\(\int \frac {-1+x^{12}}{\sqrt {1+x^4} (1+x^{12})} \, dx\) [891]

Optimal result

Integrand size = 22, antiderivative size = 67 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}} \]

[Out]

-1/3*x/(x^4+1)^(1/2)-1/9*arctan(3^(1/4)*x/(x^4+1)^(1/2))*3^(3/4)-1/9*arctanh(3^(1/4)*x/(x^4+1)^(1/2))*3^(3/4)

Rubi [C] (warning: unable to verify)

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

Time = 2.35 (sec) , antiderivative size = 1382, normalized size of antiderivative = 20.63, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6857, 226, 2098, 425, 537, 418, 1231, 1721} \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {2 x}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {2 x}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {2 i \sqrt [4]{2} \arctan \left (\frac {\sqrt {3-i \sqrt {3}} x}{\sqrt [4]{2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{\sqrt {3} \left (1-i \sqrt {3}\right )^{3/4} \left (3-i \sqrt {3}\right )^{3/2}}-\frac {2 i \sqrt [4]{2} \arctan \left (\frac {\sqrt {-3+i \sqrt {3}} x}{\sqrt [4]{2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{\sqrt {3} \left (1-i \sqrt {3}\right )^{3/4} \left (-3+i \sqrt {3}\right )^{3/2}}+\frac {2 i \sqrt [4]{2} \arctan \left (\frac {\sqrt {-3-i \sqrt {3}} x}{\sqrt [4]{2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{\sqrt {3} \left (-3-i \sqrt {3}\right )^{3/2} \left (1+i \sqrt {3}\right )^{3/4}}-\frac {2 i \sqrt [4]{2} \arctan \left (\frac {\sqrt {3+i \sqrt {3}} x}{\sqrt [4]{2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{\sqrt {3} \left (1+i \sqrt {3}\right )^{3/4} \left (3+i \sqrt {3}\right )^{3/2}}-\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (\frac {1}{6}+\frac {i}{6}\right ) \left (1-\sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\left (i+\sqrt {3}\right ) \sqrt {x^4+1}}-\frac {\left (\frac {1}{6}-\frac {i}{6}\right ) \left (1-\sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\left (i-\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {x^4+1}}+\frac {\left (3+\sqrt {3} \left (3 i+2 i \sqrt {2-2 i \sqrt {3}}\right )\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right )^2}{4 \sqrt {2 \left (1-i \sqrt {3}\right )}},2 \arctan (x),\frac {1}{2}\right )}{12 \left (3-i \sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (3 i-\sqrt {3} \left (3-2 \sqrt {2-2 i \sqrt {3}}\right )\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right )^2}{4 \sqrt {2 \left (1-i \sqrt {3}\right )}},2 \arctan (x),\frac {1}{2}\right )}{12 \left (3 i+\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (3 i+3 \sqrt {3}+2 \sqrt {6+6 i \sqrt {3}}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right )^2}{4 \sqrt {2 \left (1+i \sqrt {3}\right )}},2 \arctan (x),\frac {1}{2}\right )}{12 \left (3 i-\sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (3 i+\sqrt {3} \left (3-2 \sqrt {2+2 i \sqrt {3}}\right )\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right )^2}{4 \sqrt {2 \left (1+i \sqrt {3}\right )}},2 \arctan (x),\frac {1}{2}\right )}{12 \left (3 i-\sqrt {3}\right ) \sqrt {x^4+1}} \]

[In]

Int[(-1 + x^12)/(Sqrt[1 + x^4]*(1 + x^12)),x]

[Out]

