Integrand size = 24, antiderivative size = 134 \[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {1}{2} \arctan \left (\sqrt [4]{-1+3 x^2}\right )+\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )}{2 \sqrt {2}} \] Output:
-1/2*arctan((3*x^2-1)^(1/4))-1/4*arctan(-1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1/2 )-1/4*arctan(1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1/2)-1/2*arctanh((3*x^2-1)^(1/4 ))-1/4*arctanh(2^(1/2)*(3*x^2-1)^(1/4)/(1+(3*x^2-1)^(1/2)))*2^(1/2)
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {1}{4} \left (-2 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+3 x^2}}{\sqrt {2} \sqrt [4]{-1+3 x^2}}\right )-2 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )\right ) \] Input:
Integrate[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
Output:
(-2*ArcTan[(-1 + 3*x^2)^(1/4)] - Sqrt[2]*ArcTan[(-1 + Sqrt[-1 + 3*x^2])/(S qrt[2]*(-1 + 3*x^2)^(1/4))] - 2*ArcTanh[(-1 + 3*x^2)^(1/4)] - Sqrt[2]*ArcT anh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 + Sqrt[-1 + 3*x^2])])/4
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.30, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {354, 25, 97, 73, 755, 27, 756, 216, 219, 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.
\(\displaystyle \int \frac {1}{x \left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int -\frac {1}{x^2 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{x^2 \left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 97 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{x^2 \left (3 x^2-1\right )^{3/4}}dx^2-\frac {3}{2} \int \frac {1}{\left (2-3 x^2\right ) \left (3 x^2-1\right )^{3/4}}dx^2\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}-\frac {2}{3} \int \frac {1}{\frac {x^8}{3}+\frac {1}{3}}d\sqrt [4]{3 x^2-1}\right )\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}-\frac {2}{3} \left (\frac {1}{2} \int \frac {3 \left (1-x^4\right )}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {3 \left (x^4+1\right )}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{1-x^8}d\sqrt [4]{3 x^2-1}-\frac {2}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{2} \left (-2 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{3 x^2-1}\right )-\frac {2}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (-2 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )\right )-\frac {2}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \int \frac {x^4+1}{x^8+1}d\sqrt [4]{3 x^2-1}\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {1}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}\right )+\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x^4-1}d\left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}\right )+\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \int \frac {1-x^4}{x^8+1}d\sqrt [4]{3 x^2-1}+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (\frac {3}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt [4]{3 x^2-1}}{x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt [4]{3 x^2-1}+1}{x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1}d\sqrt [4]{3 x^2-1}\right )+\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )\right )-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (-2 \left (\frac {1}{2} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )\right )-\frac {2}{3} \left (\frac {3}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{\sqrt {2}}\right )+\frac {3}{2} \left (\frac {\log \left (x^4+\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x^4-\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
Input:
Int[1/(x*(-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]
Output:
(-2*(ArcTan[(-1 + 3*x^2)^(1/4)]/2 + ArcTanh[(-1 + 3*x^2)^(1/4)]/2) - (2*(( 3*(-(ArcTan[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4)]/Sqrt[2]) + ArcTan[1 + Sqrt[2]* (-1 + 3*x^2)^(1/4)]/Sqrt[2]))/2 + (3*(-1/2*Log[1 + x^4 - Sqrt[2]*(-1 + 3*x ^2)^(1/4)]/Sqrt[2] + Log[1 + x^4 + Sqrt[2]*(-1 + 3*x^2)^(1/4)]/(2*Sqrt[2]) ))/2))/3)/2
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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && IntegerQ [(m - 1)/2]
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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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 = 4.90 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {-\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}-\sqrt {3 x^{2}-1}-1}{\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}-\sqrt {3 x^{2}-1}-1}\right ) \sqrt {2}}{8}-\frac {\arctan \left (1+\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) \sqrt {2}}{4}-\frac {\arctan \left (-1+\sqrt {2}\, \left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) \sqrt {2}}{4}+\frac {\ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{4}-\frac {\arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{2}-\frac {\ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{4}\) | \(146\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{2}}\right )}{4}+\frac {\ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{4}\) | \(301\) |
Input:
int(1/x/(3*x^2-2)/(3*x^2-1)^(3/4),x,method=_RETURNVERBOSE)
Output:
-1/8*ln((-2^(1/2)*(3*x^2-1)^(1/4)-(3*x^2-1)^(1/2)-1)/(2^(1/2)*(3*x^2-1)^(1 /4)-(3*x^2-1)^(1/2)-1))*2^(1/2)-1/4*arctan(1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1 /2)-1/4*arctan(-1+2^(1/2)*(3*x^2-1)^(1/4))*2^(1/2)+1/4*ln(-1+(3*x^2-1)^(1/ 4))-1/2*arctan((3*x^2-1)^(1/4))-1/4*ln(1+(3*x^2-1)^(1/4))
Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {1}{2} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \] Input:
integrate(1/x/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="fricas")
Output:
-1/4*sqrt(2)*arctan(sqrt(2)*(3*x^2 - 1)^(1/4) + 1) - 1/4*sqrt(2)*arctan(sq rt(2)*(3*x^2 - 1)^(1/4) - 1) - 1/8*sqrt(2)*log(sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt( 3*x^2 - 1) + 1) - 1/2*arctan((3*x^2 - 1)^(1/4)) - 1/4*log((3*x^2 - 1)^(1/4 ) + 1) + 1/4*log((3*x^2 - 1)^(1/4) - 1)
\[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {1}{x \left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(1/x/(3*x**2-2)/(3*x**2-1)**(3/4),x)
Output:
Integral(1/(x*(3*x**2 - 2)*(3*x**2 - 1)**(3/4)), x)
\[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )} x} \,d x } \] Input:
integrate(1/x/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((3*x^2 - 1)^(3/4)*(3*x^2 - 2)*x), x)
Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {1}{2} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:
integrate(1/x/(3*x^2-2)/(3*x^2-1)^(3/4),x, algorithm="giac")
Output:
-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^2 - 1)^(1/4))) - 1/4*sqr t(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(3*x^2 - 1)^(1/4))) - 1/8*sqrt(2)*lo g(sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) + 1/8*sqrt(2)*log(-sqrt (2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 1/2*arctan((3*x^2 - 1)^(1/4 )) - 1/4*log((3*x^2 - 1)^(1/4) + 1) + 1/4*log(abs((3*x^2 - 1)^(1/4) - 1))
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{2}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (3\,x^2-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (3\,x^2-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \] Input:
int(1/(x*(3*x^2 - 1)^(3/4)*(3*x^2 - 2)),x)
Output:
(atan((3*x^2 - 1)^(1/4)*1i)*1i)/2 - atan((3*x^2 - 1)^(1/4))/2 - 2^(1/2)*at an(2^(1/2)*(3*x^2 - 1)^(1/4)*(1/2 - 1i/2))*(1/4 + 1i/4) - 2^(1/2)*atan(2^( 1/2)*(3*x^2 - 1)^(1/4)*(1/2 + 1i/2))*(1/4 - 1i/4)
\[ \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {1}{3 \left (3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}} x}d x \] Input:
int(1/x/(3*x^2-2)/(3*x^2-1)^(3/4),x)
Output:
int(1/(3*(3*x**2 - 1)**(3/4)*x**3 - 2*(3*x**2 - 1)**(3/4)*x),x)