3.23.3 \(\int \frac {(1+x^3) \sqrt {-2-x^3+x^6}}{x^4 (-1-2 x^3+x^6)} \, dx\)

Optimal. Leaf size=163 \[ \frac {\sqrt {x^6-x^3-2}}{3 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {x^6-x^3-2}}{\sqrt {2} \left (x^3+1\right )}\right )}{\sqrt {2}}+\frac {1}{3} \sqrt {4+3 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {3 \sqrt {2}-4} \sqrt {x^6-x^3-2}}{x^3+1}\right )-\frac {1}{3} \sqrt {3 \sqrt {2}-4} \tanh ^{-1}\left (\frac {\sqrt {4+3 \sqrt {2}} \sqrt {x^6-x^3-2}}{x^3+1}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.96, antiderivative size = 224, normalized size of antiderivative = 1.37, number of steps used = 26, number of rules used = 13, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {6728, 1357, 732, 843, 621, 206, 724, 204, 734, 6715, 1019, 1076, 1032} \begin {gather*} \frac {\sqrt {x^6-x^3-2}}{3 x^3}+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x^3+4}{2 \sqrt {2} \sqrt {x^6-x^3-2}}\right )-\frac {\tan ^{-1}\left (\frac {x^3+4}{2 \sqrt {2} \sqrt {x^6-x^3-2}}\right )}{6 \sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \tan ^{-1}\left (\frac {-\left (\left (1-2 \sqrt {2}\right ) x^3\right )-\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )}{3\ 2^{3/4}}-\frac {\left (1-\sqrt {2}\right ) \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) x^3\right )+\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )}{3\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-2 - x^3 + x^6]/(3*x^3) - ArcTan[(4 + x^3)/(2*Sqrt[2]*Sqrt[-2 - x^3 + x^6])]/(6*Sqrt[2]) + (Sqrt[2]*ArcTa
n[(4 + x^3)/(2*Sqrt[2]*Sqrt[-2 - x^3 + x^6])])/3 - ((1 + Sqrt[2])*ArcTan[(5 - Sqrt[2] - (1 - 2*Sqrt[2])*x^3)/(
2*2^(1/4)*Sqrt[-2 - x^3 + x^6])])/(3*2^(3/4)) - ((1 - Sqrt[2])*ArcTanh[(5 + Sqrt[2] - (1 + 2*Sqrt[2])*x^3)/(2*
2^(1/4)*Sqrt[-2 - x^3 + x^6])])/(3*2^(3/4))

Rule 204

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

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 621

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1019

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[(h*(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] - Dist[1/(2*f*(p + q + 1
)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a*(h*e - 2*g*f)*(p + q + 1) + (2*
h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x]
, x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && Ne
Q[p + q + 1, 0]

