3.12.29 \(\int \frac {(1+x^6) (-1+x^3+x^6) \sqrt {1+x^{12}}}{x^7 (-1-x^3+x^6)} \, dx\)

Optimal. Leaf size=84 \[ \log \left (\sqrt {x^{12}+1}+x^6-1\right )+\frac {\sqrt {x^{12}+1} \left (x^6+4 x^3-1\right )}{6 x^6}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^{12}+1}+x^6-x^3-1}\right )}{\sqrt {3}}-3 \log (x) \]

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Rubi [C]  time = 2.34, antiderivative size = 666, normalized size of antiderivative = 7.93, number of steps used = 45, number of rules used = 22, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6728, 275, 277, 215, 305, 220, 1196, 266, 50, 63, 207, 6715, 1729, 1209, 1198, 1217, 1707, 1248, 735, 844, 725, 206} \begin {gather*} -\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {x^{12}+1}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {x^{12}+1}+\frac {\sqrt {x^{12}+1}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^{12}+1}\right )+\frac {1}{6} \left (1+\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \left (1-\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \sinh ^{-1}\left (x^6\right )-\frac {\sqrt {x^{12}+1}}{6 x^6}+\frac {\tanh ^{-1}\left (\frac {\left (3-\sqrt {5}\right ) x^6+2}{\sqrt {6 \left (3-\sqrt {5}\right )} \sqrt {x^{12}+1}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\left (3+\sqrt {5}\right ) x^6+2}{\sqrt {6 \left (3+\sqrt {5}\right )} \sqrt {x^{12}+1}}\right )}{\sqrt {3}}+\frac {2 \sqrt {x^{12}+1}}{3 x^3}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^{12}+1}}\right )}{\sqrt {3}}+\frac {\left (5+\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {x^{12}+1}}-\frac {\left (3+\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5+\sqrt {5}\right ) \sqrt {x^{12}+1}}+\frac {\left (5-\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {x^{12}+1}}-\frac {\left (3-\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5-\sqrt {5}\right ) \sqrt {x^{12}+1}}-\frac {2 \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {x^{12}+1}}+\frac {\left (3+\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5+3 \sqrt {5}\right ) \sqrt {x^{12}+1}}+\frac {\left (3-\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5-3 \sqrt {5}\right ) \sqrt {x^{12}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^6)*(-1 + x^3 + x^6)*Sqrt[1 + x^12])/(x^7*(-1 - x^3 + x^6)),x]

[Out]

Sqrt[1 + x^12]/2 - ((1 - Sqrt[5])*Sqrt[1 + x^12])/6 - ((1 + Sqrt[5])*Sqrt[1 + x^12])/6 - Sqrt[1 + x^12]/(6*x^6
) + (2*Sqrt[1 + x^12])/(3*x^3) + ArcSinh[x^6]/6 + ((1 - Sqrt[5])*ArcSinh[x^6])/6 + ((1 + Sqrt[5])*ArcSinh[x^6]
)/6 - (2*ArcTanh[(Sqrt[3]*x^3)/Sqrt[1 + x^12]])/Sqrt[3] + ArcTanh[(2 + (3 - Sqrt[5])*x^6)/(Sqrt[6*(3 - Sqrt[5]
)]*Sqrt[1 + x^12])]/Sqrt[3] - ArcTanh[(2 + (3 + Sqrt[5])*x^6)/(Sqrt[6*(3 + Sqrt[5])]*Sqrt[1 + x^12])]/Sqrt[3]
- ArcTanh[Sqrt[1 + x^12]]/2 - (2*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])/(3*Sqrt
[1 + x^12]) - ((3 - Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])/((5 - Sqrt[
5])*Sqrt[1 + x^12]) + ((5 - Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])/(6*
Sqrt[1 + x^12]) - ((3 + Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])/((5 + S
qrt[5])*Sqrt[1 + x^12]) + ((5 + Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])
/(6*Sqrt[1 + x^12]) + ((3 - Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticPi[5/4, 2*ArcTan[x^3], 1/2
])/(2*(5 - 3*Sqrt[5])*Sqrt[1 + x^12]) + ((3 + Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticPi[5/4,
2*ArcTan[x^3], 1/2])/(2*(5 + 3*Sqrt[5])*Sqrt[1 + x^12])

