Integrand size = 18, antiderivative size = 451 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=-\frac {\sqrt [4]{3+\sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}} \]
1/20*arctan(-1+2^(3/4)*x/(3+5^(1/2))^(1/4))*(3-5^(1/2))^(1/4)*2^(1/4)*5^(1 /2)+1/20*arctan(1+2^(3/4)*x/(3+5^(1/2))^(1/4))*(3-5^(1/2))^(1/4)*2^(1/4)*5 ^(1/2)-1/40*ln(2*x^2-2*2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+1)*(3-5^(1/2))^ (1/4)*2^(1/4)*5^(1/2)+1/40*ln(2*x^2+2*2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+ 1)*(3-5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/20*arctan(-1+2^(3/4)*x/(3-5^(1/2))^ (1/4))*(3+5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/20*arctan(1+2^(3/4)*x/(3-5^(1/2 ))^(1/4))*(3+5^(1/2))^(1/4)*2^(1/4)*5^(1/2)-1/40*ln(2*x^2-2*2^(1/4)*x*(3-5 ^(1/2))^(1/4)+5^(1/2)-1)*(3+5^(1/2))^(1/4)*2^(1/4)*5^(1/2)+1/40*ln(2*x^2+2 *2^(1/4)*x*(3-5^(1/2))^(1/4)+5^(1/2)-1)*(3+5^(1/2))^(1/4)*2^(1/4)*5^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.12 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.73 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1750, 755, 27, 1476, 1082, 217, 1479, 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.
| \(\displaystyle \int \frac {x^4+1}{x^8+3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1750 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3-\sqrt {5}\right )}dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3+\sqrt {5}\right )}dx\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3-\sqrt {5}}\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3+\sqrt {5}}\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{10} \left (5-\sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}+\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )\) |
((5 - Sqrt[5])*((-(ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)]/(2^(3/4)*(3 - Sqrt[5])^(1/4))) + ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)]/(2^(3/4) *(3 - Sqrt[5])^(1/4)))/Sqrt[3 - Sqrt[5]] + (-1/4*(((3 + Sqrt[5])/2)^(1/4)* Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2]) + (((3 + Sqrt[5])/2)^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/4)/Sqrt[3 - Sqrt[5]]))/10 + ((5 + Sqrt[5])*((-(ArcTan[1 - (2^(3 /4)*x)/(3 + Sqrt[5])^(1/4)]/(2^(3/4)*(3 + Sqrt[5])^(1/4))) + ArcTan[1 + (2 ^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2^(3/4)*(3 + Sqrt[5])^(1/4)))/Sqrt[3 + Sqr t[5]] + (-1/2*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2* x^2]/(2^(3/4)*(3 + Sqrt[5])^(1/4)) + Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(2*2^(3/4)*(3 + Sqrt[5])^(1/4)))/Sqrt[3 + Sqrt [5]]))/10
3.1.10.3.1 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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 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]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 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]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && G tQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(42\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(42\) |
Time = 0.27 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} {\left (\sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} {\left (\sqrt {5} + 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} {\left (\sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} {\left (\sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (\sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (-\sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (-\sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) \]
1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*log(sqrt(10)*sqrt(sqrt(2)*sq rt(sqrt(5) - 3))*(sqrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(s qrt(5) - 3))*log(-sqrt(10)*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*(sqrt(5) + 5) + 20*x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*log(sqrt(10)*sqrt( -sqrt(2)*sqrt(sqrt(5) - 3))*(sqrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(-sq rt(2)*sqrt(sqrt(5) - 3))*log(-sqrt(10)*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*(s qrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*log(s qrt(10)*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*(sqrt(5) - 5) + 20*x) + 1/40*sqrt (10)*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*log(-sqrt(10)*sqrt(sqrt(2)*sqrt(-sqr t(5) - 3))*(sqrt(5) - 5) + 20*x) - 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(-sqrt( 5) - 3))*log(sqrt(10)*sqrt(-sqrt(2)*sqrt(-sqrt(5) - 3))*(sqrt(5) - 5) + 20 *x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(-sqrt(5) - 3))*log(-sqrt(10)*sqrt(- sqrt(2)*sqrt(-sqrt(5) - 3))*(sqrt(5) - 5) + 20*x)
Time = 0.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.05 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log {\left (25600 t^{5} + 16 t + x \right )} \right )\right )} \]
\[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\int { \frac {x^{4} + 1}{x^{8} + 3 \, x^{4} + 1} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.53 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) \]
1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(5*sqrt(5) + 5) - 1/80*( pi + 4*arctan(-x*sqrt(sqrt(5) + 1) + 1))*sqrt(5*sqrt(5) + 5) + 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(5*sqrt(5) - 5) - 1/80*(pi + 4*arct an(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(5*sqrt(5) - 5) + 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x + sqrt(sqrt(5) + 1))^2 + 16900*x^2) - 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x - sqrt(sqrt(5) + 1))^2 + 16900*x^2) + 1/40*sqrt(5*sqrt(5 ) + 5)*log(2500*(x + sqrt(sqrt(5) - 1))^2 + 2500*x^2) - 1/40*sqrt(5*sqrt(5 ) + 5)*log(2500*(x - sqrt(sqrt(5) - 1))^2 + 2500*x^2)
Time = 0.10 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.02 \[ \int \frac {1+x^4}{1+3 x^4+x^8} \, dx=\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20} \]
(2^(3/4)*5^(1/2)*atan((7*2^(3/4)*x*(- 5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))) + (3*2^(3/4) *5^(1/2)*x*(- 5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^ (1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))))*(- 5^(1/2) - 3)^(1/4))/20 - (2^(3/4 )*5^(1/2)*atan((2^(3/4)*x*(- 5^(1/2) - 3)^(1/4)*7i)/(2*(2*2^(1/2)*(- 5^(1/ 2) - 3)^(1/2) + 2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))) + (2^(3/4)*5^(1/2) *x*(- 5^(1/2) - 3)^(1/4)*3i)/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^(1/2) *5^(1/2)*(- 5^(1/2) - 3)^(1/2))))*(- 5^(1/2) - 3)^(1/4)*1i)/20 - (2^(3/4)* 5^(1/2)*atan((7*2^(3/4)*x*(5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(5^(1/2) - 3) ^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))) - (3*2^(3/4)*5^(1/2)*x*(5^( 1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1 /2) - 3)^(1/2))))*(5^(1/2) - 3)^(1/4))/20 + (2^(3/4)*5^(1/2)*atan((2^(3/4) *x*(5^(1/2) - 3)^(1/4)*7i)/(2*(2*2^(1/2)*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^( 1/2)*(5^(1/2) - 3)^(1/2))) - (2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4)*3i)/(2 *(2*2^(1/2)*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))))*( 5^(1/2) - 3)^(1/4)*1i)/20