Integrand size = 16, antiderivative size = 342 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} x}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123+55 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} x}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}\right )}{2\ 2^{3/4} \sqrt {5}} \] Output:
x+1/20*(123-55*5^(1/2))^(1/4)*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*2^(1/ 4)*5^(1/2)+1/20*(123-55*5^(1/2))^(1/4)*arctan(1+2^(3/4)*x/(3-5^(1/2))^(1/4 ))*2^(1/4)*5^(1/2)-1/20*(123+55*5^(1/2))^(1/4)*arctan(-1+2^(3/4)*x/(3+5^(1 /2))^(1/4))*2^(1/4)*5^(1/2)-1/20*(123+55*5^(1/2))^(1/4)*arctan(1+2^(3/4)*x /(3+5^(1/2))^(1/4))*2^(1/4)*5^(1/2)+1/20*(123-55*5^(1/2))^(1/4)*arctanh(2^ (3/4)*(3-5^(1/2))^(1/4)*x/(1/2*10^(1/2)-1/2*2^(1/2)+x^2*2^(1/2)))*2^(1/4)* 5^(1/2)-1/20*(123+55*5^(1/2))^(1/4)*arctanh(2^(3/4)*(3+5^(1/2))^(1/4)*x/(1 /2*10^(1/2)+1/2*2^(1/2)+x^2*2^(1/2)))*2^(1/4)*5^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.17 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[x^8/(1 + 3*x^4 + x^8),x]
Output:
x - RootSum[1 + 3*#1^4 + #1^8 & , (Log[x - #1] + 3*Log[x - #1]*#1^4)/(3*#1 ^3 + 2*#1^7) & ]/4
Time = 0.76 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.40, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1703, 1752, 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.
| \(\displaystyle \int \frac {x^8}{x^8+3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle x-\int \frac {3 x^4+1}{x^8+3 x^4+1}dx\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3-\sqrt {5}\right )}dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3+\sqrt {5}\right )}dx+x\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3-\sqrt {5}}\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3+\sqrt {5}}\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )+x\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {1}{10} \left (15-7 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}+\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )+x\) |
Input:
Int[x^8/(1 + 3*x^4 + x^8),x]
Output:
x - ((15 - 7*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 - ((15 + 7*Sqrt[5])*((-(ArcTan [1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2^(3/4)*(3 + Sqrt[5])^(1/4))) + Arc Tan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2^(3/4)*(3 + Sqrt[5])^(1/4)))/Sq rt[3 + Sqrt[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)))/Sqr t[3 + Sqrt[5]]))/10
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_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(2*n - 1)*(d*x)^(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^( p + 1)/(c*(m + 2*n*p + 1))), x] - Simp[d^(2*n)/(c*(m + 2*n*p + 1)) Int[(d *x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x ^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0 ] && IntegerQ[p]
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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(46\) |
| risch | \(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(46\) |
Input:
int(x^8/(x^8+3*x^4+1),x,method=_RETURNVERBOSE)
Output:
x+1/4*sum((-3*_R^4-1)/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.02 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=\frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-\sqrt {\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-\sqrt {\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-\sqrt {-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-\sqrt {-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {5} + 5\right )} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {5} + 5\right )} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} {\left (-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} {\left (-\frac {55}{2} \, \sqrt {5} - \frac {123}{2}\right )}^{\frac {1}{4}} + 2 \, x\right ) + x \] Input:
