3.11.47 \(\int \frac {1}{x^3 (-2+3 x^2) \sqrt [4]{-1+3 x^2}} \, dx\) [1047]

Optimal. Leaf size=191 \[ -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}} \]

[Out]

-1/4*(3*x^2-1)^(3/4)/x^2+3/4*arctan((3*x^2-1)^(1/4))-3/4*arctanh((3*x^2-1)^(1/4))-9/16*arctan(-1+(3*x^2-1)^(1/
4)*2^(1/2))*2^(1/2)-9/16*arctan(1+(3*x^2-1)^(1/4)*2^(1/2))*2^(1/2)-9/32*ln(1-(3*x^2-1)^(1/4)*2^(1/2)+(3*x^2-1)
^(1/2))*2^(1/2)+9/32*ln(1+(3*x^2-1)^(1/4)*2^(1/2)+(3*x^2-1)^(1/2))*2^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {457, 105, 162, 65, 303, 1176, 631, 210, 1179, 642, 304, 209, 212} \begin {gather*} \frac {3}{4} \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )+\frac {9 \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt {2}}-\frac {9 \text {ArcTan}\left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt {2}}-\frac {\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac {9 \log \left (\sqrt {3 x^2-1}-\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt {2}}+\frac {9 \log \left (\sqrt {3 x^2-1}+\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt {2}}-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(-1 + 3*x^2)^(3/4)/x^2 + (3*ArcTan[(-1 + 3*x^2)^(1/4)])/4 + (9*ArcTan[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4)])/(8
*Sqrt[2]) - (9*ArcTan[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4)])/(8*Sqrt[2]) - (3*ArcTanh[(-1 + 3*x^2)^(1/4)])/4 - (9*Lo
g[1 - Sqrt[2]*(-1 + 3*x^2)^(1/4) + Sqrt[-1 + 3*x^2]])/(16*Sqrt[2]) + (9*Log[1 + Sqrt[2]*(-1 + 3*x^2)^(1/4) + S
qrt[-1 + 3*x^2]])/(16*Sqrt[2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 209

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

Rule 210

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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 303

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

Rule 304

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

Rule 457

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

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

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {-\frac {9}{2}+\frac {9 x}{4}}{x (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {9}{16} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )+\frac {9}{8} \text {Subst}\left (\int \frac {1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {3}{4} \text {Subst}\left (\int \frac {x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{8} \text {Subst}\left (\int \frac {1-x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {3}{8} \text {Subst}\left (\int \frac {1+x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9}{16} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {9}{16} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt {2}}\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}+\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}\\ &=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 127, normalized size = 0.66 \begin {gather*} \frac {1}{16} \left (-\frac {4 \left (-1+3 x^2\right )^{3/4}}{x^2}+12 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-9 \sqrt {2} \tan ^{-1}\left (\frac {-1+\sqrt {-1+3 x^2}}{\sqrt {2} \sqrt [4]{-1+3 x^2}}\right )-12 \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+9 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(-1 + 3*x^2)^(3/4))/x^2 + 12*ArcTan[(-1 + 3*x^2)^(1/4)] - 9*Sqrt[2]*ArcTan[(-1 + Sqrt[-1 + 3*x^2])/(Sqrt[
2]*(-1 + 3*x^2)^(1/4))] - 12*ArcTanh[(-1 + 3*x^2)^(1/4)] + 9*Sqrt[2]*ArcTanh[(Sqrt[2]*(-1 + 3*x^2)^(1/4))/(1 +
Sqrt[-1 + 3*x^2])])/16

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 8.08, size = 315, normalized size = 1.65

