3.29.34 \(\int \frac {\sqrt {-2+3 x^4} (-1+3 x^4)}{x^3 (1-3 x^4+3 x^8)} \, dx\)

Optimal. Leaf size=288 \[ \frac {1}{2} \sqrt {3} \text {RootSum}\left [\text {$\#$1}^4-2 \text {$\#$1}^3+4 \text {$\#$1}+4\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )}{2 \text {$\#$1}^3-3 \text {$\#$1}^2+2}\& \right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-4 \text {$\#$1}+4\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )+\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )}{2 \text {$\#$1}^3+3 \text {$\#$1}^2-2}\& \right ]+\frac {\sqrt {3 x^4-2}}{2 x^2} \]

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Rubi [C]  time = 0.89, antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 16, number of rules used = 11, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {6728, 275, 277, 217, 206, 1490, 1293, 1692, 377, 205, 208} \begin {gather*} \frac {\sqrt {3 x^4-2}}{2 x^2}-\frac {\left (-\sqrt {3}+i\right ) \sqrt {\frac {3 \left (-\sqrt {3}+3 i\right )}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \left (\sqrt {3}+i\right )}} \sqrt {3 x^4-2}}\right )}{2 \left (-\sqrt {3}+3 i\right )}-\frac {\left (\sqrt {3}+i\right ) \tanh ^{-1}\left (\frac {\sqrt {-\frac {3 \left (-\sqrt {3}+i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(Sqrt[-2 + 3*x^4]*(-1 + 3*x^4))/(x^3*(1 - 3*x^4 + 3*x^8)),x]

[Out]

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

Rule 205

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

Rule 206

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

Rule 208

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 377

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

Rule 1293

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[(
e*f^2)/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[((f*x)^(m - 2)*(d + e*x^2)^(q - 1)*S
imp[a*e - (c*d - b*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]

Rule 1490

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^((2*n)/
k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[n, 0] && IntegerQ[m]

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx &=\int \left (-\frac {\sqrt {-2+3 x^4}}{x^3}+\frac {3 x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8}\right ) \, dx\\ &=3 \int \frac {x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8} \, dx-\int \frac {\sqrt {-2+3 x^4}}{x^3} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-2+3 x^2}}{x^2} \, dx,x,x^2\right )\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+3 x^2}}{1-3 x^2+3 x^4} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {3-3 x^2}{\sqrt {-2+3 x^2} \left (1-3 x^2+3 x^4\right )} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-3-3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )}+\frac {-3+3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3+i \sqrt {3}-\left (12+3 \left (-3+i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-i \sqrt {3}-\left (12+3 \left (-3-i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {3 \left (3 i-\sqrt {3}\right )}{i+\sqrt {3}}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {3 i-\sqrt {3}}{3 \left (i+\sqrt {3}\right )}} \sqrt {-2+3 x^4}}\right )}{2 \left (3 i-\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {-\frac {3 \left (i-\sqrt {3}\right )}{3 i+\sqrt {3}}} x^2}{\sqrt {-2+3 x^4}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}}\\ \end {align*}

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Mathematica [C]  time = 5.31, size = 182, normalized size = 0.63 \begin {gather*} \frac {\sqrt {3 x^4-2}}{2 x^2}+\frac {3 i \left (\tan ^{-1}\left (\frac {\sqrt {-\frac {3 \left (\sqrt {3}-i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )+\sqrt {-\sqrt {3}+i} \sqrt {-\frac {1}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \sqrt {3}+3 i}} \sqrt {3 x^4-2}}\right )\right )}{\left (\sqrt {3}-3 i\right ) \sqrt {\frac {-3 \sqrt {3}+3 i}{\sqrt {3}+3 i}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[-2 + 3*x^4]*(-1 + 3*x^4))/(x^3*(1 - 3*x^4 + 3*x^8)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 288, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-2 + 3*x^4]*(-1 + 3*x^4))/(x^3*(1 - 3*x^4 + 3*x^8)),x]

[Out]

Sqrt[-2 + 3*x^4]/(2*x^2) + (Sqrt[3]*RootSum[4 + 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4
] - #1] - Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(2 - 3*
#1^2 + 2*#1^3) & ])/2 - (Sqrt[3]*RootSum[4 - 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] -
#1] + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(-2 + 3*#1
^2 + 2*#1^3) & ])/2

