\(\int \frac {x}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1482]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )}{6 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{2-3 x^2}}{\sqrt {2}+\sqrt {2-3 x^2}}\right )}{6 \sqrt [4]{2}} \] Output:

1/12*2^(3/4)*arctan(1/2*(2^(1/2)-(-3*x^2+2)^(1/2))*2^(1/4)/(-3*x^2+2)^(1/4 
))-1/12*2^(3/4)*arctanh(2^(3/4)*(-3*x^2+2)^(1/4)/(2^(1/2)+(-3*x^2+2)^(1/2) 
))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.32 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.63, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {353, 73, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {2}{3} \int \frac {1}{x^8+2}d\sqrt [4]{2-3 x^2}\)

\(\Big \downarrow \) 755

\(\displaystyle -\frac {2}{3} \left (\frac {\int \frac {\sqrt {2}-x^4}{x^8+2}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}+\frac {\int \frac {x^4+\sqrt {2}}{x^8+2}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {2}{3} \left (\frac {\frac {1}{2} \int \frac {1}{x^4-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}}d\sqrt [4]{2-3 x^2}+\frac {1}{2} \int \frac {1}{x^4+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-x^4}{x^8+2}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {2}{3} \left (\frac {\frac {\int \frac {1}{-x^4-1}d\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{2^{3/4}}-\frac {\int \frac {1}{-x^4-1}d\left (\sqrt [4]{2} \sqrt [4]{2-3 x^2}+1\right )}{2^{3/4}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-x^4}{x^8+2}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {2}{3} \left (\frac {\int \frac {\sqrt {2}-x^4}{x^8+2}d\sqrt [4]{2-3 x^2}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{2-3 x^2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{2^{3/4}}}{2 \sqrt {2}}\right )\)

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 353

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

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

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

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {\sqrt {-3 x^{2}+2}+2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}{\sqrt {-3 x^{2}+2}-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right )+2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-1\right )\right )}{24}\)	\(100\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+3 x^{2}}{3 x^{2}-4}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}-2 \left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )-3 x^{2}}{3 x^{2}-4}\right )}{12}\)	\(190\)

Input:

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

Output:

-1/24*2^(3/4)*(ln(((-3*x^2+2)^(1/2)+2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2))/((-3 
*x^2+2)^(1/2)-2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)))+2*arctan(2^(1/4)*(-3*x^2+ 
2)^(1/4)+1)+2*arctan(2^(1/4)*(-3*x^2+2)^(1/4)-1))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/12*2^(3/4)*arctan(2^(1/4)*(-3*x^2 + 2)^(1/4) + 1) - 1/12*2^(3/4)*arctan 
(2^(1/4)*(-3*x^2 + 2)^(1/4) - 1) - 1/24*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^( 
1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 1/24*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2 
)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.27 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/12*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 1/12* 
2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 1/24*2^(3/ 
4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 1/24*2^( 
3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.27 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/12*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 1/12* 
2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 1/24*2^(3/ 
4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 1/24*2^( 
3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2))
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \] Input:

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

Output:

- 2^(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 - 1i/2))*(1/12 + 1i/12) - 2^ 
(3/4)*atan(2^(1/4)*(2 - 3*x^2)^(1/4)*(1/2 + 1i/2))*(1/12 - 1i/12)
 

Reduce [F]

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

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

Output:

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