(2*x)/(Sqrt[3]*(3*I - Sqrt[3])*Sqrt[1 + x^4]) - (2*x)/(Sqrt[3]*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((2*I)*2^(1/4)
*ArcTan[(Sqrt[3 - I*Sqrt[3]]*x)/((2*(1 - I*Sqrt[3]))^(1/4)*Sqrt[1 + x^4])])/(Sqrt[3]*(1 - I*Sqrt[3])^(3/4)*(3
- I*Sqrt[3])^(3/2)) - ((2*I)*2^(1/4)*ArcTan[(Sqrt[-3 + I*Sqrt[3]]*x)/((2*(1 - I*Sqrt[3]))^(1/4)*Sqrt[1 + x^4])
])/(Sqrt[3]*(1 - I*Sqrt[3])^(3/4)*(-3 + I*Sqrt[3])^(3/2)) + ((2*I)*2^(1/4)*ArcTan[(Sqrt[-3 - I*Sqrt[3]]*x)/((2
*(1 + I*Sqrt[3]))^(1/4)*Sqrt[1 + x^4])])/(Sqrt[3]*(-3 - I*Sqrt[3])^(3/2)*(1 + I*Sqrt[3])^(3/4)) - ((2*I)*2^(1/
4)*ArcTan[(Sqrt[3 + I*Sqrt[3]]*x)/((2*(1 + I*Sqrt[3]))^(1/4)*Sqrt[1 + x^4])])/(Sqrt[3]*(1 + I*Sqrt[3])^(3/4)*(
3 + I*Sqrt[3])^(3/2)) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(2*Sqrt[1 + x^4])
+ ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(Sqrt[3]*(3*I - Sqrt[3])*Sqrt[1 + x^4])
- ((1/6 - I/6)*(1 - Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/((I - Sqrt[3])
*Sqrt[1 + x^4]) + ((1/6 + I/6)*(1 - Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2]
)/((I + Sqrt[3])*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(Sqrt[3]
*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((1 + 1/Sqrt[(1 - I*Sqrt[3])/2])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Ellip
ticF[2*ArcTan[x], 1/2])/(Sqrt[3]*(3*I - Sqrt[3])*Sqrt[1 + x^4]) - ((1 + 1/Sqrt[(1 + I*Sqrt[3])/2])*(1 + x^2)*S
qrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(Sqrt[3]*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((3 + Sqrt[3
]*(3*I + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4*(Sqrt[2] - Sqrt
[1 - I*Sqrt[3]])^2/Sqrt[2*(1 - I*Sqrt[3])], 2*ArcTan[x], 1/2])/(12*(3 - I*Sqrt[3])*Sqrt[1 + x^4]) + ((3*I - Sq
rt[3]*(3 - 2*Sqrt[2 - (2*I)*Sqrt[3]]))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(Sqrt[2] + Sqrt[1 - I*
Sqrt[3]])^2/(4*Sqrt[2*(1 - I*Sqrt[3])]), 2*ArcTan[x], 1/2])/(12*(3*I + Sqrt[3])*Sqrt[1 + x^4]) + ((3*I + 3*Sqr
t[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4*(Sqrt[2] - Sqrt[1 + I*
Sqrt[3]])^2/Sqrt[2*(1 + I*Sqrt[3])], 2*ArcTan[x], 1/2])/(12*(3*I - Sqrt[3])*Sqrt[1 + x^4]) + ((3*I + Sqrt[3]*(
3 - 2*Sqrt[2 + (2*I)*Sqrt[3]]))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(Sqrt[2] + Sqrt[1 + I*Sqrt[3]
])^2/(4*Sqrt[2*(1 + I*Sqrt[3])]), 2*ArcTan[x], 1/2])/(12*(3*I - Sqrt[3])*Sqrt[1 + x^4])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 418

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

Rule 425

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

Rule 537

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

Rule 1231

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 1721

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))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], 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 2098