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fricas [B]  time = 0.67, size = 1041, normalized size = 3.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-2)^(1/2)*(3*x^4-1)/x^3/(3*x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/64*(4*12^(1/4)*sqrt(2)*x^2*sqrt(-4*sqrt(3) + 8)*arctan(-1/12*(432*x^10 - 576*x^6 + 192*x^2 + 2*sqrt(3*x^4 -
2)*(12^(3/4)*(sqrt(3)*sqrt(2)*(3*x^8 - 5*x^4 + 2) - sqrt(2)*(3*x^4 - 2)) - 3*12^(1/4)*(sqrt(2)*x^4 - sqrt(3)*
sqrt(2)*(x^8 - x^4)))*sqrt(-4*sqrt(3) + 8) + 8*sqrt(3)*(9*x^10 - 6*x^6 - sqrt(3)*(9*x^10 - 12*x^6 + 4*x^2)) -
12*sqrt(3)*(3*x^10 - 2*x^6) + (12*sqrt(3)*x^10 + 72*x^10 - 48*x^6 - sqrt(3*x^4 - 2)*(12^(3/4)*(sqrt(3)*sqrt(2)
*(3*x^8 - 2*x^4) + sqrt(2)*(3*x^8 - 2*x^4)) + 3*12^(1/4)*(sqrt(3)*sqrt(2)*x^8 + sqrt(2)*x^8))*sqrt(-4*sqrt(3)
+ 8))*sqrt((12*x^8 - 12*x^4 + 12^(1/4)*sqrt(3*x^4 - 2)*(sqrt(3)*sqrt(2)*(2*x^6 - x^2) + sqrt(2)*(3*x^6 - x^2))
*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3)*(3*x^8 - 2*x^4) + 4)/x^8))/(33*x^10 - 48*x^6 + 16*x^2)) + 4*12^(1/4)*sqrt(2)
*x^2*sqrt(-4*sqrt(3) + 8)*arctan(1/12*(432*x^10 - 576*x^6 + 192*x^2 - 2*sqrt(3*x^4 - 2)*(12^(3/4)*(sqrt(3)*sqr
t(2)*(3*x^8 - 5*x^4 + 2) - sqrt(2)*(3*x^4 - 2)) - 3*12^(1/4)*(sqrt(2)*x^4 - sqrt(3)*sqrt(2)*(x^8 - x^4)))*sqrt
(-4*sqrt(3) + 8) + 8*sqrt(3)*(9*x^10 - 6*x^6 - sqrt(3)*(9*x^10 - 12*x^6 + 4*x^2)) - 12*sqrt(3)*(3*x^10 - 2*x^6
) + (12*sqrt(3)*x^10 + 72*x^10 - 48*x^6 + sqrt(3*x^4 - 2)*(12^(3/4)*(sqrt(3)*sqrt(2)*(3*x^8 - 2*x^4) + sqrt(2)
*(3*x^8 - 2*x^4)) + 3*12^(1/4)*(sqrt(3)*sqrt(2)*x^8 + sqrt(2)*x^8))*sqrt(-4*sqrt(3) + 8))*sqrt((12*x^8 - 12*x^
4 - 12^(1/4)*sqrt(3*x^4 - 2)*(sqrt(3)*sqrt(2)*(2*x^6 - x^2) + sqrt(2)*(3*x^6 - x^2))*sqrt(-4*sqrt(3) + 8) + 4*
sqrt(3)*(3*x^8 - 2*x^4) + 4)/x^8))/(33*x^10 - 48*x^6 + 16*x^2)) + 12^(1/4)*(sqrt(3)*sqrt(2)*x^2 + 2*sqrt(2)*x^
2)*sqrt(-4*sqrt(3) + 8)*log(4*(12*x^8 - 12*x^4 + 12^(1/4)*sqrt(3*x^4 - 2)*(sqrt(3)*sqrt(2)*(2*x^6 - x^2) + sqr
t(2)*(3*x^6 - x^2))*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3)*(3*x^8 - 2*x^4) + 4)/x^8) - 12^(1/4)*(sqrt(3)*sqrt(2)*x^2
+ 2*sqrt(2)*x^2)*sqrt(-4*sqrt(3) + 8)*log(4*(12*x^8 - 12*x^4 - 12^(1/4)*sqrt(3*x^4 - 2)*(sqrt(3)*sqrt(2)*(2*x
^6 - x^2) + sqrt(2)*(3*x^6 - x^2))*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3)*(3*x^8 - 2*x^4) + 4)/x^8) - 32*sqrt(3*x^4
- 2))/x^2

