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

Optimal. Leaf size=123 \[ \frac {2 \sqrt {x^3+x^2+x}}{3 \left (x^2+x+1\right )}+\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{3 \sqrt {3}} \]

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Rubi [C]  time = 2.51, antiderivative size = 785, normalized size of antiderivative = 6.38, number of steps used = 43, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {2056, 6725, 716, 1103, 934, 169, 538, 537, 849, 822, 839, 1197, 1195} \begin {gather*} \frac {2 x \left (-i \sqrt {3} x-(-1)^{2/3}+1\right )}{9 \sqrt {x^3+x^2+x}}+\frac {2 x \left (i \sqrt {3} x+\sqrt [3]{-1}+1\right )}{9 \sqrt {x^3+x^2+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (-1;\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*(1 - (-1)^(2/3) - I*Sqrt[3]*x))/(9*Sqrt[x + x^2 + x^3]) + (2*x*(1 + (-1)^(1/3) + I*Sqrt[3]*x))/(9*Sqrt[x
+ x^2 + x^3]) - (Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/Sqrt[x + x^2
+ x^3] + ((1 - I*Sqrt[3])*Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/(6
*Sqrt[x + x^2 + x^3]) + ((1 + I*Sqrt[3])*Sqrt[x]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt
[x]], 1/4])/(6*Sqrt[x + x^2 + x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3]
)]*EllipticPi[-1, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x
+ x^2 + x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 - I*
Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2
+ x^3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-1 + I*Sqrt[
3])/2, ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2 + x^
3]) + (4*Sqrt[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(1 + I*Sqrt[3])/2,
ArcSin[((1 - I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/(3*(1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3])

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

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

Rule 716

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

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 839

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

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rule 934

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

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[
(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q
^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0
]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1197

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

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

Rubi steps

\begin {align*} \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+x^6}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {2}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )}\right ) \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1-x^3\right )}+\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1+x^3\right )}\right ) \, dx}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^3\right )} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1+x^3\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt {x} \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{3 (1-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(-1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(-1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+\sqrt [3]{-1} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1-(-1)^{2/3} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (-1;\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 5.02, size = 939, normalized size = 7.63 \begin {gather*} \frac {2 x \left (\sqrt {x} \left (\frac {(-1)^{2/3} \sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\left (1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2}{\left (1+(-1)^{2/3}\right ) x}-\frac {(-1)^{2/3} \sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\left (-1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2}{\left (1+(-1)^{2/3}\right ) x}-2 (-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} F\left (i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\sqrt [3]{-1} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-1;i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )-\sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-1;i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+(-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\frac {3 (-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}-\frac {3 \sqrt [3]{-1} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}\right )+1\right )}{3 \sqrt {x \left (x^2+x+1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*(1 + Sqrt[x]*(-2*(-1)^(2/3)*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticF[I*ArcSinh[(-1)^(5/6
)/Sqrt[x]], (-1)^(2/3)] + ((-1)^(2/3)*Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[
x]))]*(-1 + (-1)^(1/3)*Sqrt[x])^2*Sqrt[(-1 + (-1)^(2/3)*Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]
*Sqrt[-((1 + (-1)^(2/3)*Sqrt[x])/(-1 + (-1)^(1/3) + Sqrt[x]))]*((1 + (-1)^(1/3))*EllipticF[ArcSin[Sqrt[(1 - (-
1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3] - 2*(-1)^(1/3)*EllipticPi[-1, ArcSin[S
qrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3]))/((1 + (-1)^(2/3))*x) - Sq
rt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticPi[-1, I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)] + (-1)
^(1/3)*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticPi[-1, I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3
)] - ((-1)^(2/3)*Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*(-1 + (-1)^(1/3
)*Sqrt[x])^2*Sqrt[(-1 + (-1)^(2/3)*Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]*Sqrt[-((1 + (-1)^(2/
3)*Sqrt[x])/(-1 + (-1)^(1/3) + Sqrt[x]))]*((-1 + (-1)^(1/3))*EllipticF[ArcSin[Sqrt[(1 - (-1)^(1/3) + Sqrt[x])/
((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3] - 2*(-1)^(1/3)*EllipticPi[3, ArcSin[Sqrt[(1 - (-1)^(1/3) +
Sqrt[x])/((1 + (-1)^(1/3))*(-1 + (-1)^(1/3)*Sqrt[x]))]], -3]))/((1 + (-1)^(2/3))*x) + (-1)^(2/3)*Sqrt[1 - (-1
)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)] - (3*(-1
)^(1/3)*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticPi[-(-1)^(2/3), I*ArcSinh[(-1)^(5/6)/Sqrt[x]],
(-1)^(2/3)])/(1 + (-1)^(1/3))^2 + (3*(-1)^(2/3)*Sqrt[1 - (-1)^(2/3)/x]*Sqrt[((-1)^(1/3) + x)/x]*EllipticPi[-(
-1)^(2/3), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)])/(1 + (-1)^(1/3))^2)))/(3*Sqrt[x*(1 + x + x^2)])

