-12 x^{2}+4}d x \] Input:

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

Output:

int(sqrt(3*x**4 - 3*x**2 + 2)/(9*x**8 - 18*x**6 + 21*x**4 - 12*x**2 + 4),x 
)