Rule 1032

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx &=\int \left (-\frac {\sqrt {-2-x^3+x^6}}{x^4}+\frac {\sqrt {-2-x^3+x^6}}{x}-\frac {x^2 \left (-3+x^3\right ) \sqrt {-2-x^3+x^6}}{-1-2 x^3+x^6}\right ) \, dx\\ &=-\int \frac {\sqrt {-2-x^3+x^6}}{x^4} \, dx+\int \frac {\sqrt {-2-x^3+x^6}}{x} \, dx-\int \frac {x^2 \left (-3+x^3\right ) \sqrt {-2-x^3+x^6}}{-1-2 x^3+x^6} \, dx\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-2-x+x^2}}{x^2} \, dx,x,x^3\right )\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-2-x+x^2}}{x} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-3+x) \sqrt {-2-x+x^2}}{-1-2 x+x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {4+x}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-\frac {11}{2}-x+\frac {3 x^2}{2}}{\left (-1-2 x+x^2\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-4+2 x}{\left (-1-2 x+x^2\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-x^3}{\sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-x^3}{\sqrt {-2-x^3+x^6}}\right )+\frac {1}{3} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right )\\ &=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {\tan ^{-1}\left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )}{6 \sqrt {2}}+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1-2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {-1+2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \left (2 \left (2-\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-32+8 \left (-2-2 \sqrt {2}\right )+4 \left (-2-2 \sqrt {2}\right )^2-x^2} \, dx,x,-\frac {2 \left (5+\sqrt {2}+\left (1-2 \left (1+\sqrt {2}\right )\right ) x^3\right )}{\sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \left (2 \left (2+\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-32+8 \left (-2+2 \sqrt {2}\right )+4 \left (-2+2 \sqrt {2}\right )^2-x^2} \, dx,x,\frac {2 \left (-5+\sqrt {2}+\left (-1-2 \left (-1+\sqrt {2}\right )\right ) x^3\right )}{\sqrt {-2-x^3+x^6}}\right )\\ &=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {\tan ^{-1}\left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )}{6 \sqrt {2}}+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )-\frac {\left (1+\sqrt {2}\right ) \tan ^{-1}\left (\frac {5-\sqrt {2}-\left (1-2 \sqrt {2}\right ) x^3}{2 \sqrt [4]{2} \sqrt {-2-x^3+x^6}}\right )}{3\ 2^{3/4}}+\frac {1}{2} \tanh ^{-1}\left (\frac {1-2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {-1+2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )-\frac {\left (1-\sqrt {2}\right ) \tanh ^{-1}\left (\frac {5+\sqrt {2}-\left (1+2 \sqrt {2}\right ) x^3}{2 \sqrt [4]{2} \sqrt {-2-x^3+x^6}}\right )}{3\ 2^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 326, normalized size = 2.00 \begin {gather*} \frac {1}{12} \left (\frac {4 \sqrt {x^6-x^3-2}}{x^3}+3 \sqrt {2} \tan ^{-1}\left (\frac {x^3+4}{2 \sqrt {2} \sqrt {x^6-x^3-2}}\right )-2\ 2^{3/4} \tan ^{-1}\left (\frac {\left (2 \sqrt {2}-1\right ) x^3-\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )-2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\left (2 \sqrt {2}-1\right ) x^3-\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )-6 \tanh ^{-1}\left (\frac {1-2 x^3}{2 \sqrt {x^6-x^3-2}}\right )-6 \tanh ^{-1}\left (\frac {2 x^3-1}{2 \sqrt {x^6-x^3-2}}\right )+2\ 2^{3/4} \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) x^3\right )+\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) x^3\right )+\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[-2 - x^3 + x^6])/x^3 + 3*Sqrt[2]*ArcTan[(4 + x^3)/(2*Sqrt[2]*Sqrt[-2 - x^3 + x^6])] - 2*2^(1/4)*ArcTa
n[(5 - Sqrt[2] + (-1 + 2*Sqrt[2])*x^3)/(2*2^(1/4)*Sqrt[-2 - x^3 + x^6])] - 2*2^(3/4)*ArcTan[(5 - Sqrt[2] + (-1
+ 2*Sqrt[2])*x^3)/(2*2^(1/4)*Sqrt[-2 - x^3 + x^6])] - 6*ArcTanh[(1 - 2*x^3)/(2*Sqrt[-2 - x^3 + x^6])] - 6*Arc
Tanh[(-1 + 2*x^3)/(2*Sqrt[-2 - x^3 + x^6])] - 2*2^(1/4)*ArcTanh[(5 + Sqrt[2] - (1 + 2*Sqrt[2])*x^3)/(2*2^(1/4)
*Sqrt[-2 - x^3 + x^6])] + 2*2^(3/4)*ArcTanh[(5 + Sqrt[2] - (1 + 2*Sqrt[2])*x^3)/(2*2^(1/4)*Sqrt[-2 - x^3 + x^6
])])/12

