\(\int \frac {\sqrt {1+x^3} (2+2 x^3+x^6)}{x^7 (-1+x^6)} \, dx\) [839]

Optimal result

Integrand size = 30, antiderivative size = 63 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {\sqrt {1+x^3} \left (2+5 x^3\right )}{6 x^6}+\frac {5}{2} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]

[Out]

1/6*(x^3+1)^(1/2)*(5*x^3+2)/x^6+5/2*arctanh((x^3+1)^(1/2))-5/3*2^(1/2)*arctanh(1/2*(x^3+1)^(1/2)*2^(1/2))

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.80 (sec) , antiderivative size = 1234, normalized size of antiderivative = 19.59, number of steps used = 41, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1600, 6857, 2161, 224, 2167, 2138, 551, 585, 95, 212, 272, 44, 65, 213, 6860, 211} \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {5 i (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \arctan \left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(3-6 i)-(2-3 i) \sqrt {3}}{(4+6 i)-(2+4 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {10 \sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \arctan \left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3^{3/4} \left (3 i-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {5 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \text {arctanh}\left (\frac {\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {2} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {5}{2} \text {arctanh}\left (\sqrt {x^3+1}\right )-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {10 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (97+56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {5 \sqrt {x^3+1}}{6 x^3}+\frac {\sqrt {x^3+1}}{3 x^6} \]

[In]

Int[(Sqrt[1 + x^3]*(2 + 2*x^3 + x^6))/(x^7*(-1 + x^6)),x]

[Out]

Sqrt[1 + x^3]/(3*x^6) + (5*Sqrt[1 + x^3])/(6*x^3) + (((5*I)/3)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]
*ArcTan[(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[((3 - 6*I) - (2 - 3*I)*Sqrt[3])/((4 + 6*I) - (2 + 4*
I)*Sqrt[3])]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2])])/(Sqrt[2]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^
3]) + (10*Sqrt[((6 - 3*I) - (3 - 2*I)*Sqrt[3])/((-6 - 4*I) + (4 + 2*I)*Sqrt[3])]*(1 + x)*Sqrt[(1 - x + x^2)/(1
+ Sqrt[3] + x)^2]*ArcTan[(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2])/(Sqrt[((6 - 3*I) - (3 - 2*I)*Sqrt[3])/((
-6 - 4*I) + (4 + 2*I)*Sqrt[3])]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2])])/(3^(3/4)*(3*I - Sqrt[3])*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (5*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*ArcTanh[Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]/(Sqrt[2]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2])])/(3*Sqrt[2]*Sqrt[(1 + x)/(1 + Sqrt[
3] + x)^2]*Sqrt[1 + x^3]) + (5*ArcTanh[Sqrt[1 + x^3]])/2 - (10*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]
*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[2 + Sqrt[3]]*Sqrt[(1
+ x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (20*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^
2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*(1 + (2 - I)*Sqrt[3])*Sq
rt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (20*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3]
+ x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*(1 + (2 + I)*Sqrt[
3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (40*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + S
qrt[3] + x)^2]*EllipticPi[-((I + (1 + 2*I)*Sqrt[3])^2/(1 - (2 + I)*Sqrt[3])^2), ArcSin[(1 - Sqrt[3] + x)/(1 +
Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*(7 + I*Sqrt[3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (4
0*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticPi[-((1 + (2 + I)*Sqrt[3])^2/(I -
(1 + 2*I)*Sqrt[3])^2), ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*(7 - I*Sqrt[3])*
Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (20*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[
3] + x)^2]*EllipticPi[97 + 56*Sqrt[3], ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(3/4)*
Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 211

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

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 213

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

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 272

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 585

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2138

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

Rule 2161

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[-q/((1 + Sqrt[
3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/((1 + Sqrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*
x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2167

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Simplify[(1 +
Sqrt[3])*(f/e)]}, Dist[4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2
]/(q*Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])), Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt
[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x
]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^
3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 6857

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]

