\(\int \frac {1}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1488]

Optimal result

Integrand size = 21, antiderivative size = 148 \[ \int \frac {1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{2-3 x^2}}{2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}\right )}{4 \sqrt [4]{2} \sqrt {3}}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{2 \sqrt [4]{2} \sqrt {3}} \] Output:

1/24*2^(3/4)*arctan(1/3*(2^(3/4)-2^(1/4)*(-3*x^2+2)^(1/2))*3^(1/2)/x/(-3*x 
^2+2)^(1/4))*3^(1/2)-1/24*2^(3/4)*arctanh(3^(1/2)*x*(-3*x^2+2)^(1/4)/(2^(3 
/4)+2^(1/4)*(-3*x^2+2)^(1/2)))*3^(1/2)+1/12*2^(3/4)*InverseJacobiAM(1/2*ar 
csin(1/2*x*6^(1/2)),2^(1/2))*3^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.83 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {\sqrt {x^2} \left (\operatorname {EllipticPi}\left (-i,\arcsin \left (\sqrt [4]{1-\frac {3 x^2}{2}}\right ),-1\right )+\operatorname {EllipticPi}\left (i,\arcsin \left (\sqrt [4]{1-\frac {3 x^2}{2}}\right ),-1\right )\right )}{2 \sqrt [4]{2} \sqrt {3} x} \] Input:

Integrate[1/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
 

Output:

-1/2*(Sqrt[x^2]*(EllipticPi[-I, ArcSin[(1 - (3*x^2)/2)^(1/4)], -1] + Ellip 
ticPi[I, ArcSin[(1 - (3*x^2)/2)^(1/4)], -1]))/(2^(1/4)*Sqrt[3]*x)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {311, 230, 350}

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.

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

\(\Big \downarrow \) 311

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

\(\Big \downarrow \) 230

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

\(\Big \downarrow \) 350

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

Input:

Int[1/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 230

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] 
))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ 
[a, 0] && NegQ[b/a]
 

rule 311

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[1/c   Int[1/(a + b*x^2)^(3/4), x], x] - Simp[d/c   Int[x^2/((a + b*x^2)^( 
3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0]
 

rule 350

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] : 
> Simp[(-b/(a*d*Rt[b^2/a, 4]^3))*ArcTan[(b + Rt[b^2/a, 4]^2*Sqrt[a + b*x^2] 
)/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))], x] + Simp[(b/(a*d*Rt[b^2/a, 4]^3)) 
*ArcTanh[(b - Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2) 
^(1/4))], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a 
]
 

Maple [F]

\[\int \frac {1}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((-3*x^2 + 2)^(1/4)/(9*x^4 - 18*x^2 + 8), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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