Integrand size = 26, antiderivative size = 164 \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=-\frac {1}{2} \sqrt {2+\sqrt {3}} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \sqrt {2+\sqrt {3}} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right ) \]
-1/2*arctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2 *6^(1/2)+1/2*2^(1/2))+1/2*arctan((2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2 )-1/2*2^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))-1/4*ln(1+x^2-x*(1/2*6^(1/2)-1/2* 2^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))+1/4*ln(1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2 )))*(1/2*6^(1/2)+1/2*2^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.44 \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4+\sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4 + Sqrt[3]*Log[ x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4
Time = 0.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1753, 27, 1475, 1083, 217, 1478, 25, 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.
| \(\displaystyle \int \frac {\left (1+\sqrt {3}\right ) x^4+1}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1753 |
| \(\displaystyle \frac {\int \frac {\sqrt {3} \left (x^2+1\right )}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3} \left (1-x^2\right )}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2+1}{x^4-\sqrt {3} x^2+1}dx+\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx\right )+\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx+\frac {1}{2} \left (-\int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )-\int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}-\frac {\int -\frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}-\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}\right )\) |
(ArcTan[(-Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2 - Sqrt[3]] + ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2 - Sqrt[3]])/2 + (-1/2*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/Sqrt[2 - Sqrt[3]] + Log[1 + Sqrt [2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2 - Sqrt[3]]))/2
3.1.32.3.1 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_) + (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_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c *q*r) Int[(d*r - (d - e*q)*x^(n/2))/(q - r*x^(n/2) + x^n), x], x] + Simp[ 1/(2*c*q*r) Int[(d*r + (d - e*q)*x^(n/2))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.38
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}+2 \sqrt {3}\, \textit {\_R}^{4}+\left (1+\sqrt {3}\right ) \left (\sqrt {3}-1\right )\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{8}\) | \(62\) |
1/8*sum(1/(2*_R^7-_R^3)*(2*_R^4+2*3^(1/2)*_R^4+(1+3^(1/2))*(3^(1/2)-1))*ln (x-_R),_R=RootOf(_Z^8-_Z^4+1))
Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96 \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=-\frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (-{\left (x^{3} - \sqrt {3} x + x\right )} \sqrt {\sqrt {3} + 2}\right ) + \frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (x \sqrt {\sqrt {3} + 2}\right ) + \frac {1}{4} \, \sqrt {\sqrt {3} + 2} \log \left (\frac {x^{8} + 4 \, x^{6} + 5 \, x^{4} + 4 \, x^{2} - 2 \, \sqrt {3} {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} + 2 \, {\left (2 \, x^{7} + 5 \, x^{5} + 5 \, x^{3} - \sqrt {3} {\left (x^{7} + 3 \, x^{5} + 3 \, x^{3} + x\right )} + 2 \, x\right )} \sqrt {\sqrt {3} + 2} + 1}{x^{8} - x^{4} + 1}\right ) \]
-1/2*sqrt(sqrt(3) + 2)*arctan(-(x^3 - sqrt(3)*x + x)*sqrt(sqrt(3) + 2)) + 1/2*sqrt(sqrt(3) + 2)*arctan(x*sqrt(sqrt(3) + 2)) + 1/4*sqrt(sqrt(3) + 2)* log((x^8 + 4*x^6 + 5*x^4 + 4*x^2 - 2*sqrt(3)*(x^6 + 2*x^4 + x^2) + 2*(2*x^ 7 + 5*x^5 + 5*x^3 - sqrt(3)*(x^7 + 3*x^5 + 3*x^3 + x) + 2*x)*sqrt(sqrt(3) + 2) + 1)/(x^8 - x^4 + 1))
Exception generated. \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=\text {Exception raised: PolynomialError} \]
Exception raised: PolynomialError >> 1/(2394670008380375980290355982690325 81075191976715165250684200040290318941159424*_t**88 + 13825633739587334576 2803423705330731641326126160751478072830556473063127384064*sqrt(3)*_t**88 - 5732624312622
\[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=\int { \frac {x^{4} {\left (\sqrt {3} + 1\right )} + 1}{x^{8} - x^{4} + 1} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.75 \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=\frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
1/4*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2 ))) + 1/4*(sqrt(6) + sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/8*(sqrt(6) + sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)
Time = 8.70 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {1+\left (1+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx=0 \]