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {1+x^4}}-\frac {2}{\sqrt {1+x^4} \left (1+x^{12}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx\right )+\int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-2 \int \left (\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}-2 x^4\right ) \left (1+x^4\right )^{3/2}}+\frac {2 i}{\sqrt {3} \left (1+x^4\right )^{3/2} \left (-1+i \sqrt {3}+2 x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {(4 i) \int \frac {1}{\left (1+i \sqrt {3}-2 x^4\right ) \left (1+x^4\right )^{3/2}} \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {1}{\left (1+x^4\right )^{3/2} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx}{\sqrt {3}} \\ & = \frac {2 x}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2 x}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {(2 i) \int \frac {5-i \sqrt {3}-2 x^4}{\sqrt {1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(2 i) \int \frac {-5-i \sqrt {3}+2 x^4}{\left (1+i \sqrt {3}-2 x^4\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = \frac {2 x}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2 x}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {(8 i) \int \frac {1}{\sqrt {1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )}-\frac {(8 i) \int \frac {1}{\left (1+i \sqrt {3}-2 x^4\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = \frac {2 x}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2 x}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {1}{3} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx \\ & = \frac {2 x}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2 x}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (\left (\frac {1}{3}-\frac {i}{3}\right ) \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{i-\sqrt {3}}--\frac {\left (\left (\frac {1}{3}+\frac {i}{3}\right ) \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{i+\sqrt {3}}-\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 \left (1-\frac {2 i}{i-\sqrt {3}}\right )}+\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \int \frac {1+x^2}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx}{3 \sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )} \left (1-\frac {2}{1+i \sqrt {3}}\right )}-\frac {\left (2-\sqrt {2-2 i \sqrt {3}}\right ) \int \frac {1+x^2}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}-\frac {\left (2+\sqrt {2-2 i \sqrt {3}}\right ) \int \frac {1+x^2}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}-\frac {\left (2-\sqrt {2+2 i \sqrt {3}}\right ) \int \frac {1+x^2}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1+x^4}} \, dx}{3 \left (1-i \sqrt {3}\right )}-\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 \left (1-\frac {2 i}{i+\sqrt {3}}\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}} \]

[In]

Integrate[(-1 + x^12)/(Sqrt[1 + x^4]*(1 + x^12)),x]

[Out]

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

Maple [A] (verified)

Time = 21.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {x}{3 \sqrt {x^{4}+1}}-\frac {3^{\frac {3}{4}} \left (-2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}+1}}{3 x}\right )+\ln \left (\frac {-3^{\frac {1}{4}} x -\sqrt {x^{4}+1}}{3^{\frac {1}{4}} x -\sqrt {x^{4}+1}}\right )\right )}{18}\)	\(71\)
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {x^{4}+1}}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}+1}}{3 x}\right )-\ln \left (\frac {\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}+\frac {3^{\frac {1}{4}} \sqrt {2}}{2}}{\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\frac {3^{\frac {1}{4}} \sqrt {2}}{2}}\right )\right )}{18}\right ) \sqrt {2}}{2}\)	\(101\)
default	\(\frac {-\left (x^{4}+1\right ) \left (\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )+\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )\right ) 3^{\frac {3}{4}}-6 \sqrt {x^{4}+1}\, x}{18 \left (x^{2}+x \sqrt {2}+1\right ) \left (x^{2}-x \sqrt {2}+1\right )}\)	\(178\)
pseudoelliptic	\(\frac {-\left (x^{4}+1\right ) \left (\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )+\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )\right ) 3^{\frac {3}{4}}-6 \sqrt {x^{4}+1}\, x}{18 \left (x^{2}+x \sqrt {2}+1\right ) \left (x^{2}-x \sqrt {2}+1\right )}\)	\(178\)
trager	\(-\frac {x}{3 \sqrt {x^{4}+1}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \sqrt {x^{4}+1}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}-3 x^{4}-3}\right )}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}+18 \sqrt {x^{4}+1}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{3 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+3}\right )}{18}\)	\(195\)

[In]

int((x^12-1)/(x^4+1)^(1/2)/(x^12+1),x,method=_RETURNVERBOSE)

[Out]

-1/3*x/(x^4+1)^(1/2)-1/18*3^(3/4)*(-2*arctan(1/3*3^(3/4)/x*(x^4+1)^(1/2))+ln((-3^(1/4)*x-(x^4+1)^(1/2))/(3^(1/
4)*x-(x^4+1)^(1/2))))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 326, normalized size of antiderivative = 4.87 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (-\frac {3^{\frac {3}{4}} {\left (x^{8} + 5 \, x^{4} + 1\right )} + 6 \, {\left (x^{5} + \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} + 6 \cdot 3^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (x^{8} + 5 \, x^{4} + 1\right )} - 6 \, {\left (x^{5} + \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} + 6 \cdot 3^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (i \, x^{8} + 5 i \, x^{4} + i\right )} - 6 \, {\left (x^{5} - \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} - 6 \cdot 3^{\frac {1}{4}} {\left (i \, x^{6} + i \, x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (-i \, x^{8} - 5 i \, x^{4} - i\right )} - 6 \, {\left (x^{5} - \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} - 6 \cdot 3^{\frac {1}{4}} {\left (-i \, x^{6} - i \, x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) + 12 \, \sqrt {x^{4} + 1} x}{36 \, {\left (x^{4} + 1\right )}} \]

[In]

integrate((x^12-1)/(x^4+1)^(1/2)/(x^12+1),x, algorithm="fricas")

[Out]

-1/36*(3^(3/4)*(x^4 + 1)*log(-(3^(3/4)*(x^8 + 5*x^4 + 1) + 6*(x^5 + sqrt(3)*x^3 + x)*sqrt(x^4 + 1) + 6*3^(1/4)
*(x^6 + x^2))/(x^8 - x^4 + 1)) - 3^(3/4)*(x^4 + 1)*log((3^(3/4)*(x^8 + 5*x^4 + 1) - 6*(x^5 + sqrt(3)*x^3 + x)*
sqrt(x^4 + 1) + 6*3^(1/4)*(x^6 + x^2))/(x^8 - x^4 + 1)) - 3^(3/4)*(I*x^4 + I)*log((3^(3/4)*(I*x^8 + 5*I*x^4 +
I) - 6*(x^5 - sqrt(3)*x^3 + x)*sqrt(x^4 + 1) - 6*3^(1/4)*(I*x^6 + I*x^2))/(x^8 - x^4 + 1)) - 3^(3/4)*(-I*x^4 -
I)*log((3^(3/4)*(-I*x^8 - 5*I*x^4 - I) - 6*(x^5 - sqrt(3)*x^3 + x)*sqrt(x^4 + 1) - 6*3^(1/4)*(-I*x^6 - I*x^2)
)/(x^8 - x^4 + 1)) + 12*sqrt(x^4 + 1)*x)/(x^4 + 1)

Sympy [F]

\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\left (x^{4} + 1\right )^{\frac {3}{2}} \left (x^{8} - x^{4} + 1\right )}\, dx \]

[In]

integrate((x**12-1)/(x**4+1)**(1/2)/(x**12+1),x)

[Out]

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

Maxima [F]

\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int { \frac {x^{12} - 1}{{\left (x^{12} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^12-1)/(x^4+1)^(1/2)/(x^12+1),x, algorithm="maxima")

[Out]

integrate((x^12 - 1)/((x^12 + 1)*sqrt(x^4 + 1)), x)

Giac [F]

\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int { \frac {x^{12} - 1}{{\left (x^{12} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^12-1)/(x^4+1)^(1/2)/(x^12+1),x, algorithm="giac")

[Out]

integrate((x^12 - 1)/((x^12 + 1)*sqrt(x^4 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int \frac {x^{12}-1}{\sqrt {x^4+1}\,\left (x^{12}+1\right )} \,d x \]

[In]

int((x^12 - 1)/((x^4 + 1)^(1/2)*(x^12 + 1)),x)

[Out]

int((x^12 - 1)/((x^4 + 1)^(1/2)*(x^12 + 1)), x)