method	result	size
trager	\(-\frac {\left (3 x^{2}-1\right )^{\frac {3}{4}}}{4 x^{2}}+\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}}{x^{2}}\right )}{16}+\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{16}-\frac {3 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}\)	\(315\)
risch	\(-\frac {\left (3 x^{2}-1\right )^{\frac {3}{4}}}{4 x^{2}}+\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{16}-\frac {9 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{2}}\right )}{16}-\frac {3 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}\)	\(315\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*x^2-1)^(3/4)/x^2+9/16*RootOf(_Z^4+1)^3*ln(-(3*RootOf(_Z^4+1)^3*x^2-2*RootOf(_Z^4+1)^3+2*RootOf(_Z^4+1)
^2*(3*x^2-1)^(1/4)-2*(3*x^2-1)^(1/2)*RootOf(_Z^4+1)+2*(3*x^2-1)^(3/4))/x^2)+9/16*RootOf(_Z^4+1)*ln((2*(3*x^2-1
)^(1/2)*RootOf(_Z^4+1)^3+2*RootOf(_Z^4+1)^2*(3*x^2-1)^(1/4)-3*RootOf(_Z^4+1)*x^2-2*(3*x^2-1)^(3/4)+2*RootOf(_Z
^4+1))/x^2)-3/8*ln(-(2*(3*x^2-1)^(3/4)+2*(3*x^2-1)^(1/2)+3*x^2+2*(3*x^2-1)^(1/4))/(3*x^2-2))+3/8*RootOf(_Z^4+1
)^2*ln(-(2*RootOf(_Z^4+1)^2*(3*x^2-1)^(1/2)-3*RootOf(_Z^4+1)^2*x^2+2*(3*x^2-1)^(3/4)-2*(3*x^2-1)^(1/4))/(3*x^2
-2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.55, size = 252, normalized size = 1.32 \begin {gather*} \frac {36 \, \sqrt {2} x^{2} \arctan \left (\sqrt {2} \sqrt {\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}} - 1\right ) + 36 \, \sqrt {2} x^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4} - \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + 9 \, \sqrt {2} x^{2} \log \left (4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4\right ) - 9 \, \sqrt {2} x^{2} \log \left (-4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4\right ) + 24 \, x^{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) - 8 \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}}}{32 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(36*sqrt(2)*x^2*arctan(sqrt(2)*sqrt(sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - sqrt(2)*(3*x^2 - 1
)^(1/4) - 1) + 36*sqrt(2)*x^2*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*(3*x^2 - 1)^(1/4) + 4*sqrt(3*x^2 - 1) + 4) -
sqrt(2)*(3*x^2 - 1)^(1/4) + 1) + 9*sqrt(2)*x^2*log(4*sqrt(2)*(3*x^2 - 1)^(1/4) + 4*sqrt(3*x^2 - 1) + 4) - 9*sq
rt(2)*x^2*log(-4*sqrt(2)*(3*x^2 - 1)^(1/4) + 4*sqrt(3*x^2 - 1) + 4) + 24*x^2*arctan((3*x^2 - 1)^(1/4)) - 12*x^
2*log((3*x^2 - 1)^(1/4) + 1) + 12*x^2*log((3*x^2 - 1)^(1/4) - 1) - 8*(3*x^2 - 1)^(3/4))/x^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \cdot \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.75, size = 169, normalized size = 0.88 \begin {gather*} -\frac {9}{16} \, \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 {9}{16} \, \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 {9}{32} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {9}{32} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}}}{4 \, x^{2}} + \frac {3}{4} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{8} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{8} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(3*x^2 - 1)^(1/4))) - 9/16*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)
- 2*(3*x^2 - 1)^(1/4))) + 9/32*sqrt(2)*log(sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 9/32*sqrt(2)*log
(-sqrt(2)*(3*x^2 - 1)^(1/4) + sqrt(3*x^2 - 1) + 1) - 1/4*(3*x^2 - 1)^(3/4)/x^2 + 3/4*arctan((3*x^2 - 1)^(1/4))
- 3/8*log((3*x^2 - 1)^(1/4) + 1) + 3/8*log(abs((3*x^2 - 1)^(1/4) - 1))

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Mupad [B]
time = 0.12, size = 82, normalized size = 0.43 \begin {gather*} \frac {3\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{4}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4}-\frac {{\left (3\,x^2-1\right )}^{3/4}}{4\,x^2}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{8}-\frac {{\left (-1\right )}^{3/4}\,\mathrm {atan}\left ({\left (-1\right )}^{3/4}\,{\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*atan((3*x^2 - 1)^(1/4)))/4 + (atan((3*x^2 - 1)^(1/4)*1i)*3i)/4 - (3*x^2 - 1)^(3/4)/(4*x^2) - ((-1)^(1/4)*at
an((-1)^(1/4)*(3*x^2 - 1)^(1/4)*1i)*9i)/8 - ((-1)^(3/4)*atan((-1)^(3/4)*(3*x^2 - 1)^(1/4)*1i)*9i)/8

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