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IntegrateAlgebraic [A]  time = 0.23, size = 123, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x+x^2+x^3}}{3 \left (1+x+x^2\right )}+\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x + x^2 + x^3])/(3*(1 + x + x^2)) + ArcTan[Sqrt[x + x^2 + x^3]/(1 + x + x^2)]/3 + (Sqrt[2]*ArcTanh[(Sq
rt[2]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)])/3 + ArcTanh[(Sqrt[3]*Sqrt[x + x^2 + x^3])/(1 + x + x^2)]/(3*Sqrt[3]
)

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fricas [A]  time = 0.48, size = 196, normalized size = 1.59 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 14 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \sqrt {3} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 20 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 6 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + 24 \, \sqrt {x^{3} + x^{2} + x}}{36 \, {\left (x^{2} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(3*sqrt(2)*(x^2 + x + 1)*log((x^4 + 14*x^3 + 4*sqrt(2)*sqrt(x^3 + x^2 + x)*(x^2 + 3*x + 1) + 19*x^2 + 14*
x + 1)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + sqrt(3)*(x^2 + x + 1)*log((x^4 + 20*x^3 + 4*sqrt(3)*sqrt(x^3 + x^2 +
x)*(x^2 + 4*x + 1) + 30*x^2 + 20*x + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 6*(x^2 + x + 1)*arctan(1/2*(x^2 +
1)/sqrt(x^3 + x^2 + x)) + 24*sqrt(x^3 + x^2 + x))/(x^2 + x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.26, size = 181, normalized size = 1.47

method	result	size
trager	\(\frac {2 \sqrt {x^{3}+x^{2}+x}}{3 \left (x^{2}+x +1\right )}+\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +\RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{18}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )-2 \sqrt {x^{3}+x^{2}+x}}{\left (1+x \right )^{2}}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +1}\right )}{6}\)	\(181\)
default	\(-\frac {4 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}+\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{8}+\frac {3 i \sqrt {3}}{8}-\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}\right ) \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{8}+\frac {i \sqrt {3}}{8}-\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}\right ) \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}\)	\(736\)
risch	\(-\frac {4 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}+\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{8}+\frac {3 i \sqrt {3}}{8}-\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}\right ) \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{8}+\frac {i \sqrt {3}}{8}-\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{4}\right ) \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}\)	\(736\)
elliptic	\(\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}-\frac {2 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {2 i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{18 \sqrt {x^{3}+x^{2}+x}}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}}-\frac {2 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{9 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\)	\(1330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)/(x^3+x^2+x)^(1/2)/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

2/3*(x^3+x^2+x)^(1/2)/(x^2+x+1)+1/18*RootOf(_Z^2-3)*ln((RootOf(_Z^2-3)*x^2+4*RootOf(_Z^2-3)*x+RootOf(_Z^2-3)+6
*(x^3+x^2+x)^(1/2))/(-1+x)^2)-1/6*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^2+RootOf(_Z^2+1)-2*(x^3+x^2+x)^(1/2))/(
1+x)^2)-1/6*RootOf(_Z^2-2)*ln((-RootOf(_Z^2-2)*x^2-3*RootOf(_Z^2-2)*x+4*(x^3+x^2+x)^(1/2)-RootOf(_Z^2-2))/(x^2
-x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.88, size = 1195, normalized size = 9.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2)
)^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(-1, asin((x/((3^(1/2)*1i)/2 - 1/2
))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i
)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((
3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticF(asin((x/((3^(1
/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/
2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) - (2*((3^(1/2)*1i)/6 - 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((ellipticE(asin
((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) - ((x/((3^(1/2)*1i)/2 - 1/
2))^(1/2)*(x/((3^(1/2)*1i)/2 + 1/2) + 1)^(1/2))/(1 - x/((3^(1/2)*1i)/2 - 1/2))^(1/2))/(((3^(1/2)*1i)/2 - 1/2)/
((3^(1/2)*1i)/2 + 1/2) + 1) - ellipticF(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1
/2)*1i)/2 + 1/2)))*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^
(1/2)*1i)/2 + 1/2))^(1/2))/(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) + (2*((3^(1/2)*
1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x +
(3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(1/2 - (3^(1/2)*1i)/2, asin((x/((3^(1/2)*1i)/2
- 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(3*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^
(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2
+ 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi((3^(
1/2)*1i)/2 - 1/2, asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(3*
(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)
/6 + 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (3
^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2), as
in((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)))/(((3^(1/2)*1i)/2 + 1/2)
*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (2*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i
)/6 + 1/6)*(x/((3^(1/2)*1i)/2 - 1/2))^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 1/2))^(1/2)*((x + (
3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 1/2))^(1/2)*(ellipticE(asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)), -((3^(1/
2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 + 1/2)) + (sin(2*asin((x/((3^(1/2)*1i)/2 - 1/2))^(1/2)))*((3^(1/2)*1i)/2 - 1/2
))/(2*((3^(1/2)*1i)/2 + 1/2)*(x/((3^(1/2)*1i)/2 + 1/2) + 1)^(1/2))))/(((3^(1/2)*1i)/2 + 1/2)*(((3^(1/2)*1i)/2
- 1/2)/((3^(1/2)*1i)/2 + 1/2) + 1)*(x^2 + x^3 - x*((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{6}}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx - \int \frac {1}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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