Integrand size = 8, antiderivative size = 334 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=-4 x-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+x \log \left (\frac {1}{x^4}+x^4\right ) \]
-4*x+x*ln(1/x^4+x^4)-arctan((-2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2 -2^(1/2))^(1/2)+arctan((2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2-2^(1/ 2))^(1/2)-1/2*ln(1+x^2-x*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/2*ln(1+x^2 +x*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)-arctan((-2*x+(2-2^(1/2))^(1/2))/(2 +2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+arctan((2*x+(2-2^(1/2))^(1/2))/(2+2^(1/ 2))^(1/2))*(2+2^(1/2))^(1/2)-1/2*ln(1+x^2-x*(2+2^(1/2))^(1/2))*(2+2^(1/2)) ^(1/2)+1/2*ln(1+x^2+x*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=-4 x+8 x \operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-x^8\right )+x \log \left (\frac {1}{x^4}+x^4\right ) \]
Time = 0.61 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {3003, 27, 913, 757, 1483, 1142, 25, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \log \left (x^4+\frac {1}{x^4}\right ) \, dx\) |
| \(\Big \downarrow \) 3003 |
| \(\displaystyle x \log \left (x^4+\frac {1}{x^4}\right )-\int -\frac {4 \left (1-x^8\right )}{x^8+1}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {1-x^8}{x^8+1}dx+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle 4 \left (2 \int \frac {1}{x^8+1}dx-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 757 |
| \(\displaystyle 4 \left (2 \left (\frac {\int \frac {\sqrt {2}-x^2}{x^4-\sqrt {2} x^2+1}dx}{2 \sqrt {2}}+\frac {\int \frac {x^2+\sqrt {2}}{x^4+\sqrt {2} x^2+1}dx}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {\int \frac {\left (1-\sqrt {2}\right ) x+\sqrt {2 \left (2-\sqrt {2}\right )}}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\left (1-\sqrt {2}\right ) x}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}-\left (1+\sqrt {2}\right ) x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\left (1+\sqrt {2}\right ) x+\sqrt {2 \left (2+\sqrt {2}\right )}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2-\sqrt {2-\sqrt {2}} x+1}dx+\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2+\sqrt {2-\sqrt {2}} x+1}dx-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2-\sqrt {2+\sqrt {2}} x+1}dx-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2+\sqrt {2+\sqrt {2}} x+1}dx+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2-\sqrt {2-\sqrt {2}} x+1}dx-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2+\sqrt {2-\sqrt {2}} x+1}dx-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2-\sqrt {2+\sqrt {2}} x+1}dx+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2+\sqrt {2+\sqrt {2}} x+1}dx+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx-\sqrt {2+\sqrt {2}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {2}}\right )^2-\sqrt {2}-2}d\left (2 x-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx-\sqrt {2+\sqrt {2}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {2}}\right )^2-\sqrt {2}-2}d\left (2 x+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx-\sqrt {2-\sqrt {2}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {2}}\right )^2+\sqrt {2}-2}d\left (2 x-\sqrt {2+\sqrt {2}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx-\sqrt {2-\sqrt {2}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {2}}\right )^2+\sqrt {2}-2}d\left (2 x+\sqrt {2+\sqrt {2}}\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx+\arctan \left (\frac {2 x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx+\arctan \left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (2 \left (\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-x\right )+x \log \left (x^4+\frac {1}{x^4}\right )\) |
4*(-x + 2*(((ArcTan[(-Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/2)/(2*Sqrt[2 - Sqrt[2]]) + (A rcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2])*Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/2)/(2*Sqrt[2 - Sqrt[2]]))/(2*Sqrt[2]) + ((Ar cTan[(-Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]] - ((1 + Sqrt[2])*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/2)/(2*Sqrt[2 + Sqrt[2]]) + (ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]] + ((1 + Sqrt[2])*Log[1 + Sqrt[2 + Sqrt [2]]*x + x^2])/2)/(2*Sqrt[2 + Sqrt[2]]))/(2*Sqrt[2]))) + x*Log[x^(-4) + x^ 4]
3.1.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[r/(2*Sqrt[2]*a) Int[(Sqrt[2]*r - s*x^(n/4))/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] + Simp[r/(2*S qrt[2]*a) Int[(Sqrt[2]*r + s*x^(n/4))/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^ (n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && GtQ[a/b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Simp[b*n*p Int[SimplifyIntegrand[x*(a + b*Log[c* RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] && Ra tionalFunctionQ[RFx, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.10
| method | result | size |
| risch | \(x \ln \left (\frac {1}{x^{4}}+x^{4}\right )-4 x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )\) | \(34\) |
| parts | \(x \ln \left (\frac {1}{x^{4}}+x^{4}\right )-4 x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )\) | \(34\) |
| default | \(x \ln \left (\frac {x^{8}+1}{x^{4}}\right )-4 x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )\) | \(36\) |
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.45 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=x \log \left (\frac {x^{8} + 1}{x^{4}}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (2 \, x + \left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (2 \, x - \left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (2 \, x + \left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (2 \, x - \left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}}\right ) + \left (-1\right )^{\frac {1}{8}} \log \left (x + \left (-1\right )^{\frac {1}{8}}\right ) + i \, \left (-1\right )^{\frac {1}{8}} \log \left (x + i \, \left (-1\right )^{\frac {1}{8}}\right ) - i \, \left (-1\right )^{\frac {1}{8}} \log \left (x - i \, \left (-1\right )^{\frac {1}{8}}\right ) - \left (-1\right )^{\frac {1}{8}} \log \left (x - \left (-1\right )^{\frac {1}{8}}\right ) - 4 \, x \]
x*log((x^8 + 1)/x^4) + (1/2*I + 1/2)*sqrt(2)*(-1)^(1/8)*log(2*x + (I + 1)* sqrt(2)*(-1)^(1/8)) - (1/2*I - 1/2)*sqrt(2)*(-1)^(1/8)*log(2*x - (I - 1)*s qrt(2)*(-1)^(1/8)) + (1/2*I - 1/2)*sqrt(2)*(-1)^(1/8)*log(2*x + (I - 1)*sq rt(2)*(-1)^(1/8)) - (1/2*I + 1/2)*sqrt(2)*(-1)^(1/8)*log(2*x - (I + 1)*sqr t(2)*(-1)^(1/8)) + (-1)^(1/8)*log(x + (-1)^(1/8)) + I*(-1)^(1/8)*log(x + I *(-1)^(1/8)) - I*(-1)^(1/8)*log(x - I*(-1)^(1/8)) - (-1)^(1/8)*log(x - (-1 )^(1/8)) - 4*x
Time = 1.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.08 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=x \log {\left (x^{4} + \frac {1}{x^{4}} \right )} - 4 x - \operatorname {RootSum} {\left (t^{8} + 1, \left ( t \mapsto t \log {\left (- t + x \right )} \right )\right )} \]
\[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=\int { \log \left (x^{4} + \frac {1}{x^{4}}\right ) \,d x } \]
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.74 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=x \log \left (x^{4} + \frac {1}{x^{4}}\right ) + \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) - 4 \, x \]
x*log(x^4 + 1/x^4) + sqrt(sqrt(2) + 2)*arctan((2*x + sqrt(-sqrt(2) + 2))/s qrt(sqrt(2) + 2)) + sqrt(sqrt(2) + 2)*arctan((2*x - sqrt(-sqrt(2) + 2))/sq rt(sqrt(2) + 2)) + sqrt(-sqrt(2) + 2)*arctan((2*x + sqrt(sqrt(2) + 2))/sqr t(-sqrt(2) + 2)) + sqrt(-sqrt(2) + 2)*arctan((2*x - sqrt(sqrt(2) + 2))/sqr t(-sqrt(2) + 2)) + 1/2*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1 ) - 1/2*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) + 1/2*sqrt(-s qrt(2) + 2)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) - 1/2*sqrt(-sqrt(2) + 2)*l og(x^2 - x*sqrt(-sqrt(2) + 2) + 1) - 4*x
Time = 0.27 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.94 \[ \int \log \left (\frac {1}{x^4}+x^4\right ) \, dx=x\,\ln \left (\frac {1}{x^4}+x^4\right )-4\,x+\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}\right )\,\left (\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}-\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}+\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )+\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{2}-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}-\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{2}-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \]
x*log(1/x^4 + x^4) - 4*x + atan((x*(- 2^(1/2) - 2)^(1/2)*2097152i)/(209715 2*(2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2)^(1/2) + 2097152*2^(1/2)) - (x*(2 - 2 ^(1/2))^(1/2)*2097152i)/(2097152*(2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2)^(1/2) + 2097152*2^(1/2)))*((- 2^(1/2) - 2)^(1/2)*1i - (2 - 2^(1/2))^(1/2)*1i) - atan((x*(2^(1/2) - 2)^(1/2)*2097152i)/(2097152*2^(1/2) + 2097152*(2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)) + (x*(2^(1/2) + 2)^(1/2)*2097152i)/(20971 52*2^(1/2) + 2097152*(2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)))*((2^(1/2) - 2)^(1/2)*1i + (2^(1/2) + 2)^(1/2)*1i) + atan(x*(2^(1/2) + 2)^(1/2)*(1/2 + 1i/2) - (2^(1/2)*x*(2^(1/2) + 2)^(1/2))/2)*((2^(1/2)*1i)/2 - (1/2 + 1i/2) )*(2^(1/2) + 2)^(1/2)*2i - atan(x*(2^(1/2) + 2)^(1/2)*(1/2 - 1i/2) + (2^(1 /2)*x*(2^(1/2) + 2)^(1/2)*1i)/2)*(2^(1/2)/2 - (1/2 - 1i/2))*(2^(1/2) + 2)^ (1/2)*2i