Rule 6860

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*} \text {integral}& = \int \frac {2+2 x^3+x^6}{x^7 \left (-1+x^3\right ) \sqrt {1+x^3}} \, dx \\ & = \int \left (\frac {5}{3 (-1+x) \sqrt {1+x^3}}-\frac {2}{x^7 \sqrt {1+x^3}}-\frac {4}{x^4 \sqrt {1+x^3}}-\frac {5}{x \sqrt {1+x^3}}+\frac {5 (1+2 x)}{3 \left (1+x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = \frac {5}{3} \int \frac {1}{(-1+x) \sqrt {1+x^3}} \, dx+\frac {5}{3} \int \frac {1+2 x}{\left (1+x+x^2\right ) \sqrt {1+x^3}} \, dx-2 \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx-4 \int \frac {1}{x^4 \sqrt {1+x^3}} \, dx-5 \int \frac {1}{x \sqrt {1+x^3}} \, dx \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\frac {5}{3} \int \left (\frac {2}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {2}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx-\frac {5}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )-\frac {5 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {5 \int \frac {1+\sqrt {3}+x}{(-1+x) \sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )} \\ & = \frac {\sqrt {1+x^3}}{3 x^6}+\frac {4 \sqrt {1+x^3}}{3 x^3}-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+\frac {10}{3} \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {10}{3} \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx-\frac {10}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )+\frac {\left (20 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (2-\sqrt {3}+\left (2+\sqrt {3}\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+\frac {10}{3} \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}-\frac {\left (20 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (20 \left (2-\sqrt {3}\right )^{3/2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+2 \text {arctanh}\left (\sqrt {1+x^3}\right )-\frac {10 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (97+56 \sqrt {3},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )-\frac {\left (10 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {1}{6} \left (\frac {\sqrt {1+x^3} \left (2+5 x^3\right )}{x^6}+15 \text {arctanh}\left (\sqrt {1+x^3}\right )-10 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right )\right ) \]

[In]

Integrate[(Sqrt[1 + x^3]*(2 + 2*x^3 + x^6))/(x^7*(-1 + x^6)),x]

[Out]

((Sqrt[1 + x^3]*(2 + 5*x^3))/x^6 + 15*ArcTanh[Sqrt[1 + x^3]] - 10*Sqrt[2]*ArcTanh[Sqrt[1 + x^3]/Sqrt[2]])/6

Maple [A] (verified)

Time = 3.69 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {5 x^{6}+7 x^{3}+2}{6 x^{6} \sqrt {x^{3}+1}}-\frac {5 \ln \left (\sqrt {x^{3}+1}-1\right )}{4}-\frac {5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}\right )}{3}+\frac {5 \ln \left (\sqrt {x^{3}+1}+1\right )}{4}\)	\(68\)
trager	\(\frac {\sqrt {x^{3}+1}\, \left (5 x^{3}+2\right )}{6 x^{6}}+\frac {5 \ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+1}+2}{x^{3}}\right )}{4}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}+4 \sqrt {x^{3}+1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}{6}\)	\(96\)
default	\(\frac {-20 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}\right ) \sqrt {2}\, x^{6}-15 \ln \left (\sqrt {x^{3}+1}-1\right ) x^{6}+15 \ln \left (\sqrt {x^{3}+1}+1\right ) x^{6}+10 \sqrt {x^{3}+1}\, x^{3}+4 \sqrt {x^{3}+1}}{12 \left (\sqrt {x^{3}+1}-1\right )^{2} \left (\sqrt {x^{3}+1}+1\right )^{2}}\)	\(98\)
pseudoelliptic	\(\frac {-20 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{3}+1}\, \sqrt {2}}{2}\right ) \sqrt {2}\, x^{6}-15 \ln \left (\sqrt {x^{3}+1}-1\right ) x^{6}+15 \ln \left (\sqrt {x^{3}+1}+1\right ) x^{6}+10 \sqrt {x^{3}+1}\, x^{3}+4 \sqrt {x^{3}+1}}{12 \left (\sqrt {x^{3}+1}-1\right )^{2} \left (\sqrt {x^{3}+1}+1\right )^{2}}\)	\(98\)
elliptic	\(\text {Expression too large to display}\)	\(920\)

[In]

int((x^3+1)^(1/2)*(x^6+2*x^3+2)/x^7/(x^6-1),x,method=_RETURNVERBOSE)

[Out]

1/6*(5*x^6+7*x^3+2)/x^6/(x^3+1)^(1/2)-5/4*ln((x^3+1)^(1/2)-1)-5/3*2^(1/2)*arctanh(1/2*(x^3+1)^(1/2)*2^(1/2))+5
/4*ln((x^3+1)^(1/2)+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {10 \, \sqrt {2} x^{6} \log \left (\frac {x^{3} - 2 \, \sqrt {2} \sqrt {x^{3} + 1} + 3}{x^{3} - 1}\right ) + 15 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 15 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) + 2 \, {\left (5 \, x^{3} + 2\right )} \sqrt {x^{3} + 1}}{12 \, x^{6}} \]

[In]

integrate((x^3+1)^(1/2)*(x^6+2*x^3+2)/x^7/(x^6-1),x, algorithm="fricas")

[Out]

1/12*(10*sqrt(2)*x^6*log((x^3 - 2*sqrt(2)*sqrt(x^3 + 1) + 3)/(x^3 - 1)) + 15*x^6*log(sqrt(x^3 + 1) + 1) - 15*x
^6*log(sqrt(x^3 + 1) - 1) + 2*(5*x^3 + 2)*sqrt(x^3 + 1))/x^6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (58) = 116\).

Time = 20.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {5 \sqrt {2} \left (\log {\left (\sqrt {x^{3} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {x^{3} + 1} + \sqrt {2} \right )}\right )}{6} - \frac {5 \log {\left (\sqrt {x^{3} + 1} - 1 \right )}}{4} + \frac {5 \log {\left (\sqrt {x^{3} + 1} + 1 \right )}}{4} + \frac {5}{12 \left (\sqrt {x^{3} + 1} + 1\right )} - \frac {1}{12 \left (\sqrt {x^{3} + 1} + 1\right )^{2}} + \frac {5}{12 \left (\sqrt {x^{3} + 1} - 1\right )} + \frac {1}{12 \left (\sqrt {x^{3} + 1} - 1\right )^{2}} \]

[In]

integrate((x**3+1)**(1/2)*(x**6+2*x**3+2)/x**7/(x**6-1),x)

[Out]

5*sqrt(2)*(log(sqrt(x**3 + 1) - sqrt(2)) - log(sqrt(x**3 + 1) + sqrt(2)))/6 - 5*log(sqrt(x**3 + 1) - 1)/4 + 5*
log(sqrt(x**3 + 1) + 1)/4 + 5/(12*(sqrt(x**3 + 1) + 1)) - 1/(12*(sqrt(x**3 + 1) + 1)**2) + 5/(12*(sqrt(x**3 +
1) - 1)) + 1/(12*(sqrt(x**3 + 1) - 1)**2)

Maxima [F]

\[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 2\right )} \sqrt {x^{3} + 1}}{{\left (x^{6} - 1\right )} x^{7}} \,d x } \]

[In]

integrate((x^3+1)^(1/2)*(x^6+2*x^3+2)/x^7/(x^6-1),x, algorithm="maxima")

[Out]

integrate((x^6 + 2*x^3 + 2)*sqrt(x^3 + 1)/((x^6 - 1)*x^7), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\frac {5}{6} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{3} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{3} + 1}\right )}}\right ) + \frac {5 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{3} + 1}}{6 \, x^{6}} + \frac {5}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{4} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \]

[In]

integrate((x^3+1)^(1/2)*(x^6+2*x^3+2)/x^7/(x^6-1),x, algorithm="giac")

[Out]

5/6*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(x^3 + 1))/(sqrt(2) + sqrt(x^3 + 1))) + 1/6*(5*(x^3 + 1)^(3/2) - 3*
sqrt(x^3 + 1))/x^6 + 5/4*log(sqrt(x^3 + 1) + 1) - 5/4*log(abs(sqrt(x^3 + 1) - 1))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 724, normalized size of antiderivative = 11.49 \[ \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx=\text {Too large to display} \]

[In]

int(((x^3 + 1)^(1/2)*(2*x^3 + x^6 + 2))/(x^7*(x^6 - 1)),x)

[Out]

(5*(x^3 + 1)^(1/2))/(6*x^3) + (x^3 + 1)^(1/2)/(3*x^6) + (15*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)
/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i
)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i
)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(2*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2
)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - (5*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2
)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2
))^(1/2)*ellipticPi((3^(1/2)*1i)/4 + 3/4, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2
)/((3^(1/2)*1i)/2 - 3/2)))/(3*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 -
1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - (10*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2
- 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)
*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 + 1/2), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^
(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*((3^(1/2)*1i)/2 + 1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/
2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) + (10*((3^(1/2)*1i)/2 + 3/2)*((x
+ (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2
- x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 1/2), asin(((x +
1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*((3^(1/2)*1i)/2 - 1/2)
*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))
^(1/2))