Rule 50

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

Rule 63

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 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 207

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

Rule 215

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

Rule 220

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

Rule 266

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 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 305

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

Rule 725

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

Rule 735

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

Rule 844

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

Rule 1196

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

Rule 1198

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

Rule 1209

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

Rule 1217

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

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1707

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

Rule 1729

Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x
], x] - Dist[e, Int[(x*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]

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^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx &=\int \left (\frac {\sqrt {1+x^{12}}}{x^7}-\frac {2 \sqrt {1+x^{12}}}{x^4}+\frac {3 \sqrt {1+x^{12}}}{x}-\frac {2 x^2 \left (-3+x^3\right ) \sqrt {1+x^{12}}}{-1-x^3+x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^{12}}}{x^4} \, dx\right )-2 \int \frac {x^2 \left (-3+x^3\right ) \sqrt {1+x^{12}}}{-1-x^3+x^6} \, dx+3 \int \frac {\sqrt {1+x^{12}}}{x} \, dx+\int \frac {\sqrt {1+x^{12}}}{x^7} \, dx\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,x^6\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^{12}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^2} \, dx,x,x^3\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {(-3+x) \sqrt {1+x^4}}{-1-x+x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^6\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^{12}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \left (\frac {\left (1-\sqrt {5}\right ) \sqrt {1+x^4}}{-1-\sqrt {5}+2 x}+\frac {\left (1+\sqrt {5}\right ) \sqrt {1+x^4}}{-1+\sqrt {5}+2 x}\right ) \, dx,x,x^3\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^{12}}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^3\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,x^3\right )-\frac {1}{3} \left (2 \left (1-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{-1-\sqrt {5}+2 x} \, dx,x,x^3\right )-\frac {1}{3} \left (2 \left (1+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{-1+\sqrt {5}+2 x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {8}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{\left (-1-\sqrt {5}\right )^2-4 x^2} \, dx,x,x^3\right )-\frac {8}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{\left (-1+\sqrt {5}\right )^2-4 x^2} \, dx,x,x^3\right )+\frac {1}{3} \left (4 \left (1-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {1+x^4}}{\left (-1-\sqrt {5}\right )^2-4 x^2} \, dx,x,x^3\right )+\frac {1}{3} \left (4 \left (1+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {1+x^4}}{\left (-1+\sqrt {5}\right )^2-4 x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {\left (-1-\sqrt {5}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx,x,x^3\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {\left (-1+\sqrt {5}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx,x,x^3\right )+\frac {1}{3} \left (2 \left (1-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1-\sqrt {5}\right )^2-4 x} \, dx,x,x^6\right )-\left (4 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,x^3\right )+\frac {1}{3} \left (2 \left (1+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1+\sqrt {5}\right )^2-4 x} \, dx,x,x^6\right )-\left (4 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-2 \left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,x^3\right )\right )+\frac {1}{6} \left (-1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {-4-2 \left (3-\sqrt {5}\right ) x}{\left (\left (-1+\sqrt {5}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^6\right )+\frac {1}{3} \left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^3\right )+\frac {1}{6} \left (-1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {-4-2 \left (3+\sqrt {5}\right ) x}{\left (\left (-1-\sqrt {5}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^6\right )-\frac {\left (2 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^3\right )}{5+\sqrt {5}}-\frac {\left (8 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1-\sqrt {5}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,x^3\right )}{5+\sqrt {5}}+\frac {1}{3} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^3\right )-\frac {\left (4 \left (3-\sqrt {5}\right ) \left (-4+\left (-1+\sqrt {5}\right )^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^3\right )}{-16+\left (-1+\sqrt {5}\right )^4}-\frac {\left (16 \left (3-\sqrt {5}\right ) \left (-4+\left (-1+\sqrt {5}\right )^2\right )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (\left (-1+\sqrt {5}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx,x,x^3\right )}{-16+\left (-1+\sqrt {5}\right )^4}\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{\sqrt {3}}+\frac {\sqrt {\frac {5}{3}} \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{5-3 \sqrt {5}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-2 \left (-\frac {2 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}\right )-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5-\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}-\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5+\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}+\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5-3 \sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5+3 \sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {1}{6} \left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^6\right )-\left (2 \left (1-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+\sqrt {5}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^6\right )+\frac {1}{6} \left (1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^6\right )-\left (2 \left (1+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1-\sqrt {5}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \left (1-\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \left (1+\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{\sqrt {3}}+\frac {\sqrt {\frac {5}{3}} \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{5-3 \sqrt {5}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-2 \left (-\frac {2 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}\right )-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5-\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}-\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5+\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}+\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5-3 \sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5+3 \sqrt {5}\right ) \sqrt {1+x^{12}}}+\left (2 \left (1-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1+\sqrt {5}\right )^4-x^2} \, dx,x,\frac {-4-\left (-1+\sqrt {5}\right )^2 x^6}{\sqrt {1+x^{12}}}\right )+\left (2 \left (1+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1-\sqrt {5}\right )^4-x^2} \, dx,x,-\frac {2 \left (2+\left (3+\sqrt {5}\right ) x^6\right )}{\sqrt {1+x^{12}}}\right )\\ &=\frac {\sqrt {1+x^{12}}}{2}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {1+x^{12}}-\frac {\sqrt {1+x^{12}}}{6 x^6}+\frac {2 \sqrt {1+x^{12}}}{3 x^3}-\frac {4 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {1}{6} \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \left (1-\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )+\frac {1}{6} \left (1+\sqrt {5}\right ) \sinh ^{-1}\left (x^6\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{\sqrt {3}}+\frac {\sqrt {\frac {5}{3}} \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {1+x^{12}}}\right )}{5-3 \sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {2+\left (3-\sqrt {5}\right ) x^6}{\sqrt {6 \left (3-\sqrt {5}\right )} \sqrt {1+x^{12}}}\right )}{\sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2+\left (3+\sqrt {5}\right ) x^6}{\sqrt {6 \left (3+\sqrt {5}\right )} \sqrt {1+x^{12}}}\right )}{\sqrt {3}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^{12}}\right )+\frac {4 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-2 \left (-\frac {2 x^3 \sqrt {1+x^{12}}}{3 \left (1+x^6\right )}+\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}\right )-\frac {2 \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{3 \sqrt {1+x^{12}}}-\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5-\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}-\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{\left (5+\sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (5+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^{12}}}+\frac {\left (3-\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5-3 \sqrt {5}\right ) \sqrt {1+x^{12}}}+\frac {\left (3+\sqrt {5}\right ) \left (1+x^6\right ) \sqrt {\frac {1+x^{12}}{\left (1+x^6\right )^2}} \Pi \left (\frac {5}{4};2 \tan ^{-1}\left (x^3\right )|\frac {1}{2}\right )}{2 \left (5+3 \sqrt {5}\right ) \sqrt {1+x^{12}}}\\ \end {align*}

