Integrand size = 22, antiderivative size = 143 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {\arctan \left (1+\sqrt [4]{4-6 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{12 \sqrt [4]{2}} \]
-1/12*2^(3/4)*arctan(1+(-6*x^2+4)^(1/4))-1/12*2^(3/4)*arctan(-1+2^(1/4)*(- 3*x^2+2)^(1/4))+1/24*2^(3/4)*ln(-2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x^2+ 2)^(1/2))-1/24*2^(3/4)*ln(2^(3/4)*(-3*x^2+2)^(1/4)+2^(1/2)+(-3*x^2+2)^(1/2 ))
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2}} \]
(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))
Time = 0.32 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06, 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 |
| \(\displaystyle -\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{2} \sqrt [4]{2-3 x^2}+1\right )}{2^{3/4}}-\frac {\arctan \left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{2^{3/4}}}{2 \sqrt {2}}+\frac {\frac {\log \left (x^4+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{2\ 2^{3/4}}-\frac {\log \left (x^4-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{2\ 2^{3/4}}}{2 \sqrt {2}}\right )\) |
(-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
3.11.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]]))
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]
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]
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]
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]
Time = 3.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.70
| method | result | size |
| pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (\ln \left (\frac {2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right )+2 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right )+2 \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right )\right )}{24}\) | \(100\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}-3 x^{2}}{3 x^{2}-4}\right )}{12}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (-3 x^{2}+2\right )^{\frac {1}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \sqrt {-3 x^{2}+2}+3 x^{2}}{3 x^{2}-4}\right )}{12}\) | \(190\) |
-1/24*2^(3/4)*(ln((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)+(-3*x^2+2)^(1/2)))+2*arctan(2^(1/4)*(-3*x^2 +2)^(1/4)+1)+2*arctan(-1+2^(1/4)*(-3*x^2+2)^(1/4)))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} + 8 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} + 8 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{96} i - \frac {1}{96}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} + 8 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{96} i + \frac {1}{96}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} + 8 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) \]
-(1/96*I + 1/96)*8^(3/4)*sqrt(2)*log((I + 1)*8^(3/4)*sqrt(2) + 8*(-3*x^2 + 2)^(1/4)) + (1/96*I - 1/96)*8^(3/4)*sqrt(2)*log(-(I - 1)*8^(3/4)*sqrt(2) + 8*(-3*x^2 + 2)^(1/4)) - (1/96*I - 1/96)*8^(3/4)*sqrt(2)*log((I - 1)*8^(3 /4)*sqrt(2) + 8*(-3*x^2 + 2)^(1/4)) + (1/96*I + 1/96)*8^(3/4)*sqrt(2)*log( -(I + 1)*8^(3/4)*sqrt(2) + 8*(-3*x^2 + 2)^(1/4))
\[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \]
-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))
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{24} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) \]
-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))
Time = 5.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.34 \[ \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \]