Integrand size = 11, antiderivative size = 344 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=-\frac {1}{x}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right ) \]
-1/x-1/16*ln(1+x^2-x*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/16*ln(1+x^2+x* (2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/4*arctan((-2*x+(2-2^(1/2))^(1/2))/( 2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/4*arctan((2*x+(2-2^(1/2))^(1/2))/( 2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/16*ln(1+x^2-x*(2+2^(1/2))^(1/2))*( 2+2^(1/2))^(1/2)+1/16*ln(1+x^2+x*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/4* arctan((-2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)-1/4 *arctan((2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)
Time = 0.01 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=-\frac {1}{x}-\frac {1}{4} \arctan \left (\sec \left (\frac {\pi }{8}\right ) \left (x-\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{4} \arctan \left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right )+\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right )-\frac {1}{4} \arctan \left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{4} \arctan \left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{8} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{8} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right ) \]
-x^(-1) - (ArcTan[Sec[Pi/8]*(x - Sin[Pi/8])]*Cos[Pi/8])/4 - (ArcTan[Sec[Pi /8]*(x + Sin[Pi/8])]*Cos[Pi/8])/4 - (Cos[Pi/8]*Log[1 + x^2 - 2*x*Cos[Pi/8] ])/8 + (Cos[Pi/8]*Log[1 + x^2 + 2*x*Cos[Pi/8]])/8 - (ArcTan[(x - Cos[Pi/8] )*Csc[Pi/8]]*Sin[Pi/8])/4 - (ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*Sin[Pi/8])/ 4 - (Log[1 + x^2 - 2*x*Sin[Pi/8]]*Sin[Pi/8])/8 + (Log[1 + x^2 + 2*x*Sin[Pi /8]]*Sin[Pi/8])/8
Time = 0.62 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {847, 828, 1442, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (x^8+1\right )} \, dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\int \frac {x^6}{x^8+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle -\frac {\int \frac {x^4}{x^4-\sqrt {2} x^2+1}dx}{2 \sqrt {2}}+\frac {\int \frac {x^4}{x^4+\sqrt {2} x^2+1}dx}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle -\frac {x-\int \frac {1-\sqrt {2} x^2}{x^4-\sqrt {2} x^2+1}dx}{2 \sqrt {2}}+\frac {x-\int \frac {\sqrt {2} x^2+1}{x^4+\sqrt {2} x^2+1}dx}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\left (1-\sqrt {2}\right ) x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\left (1+\sqrt {2}\right ) x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2-\sqrt {2-\sqrt {2}} x+1}dx-\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2+\sqrt {2-\sqrt {2}} x+1}dx+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2-\sqrt {2+\sqrt {2}} x+1}dx-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2-\sqrt {2-\sqrt {2}} x+1}dx+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{x^2+\sqrt {2-\sqrt {2}} x+1}dx+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2-\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{x^2+\sqrt {2+\sqrt {2}} x+1}dx}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx-\sqrt {2+\sqrt {2}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {2}}\right )^2-\sqrt {2}-2}d\left (2 x-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx-\sqrt {2+\sqrt {2}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {2}}\right )^2-\sqrt {2}-2}d\left (2 x+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx+\sqrt {2-\sqrt {2}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {2}}\right )^2+\sqrt {2}-2}d\left (2 x-\sqrt {2+\sqrt {2}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx+\sqrt {2-\sqrt {2}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {2}}\right )^2+\sqrt {2}-2}d\left (2 x+\sqrt {2+\sqrt {2}}\right )}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 x}{x^2-\sqrt {2-\sqrt {2}} x+1}dx+\arctan \left (\frac {2 x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {2 x+\sqrt {2-\sqrt {2}}}{x^2+\sqrt {2-\sqrt {2}} x+1}dx+\arctan \left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 x}{x^2-\sqrt {2+\sqrt {2}} x+1}dx-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {2 x+\sqrt {2+\sqrt {2}}}{x^2+\sqrt {2+\sqrt {2}} x+1}dx-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )}{2 \sqrt {2-\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {-\frac {-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+x}{2 \sqrt {2}}-\frac {1}{x}\) |
-x^(-1) + (x - (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]]) - (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 - (-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]]) - (-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])
3.16.5.3.1 Defintions of rubi rules used
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r) Int[x^ (m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S qrt[2]*b*r) Int[x^(m - 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, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.08
| method | result | size |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{8}-\frac {1}{x}\) | \(28\) |
| risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{7}+x \right )\right )}{8}\) | \(28\) |
| meijerg | \(-\frac {1}{x}-\frac {x^{7} \left (\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}+\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}-\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}-\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {7}{8}}}\right )}{8}\) | \(277\) |
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=\frac {\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x \log \left (\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, x\right ) - \left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x \log \left (-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, x\right ) + \left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x \log \left (\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, x\right ) - \left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x \log \left (-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, x\right ) - 2 \, \left (-1\right )^{\frac {1}{8}} x \log \left (x + \left (-1\right )^{\frac {7}{8}}\right ) + 2 i \, \left (-1\right )^{\frac {1}{8}} x \log \left (x + i \, \left (-1\right )^{\frac {7}{8}}\right ) - 2 i \, \left (-1\right )^{\frac {1}{8}} x \log \left (x - i \, \left (-1\right )^{\frac {7}{8}}\right ) + 2 \, \left (-1\right )^{\frac {1}{8}} x \log \left (x - \left (-1\right )^{\frac {7}{8}}\right ) - 16}{16 \, x} \]
1/16*((I - 1)*sqrt(2)*(-1)^(1/8)*x*log((I + 1)*sqrt(2)*(-1)^(7/8) + 2*x) - (I + 1)*sqrt(2)*(-1)^(1/8)*x*log(-(I - 1)*sqrt(2)*(-1)^(7/8) + 2*x) + (I + 1)*sqrt(2)*(-1)^(1/8)*x*log((I - 1)*sqrt(2)*(-1)^(7/8) + 2*x) - (I - 1)* sqrt(2)*(-1)^(1/8)*x*log(-(I + 1)*sqrt(2)*(-1)^(7/8) + 2*x) - 2*(-1)^(1/8) *x*log(x + (-1)^(7/8)) + 2*I*(-1)^(1/8)*x*log(x + I*(-1)^(7/8)) - 2*I*(-1) ^(1/8)*x*log(x - I*(-1)^(7/8)) + 2*(-1)^(1/8)*x*log(x - (-1)^(7/8)) - 16)/ x
Time = 1.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.06 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} + 1, \left ( t \mapsto t \log {\left (- 2097152 t^{7} + x \right )} \right )\right )} - \frac {1}{x} \]
\[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + 1\right )} x^{2}} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=-\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) + \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) - \frac {1}{x} \]
-1/8*sqrt(sqrt(2) + 2)*arctan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2) ) - 1/8*sqrt(sqrt(2) + 2)*arctan((2*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/8*sqrt(-sqrt(2) + 2)*arctan((2*x + sqrt(sqrt(2) + 2))/sqrt(-sqrt( 2) + 2)) - 1/8*sqrt(-sqrt(2) + 2)*arctan((2*x - sqrt(sqrt(2) + 2))/sqrt(-s qrt(2) + 2)) + 1/16*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/16*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) + 1/16*sqrt(-sq rt(2) + 2)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) - 1/16*sqrt(-sqrt(2) + 2)*l og(x^2 - x*sqrt(-sqrt(2) + 2) + 1) - 1/x
Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (1+x^8\right )} \, dx=-\mathrm {atan}\left (-\frac {x\,1{}\mathrm {i}}{\sqrt {\sqrt {2}-2}}+\frac {x\,1{}\mathrm {i}}{\sqrt {\sqrt {2}+2}}+\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}-2}}+\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}+2}}\right )\,\left (\frac {\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {-\sqrt {2}-2}}+\frac {x\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {-\sqrt {2}-2}}-\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {2-\sqrt {2}}}\right )\,\left (\frac {\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )-\frac {1}{x}-\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{16}-\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}+\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \]
atan((x*1i)/(- 2^(1/2) - 2)^(1/2) + (x*1i)/(2 - 2^(1/2))^(1/2) + (2^(1/2)* x*1i)/(2*(- 2^(1/2) - 2)^(1/2)) - (2^(1/2)*x*1i)/(2*(2 - 2^(1/2))^(1/2)))* (((- 2^(1/2) - 2)^(1/2)*1i)/8 - ((2 - 2^(1/2))^(1/2)*1i)/8) - atan((x*1i)/ (2^(1/2) + 2)^(1/2) - (x*1i)/(2^(1/2) - 2)^(1/2) + (2^(1/2)*x*1i)/(2*(2^(1 /2) - 2)^(1/2)) + (2^(1/2)*x*1i)/(2*(2^(1/2) + 2)^(1/2)))*(((2^(1/2) - 2)^ (1/2)*1i)/8 + ((2^(1/2) + 2)^(1/2)*1i)/8) - 1/x - 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)/16 - (1 /16 + 1i/16))*(2^(1/2) + 2)^(1/2)*2i + atan(x*(2^(1/2) + 2)^(1/2)*(1/2 + 1 i/2) - (2^(1/2)*x*(2^(1/2) + 2)^(1/2)*1i)/2)*(2^(1/2)/16 - (1/16 - 1i/16)) *(2^(1/2) + 2)^(1/2)*2i