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Mathematica [F]  time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^6)*(-1 + x^3 + x^6)*Sqrt[1 + x^12])/(x^7*(-1 - x^3 + x^6)),x]

[Out]

Integrate[((1 + x^6)*(-1 + x^3 + x^6)*Sqrt[1 + x^12])/(x^7*(-1 - x^3 + x^6)), x]

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IntegrateAlgebraic [A]  time = 37.05, size = 84, normalized size = 1.00 \begin {gather*} \frac {\left (-1+4 x^3+x^6\right ) \sqrt {1+x^{12}}}{6 x^6}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{-1-x^3+x^6+\sqrt {1+x^{12}}}\right )}{\sqrt {3}}-3 \log (x)+\log \left (-1+x^6+\sqrt {1+x^{12}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^6)*(-1 + x^3 + x^6)*Sqrt[1 + x^12])/(x^7*(-1 - x^3 + x^6)),x]

[Out]

((-1 + 4*x^3 + x^6)*Sqrt[1 + x^12])/(6*x^6) - (4*ArcTanh[(Sqrt[3]*x^3)/(-1 - x^3 + x^6 + Sqrt[1 + x^12])])/Sqr
t[3] - 3*Log[x] + Log[-1 + x^6 + Sqrt[1 + x^12]]

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fricas [A]  time = 0.56, size = 120, normalized size = 1.43 \begin {gather*} \frac {2 \, \sqrt {3} x^{6} \log \left (\frac {2 \, x^{12} + 2 \, x^{9} + x^{6} - 2 \, x^{3} - \sqrt {3} \sqrt {x^{12} + 1} {\left (x^{6} + 2 \, x^{3} - 1\right )} + 2}{x^{12} - 2 \, x^{9} - x^{6} + 2 \, x^{3} + 1}\right ) + 6 \, x^{6} \log \left (\frac {x^{6} + \sqrt {x^{12} + 1} - 1}{x^{3}}\right ) + \sqrt {x^{12} + 1} {\left (x^{6} + 4 \, x^{3} - 1\right )}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)*(x^12+1)^(1/2)/x^7/(x^6-x^3-1),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(3)*x^6*log((2*x^12 + 2*x^9 + x^6 - 2*x^3 - sqrt(3)*sqrt(x^12 + 1)*(x^6 + 2*x^3 - 1) + 2)/(x^12 - 2
*x^9 - x^6 + 2*x^3 + 1)) + 6*x^6*log((x^6 + sqrt(x^12 + 1) - 1)/x^3) + sqrt(x^12 + 1)*(x^6 + 4*x^3 - 1))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{12} + 1} {\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)*(x^12+1)^(1/2)/x^7/(x^6-x^3-1),x, algorithm="giac")

[Out]

integrate(sqrt(x^12 + 1)*(x^6 + x^3 - 1)*(x^6 + 1)/((x^6 - x^3 - 1)*x^7), x)

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maple [C]  time = 1.94, size = 103, normalized size = 1.23

method	result	size
trager	\(\frac {\left (x^{6}+4 x^{3}-1\right ) \sqrt {x^{12}+1}}{6 x^{6}}+\ln \left (\frac {-1+x^{6}+\sqrt {x^{12}+1}}{x^{3}}\right )+\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}-3\right )-3 \sqrt {x^{12}+1}}{x^{6}-x^{3}-1}\right )}{3}\)	\(103\)
risch	\(\frac {4 x^{15}-x^{12}+4 x^{3}-1}{6 x^{6} \sqrt {x^{12}+1}}+\frac {\sqrt {x^{12}+1}}{6}-\ln \left (\frac {-x^{6}+\sqrt {x^{12}+1}+1}{x^{3}}\right )-\frac {2 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {x^{12}+1}}{x^{6}-x^{3}-1}\right )}{3}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)*(x^6+x^3-1)*(x^12+1)^(1/2)/x^7/(x^6-x^3-1),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^6+4*x^3-1)*(x^12+1)^(1/2)/x^6+ln((-1+x^6+(x^12+1)^(1/2))/x^3)+2/3*RootOf(_Z^2-3)*ln(-(RootOf(_Z^2-3)*x^
6+2*RootOf(_Z^2-3)*x^3-RootOf(_Z^2-3)-3*(x^12+1)^(1/2))/(x^6-x^3-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{12} + 1} {\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)*(x^12+1)^(1/2)/x^7/(x^6-x^3-1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^12 + 1)*(x^6 + x^3 - 1)*(x^6 + 1)/((x^6 - x^3 - 1)*x^7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^6+1\right )\,\sqrt {x^{12}+1}\,\left (x^6+x^3-1\right )}{x^7\,\left (-x^6+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 + 1)*(x^12 + 1)^(1/2)*(x^3 + x^6 - 1))/(x^7*(x^3 - x^6 + 1)),x)

[Out]

int(-((x^6 + 1)*(x^12 + 1)^(1/2)*(x^3 + x^6 - 1))/(x^7*(x^3 - x^6 + 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((x**6+1)*(x**6+x**3-1)*(x**12+1)**(1/2)/x**7/(x**6-x**3-1),x)

[Out]

Timed out

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