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giac [B]  time = 1.66, size = 178, normalized size = 0.62 \begin {gather*} \frac {1}{16} \, \sqrt {3} {\left ({\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} - 8 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 4\right )} \log \left ({\left | -{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{8} + 4 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{6} - 24 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} + 16 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} - 16 \right |}\right ) + \frac {2 \, \sqrt {3}}{{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-2)^(1/2)*(3*x^4-1)/x^3/(3*x^8-3*x^4+1),x, algorithm="giac")

[Out]

1/16*sqrt(3)*((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 - 8*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^2 + 4)*log(abs(-(sqrt(3)*x
^2 - sqrt(3*x^4 - 2))^8 + 4*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^6 - 24*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 + 16*(sqr
t(3)*x^2 - sqrt(3*x^4 - 2))^2 - 16)) + 2*sqrt(3)/((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^2 + 2)

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maple [B]  time = 8.00, size = 92, normalized size = 0.32

method	result	size
risch	\(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+24 \textit {\_Z}^{2}-16 \textit {\_Z} +16\right )}{\sum }\frac {\left (\textit {\_R}^{2}-8 \textit {\_R} +4\right ) \ln \left (\left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+12 \textit {\_R} -4}\right )}{4}\)	\(92\)
elliptic	\(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+24 \textit {\_Z}^{2}-16 \textit {\_Z} +16\right )}{\sum }\frac {\left (\textit {\_R}^{2}-8 \textit {\_R} +4\right ) \ln \left (\left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+12 \textit {\_R} -4}\right )}{4}\)	\(92\)
default	\(-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-4 \textit {\_Z} +4\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}-2}\right )}{2}-\frac {\sqrt {3}\, \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )}{2}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+4 \textit {\_Z} +4\right )}{\sum }\frac {\left (\textit {\_R}^{2}-\textit {\_R} -2\right ) \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2}\right )}{2}-\frac {\left (3 x^{4}-2\right )^{\frac {3}{2}}}{4 x^{2}}+\frac {3 x^{2} \sqrt {3 x^{4}-2}}{4}-\frac {\ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) \sqrt {3}}{2}\)	\(213\)
trager	\(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) \ln \left (-\frac {-55296 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{4} x^{4} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )-720 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) x^{4}+1872 \sqrt {3 x^{4}-2}\, \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+55296 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{4}+39 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) x^{4}-47 x^{2} \sqrt {3 x^{4}-2}+1104 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )-26 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )}{144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{4}-144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+1}\right )}{4}-3 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-165888 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x^{4}+9072 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{4}+468 \sqrt {3 x^{4}-2}\, \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+165888 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5}+2 x^{2} \sqrt {3 x^{4}-2}-10224 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}+63 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )}{\left (1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{2}+144 x^{2} \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}-24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x^{2}-144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3 x^{2}+24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )+2\right ) \left (1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{2}-144 x^{2} \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}-24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+3 x^{2}+24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )-2\right )}\right )\)	\(730\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4-2)^(1/2)*(3*x^4-1)/x^3/(3*x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(3*x^4-2)^(1/2)/x^2-1/4*3^(1/2)*sum((_R^2-8*_R+4)/(_R^3-3*_R^2+12*_R-4)*ln(((3*x^4-2)^(1/2)-3^(1/2)*x^2)^2
-_R),_R=RootOf(_Z^4-4*_Z^3+24*_Z^2-16*_Z+16))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{4} - 1\right )} \sqrt {3 \, x^{4} - 2}}{{\left (3 \, x^{8} - 3 \, x^{4} + 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4-2)^(1/2)*(3*x^4-1)/x^3/(3*x^8-3*x^4+1),x, algorithm="maxima")

[Out]

integrate((3*x^4 - 1)*sqrt(3*x^4 - 2)/((3*x^8 - 3*x^4 + 1)*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (3\,x^4-1\right )\,\sqrt {3\,x^4-2}}{x^3\,\left (3\,x^8-3\,x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4 - 1)*(3*x^4 - 2)^(1/2))/(x^3*(3*x^8 - 3*x^4 + 1)),x)

[Out]

int(((3*x^4 - 1)*(3*x^4 - 2)^(1/2))/(x^3*(3*x^8 - 3*x^4 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**4-2)**(1/2)*(3*x**4-1)/x**3/(3*x**8-3*x**4+1),x)

[Out]

Timed out

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