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.55, size = 162, normalized size = 0.99 \begin {gather*} \frac {\sqrt {-2-x^3+x^6}}{3 x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-2-x^3+x^6}}{-2+x^3}\right )}{\sqrt {2}}-\frac {1}{3} \sqrt {4+3 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\frac {3}{\sqrt {2}}} \sqrt {-2-x^3+x^6}}{-2+x^3}\right )-\frac {1}{3} \sqrt {-4+3 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {-2+\frac {3}{\sqrt {2}}} \sqrt {-2-x^3+x^6}}{-2+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.54, size = 289, normalized size = 1.77 \begin {gather*} -\frac {3 \, \sqrt {2} x^{3} \arctan \left (-\frac {1}{2} \, \sqrt {2} x^{3} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{6} - x^{3} - 2}\right ) - 4 \, x^{3} \sqrt {3 \, \sqrt {2} + 4} \arctan \left (\frac {1}{2} \, \sqrt {2 \, x^{6} - 3 \, x^{3} + \sqrt {2} {\left (2 \, x^{3} - 1\right )} - 2 \, \sqrt {x^{6} - x^{3} - 2} {\left (x^{3} + \sqrt {2} - 1\right )} + 1} \sqrt {3 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} - \frac {1}{2} \, {\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - 3\right )} + \sqrt {x^{6} - x^{3} - 2} {\left (\sqrt {2} - 2\right )} - 4\right )} \sqrt {3 \, \sqrt {2} + 4}\right ) + x^{3} \sqrt {3 \, \sqrt {2} - 4} \log \left (-x^{3} + \sqrt {3 \, \sqrt {2} - 4} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) - x^{3} \sqrt {3 \, \sqrt {2} - 4} \log \left (-x^{3} - \sqrt {3 \, \sqrt {2} - 4} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) - 2 \, x^{3} - 2 \, \sqrt {x^{6} - x^{3} - 2}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),x, algorithm="fricas")

[Out]

-1/6*(3*sqrt(2)*x^3*arctan(-1/2*sqrt(2)*x^3 + 1/2*sqrt(2)*sqrt(x^6 - x^3 - 2)) - 4*x^3*sqrt(3*sqrt(2) + 4)*arc
tan(1/2*sqrt(2*x^6 - 3*x^3 + sqrt(2)*(2*x^3 - 1) - 2*sqrt(x^6 - x^3 - 2)*(x^3 + sqrt(2) - 1) + 1)*sqrt(3*sqrt(
2) + 4)*(sqrt(2) - 2) - 1/2*(2*x^3 - sqrt(2)*(x^3 - 3) + sqrt(x^6 - x^3 - 2)*(sqrt(2) - 2) - 4)*sqrt(3*sqrt(2)
+ 4)) + x^3*sqrt(3*sqrt(2) - 4)*log(-x^3 + sqrt(3*sqrt(2) - 4)*(sqrt(2) + 1) + sqrt(2) + sqrt(x^6 - x^3 - 2)
+ 1) - x^3*sqrt(3*sqrt(2) - 4)*log(-x^3 - sqrt(3*sqrt(2) - 4)*(sqrt(2) + 1) + sqrt(2) + sqrt(x^6 - x^3 - 2) +
1) - 2*x^3 - 2*sqrt(x^6 - x^3 - 2))/x^3

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),x, algorithm="giac")

[Out]

Exception raised: AttributeError >> type

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+1\right ) \sqrt {x^{6}-x^{3}-2}}{x^{4} \left (x^{6}-2 x^{3}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),x)

[Out]

int((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - x^{3} - 2} {\left (x^{3} + 1\right )}}{{\left (x^{6} - 2 \, x^{3} - 1\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - x^3 - 2)*(x^3 + 1)/((x^6 - 2*x^3 - 1)*x^4), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^3+1\right )\,\sqrt {x^6-x^3-2}}{x^4\,\left (-x^6+2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 1)*(x^6 - x^3 - 2)^(1/2))/(x^4*(2*x^3 - x^6 + 1)),x)

[Out]

int(-((x^3 + 1)*(x^6 - x^3 - 2)^(1/2))/(x^4*(2*x^3 - x^6 + 1)), x)

________________________________________________________________________________________

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((x**3+1)*(x**6-x**3-2)**(1/2)/x**4/(x**6-2*x**3-1),x)

[Out]

Timed out

________________________________________________________________________________________