integrate(x^8/(x^8+3*x^4+1),x, algorithm="fricas")
Output:
1/4*sqrt(1/5)*sqrt(-sqrt(55/2*sqrt(5) - 123/2))*log(sqrt(1/5)*(3*sqrt(5) + 5)*sqrt(-sqrt(55/2*sqrt(5) - 123/2)) + 2*x) - 1/4*sqrt(1/5)*sqrt(-sqrt(55 /2*sqrt(5) - 123/2))*log(-sqrt(1/5)*(3*sqrt(5) + 5)*sqrt(-sqrt(55/2*sqrt(5 ) - 123/2)) + 2*x) - 1/4*sqrt(1/5)*sqrt(-sqrt(-55/2*sqrt(5) - 123/2))*log( sqrt(1/5)*(3*sqrt(5) - 5)*sqrt(-sqrt(-55/2*sqrt(5) - 123/2)) + 2*x) + 1/4* sqrt(1/5)*sqrt(-sqrt(-55/2*sqrt(5) - 123/2))*log(-sqrt(1/5)*(3*sqrt(5) - 5 )*sqrt(-sqrt(-55/2*sqrt(5) - 123/2)) + 2*x) + 1/4*sqrt(1/5)*(55/2*sqrt(5) - 123/2)^(1/4)*log(sqrt(1/5)*(55/2*sqrt(5) - 123/2)^(1/4)*(3*sqrt(5) + 5) + 2*x) - 1/4*sqrt(1/5)*(55/2*sqrt(5) - 123/2)^(1/4)*log(-sqrt(1/5)*(55/2*s qrt(5) - 123/2)^(1/4)*(3*sqrt(5) + 5) + 2*x) - 1/4*sqrt(1/5)*(-55/2*sqrt(5 ) - 123/2)^(1/4)*log(sqrt(1/5)*(3*sqrt(5) - 5)*(-55/2*sqrt(5) - 123/2)^(1/ 4) + 2*x) + 1/4*sqrt(1/5)*(-55/2*sqrt(5) - 123/2)^(1/4)*log(-sqrt(1/5)*(3* sqrt(5) - 5)*(-55/2*sqrt(5) - 123/2)^(1/4) + 2*x) + x
Time = 1.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.08 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x + \operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \] Input:
integrate(x**8/(x**8+3*x**4+1),x)
Output:
x + RootSum(40960000*_t**8 + 787200*_t**4 + 1, Lambda(_t, _t*log(15360*_t* *5/11 + 1288*_t/55 + x)))
\[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=\int { \frac {x^{8}}{x^{8} + 3 \, x^{4} + 1} \,d x } \] Input:
integrate(x^8/(x^8+3*x^4+1),x, algorithm="maxima")
Output:
x - integrate((3*x^4 + 1)/(x^8 + 3*x^4 + 1), x)
Time = 0.20 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.70 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=-\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (722500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 722500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (2992900 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2992900 \, x^{2}\right ) + x \] Input:
integrate(x^8/(x^8+3*x^4+1),x, algorithm="giac")
Output:
-1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) + 1))*sqrt(25*sqrt(5) + 55) + 1/8 0*(pi + 4*arctan(-x*sqrt(sqrt(5) - 1) + 1))*sqrt(25*sqrt(5) + 55) + 1/80*( pi + 4*arctan(x*sqrt(sqrt(5) + 1) - 1))*sqrt(25*sqrt(5) - 55) - 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) - 1))*sqrt(25*sqrt(5) - 55) - 1/40*sqrt(25* sqrt(5) + 55)*log(722500*(x + sqrt(sqrt(5) + 1))^2 + 722500*x^2) + 1/40*sq rt(25*sqrt(5) + 55)*log(722500*(x - sqrt(sqrt(5) + 1))^2 + 722500*x^2) + 1 /40*sqrt(25*sqrt(5) - 55)*log(2992900*(x + sqrt(sqrt(5) - 1))^2 + 2992900* x^2) - 1/40*sqrt(25*sqrt(5) - 55)*log(2992900*(x - sqrt(sqrt(5) - 1))^2 + 2992900*x^2) + x
Time = 18.97 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.63 \[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=x-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {3\,2^{1/4}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,3{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {5}\,x\,1{}\mathrm {i}}{2\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \] Input:
int(x^8/(3*x^4 + x^8 + 1),x)
Output:
x - (2^(3/4)*5^(1/2)*atan((3*2^(1/4)*x)/(2*(- 55*5^(1/2) - 123)^(1/4)) + ( 2^(1/4)*5^(1/2)*x)/(2*(- 55*5^(1/2) - 123)^(1/4)))*(- 55*5^(1/2) - 123)^(1 /4))/20 + (2^(3/4)*5^(1/2)*atan((3*2^(1/4)*x)/(2*(55*5^(1/2) - 123)^(1/4)) - (2^(1/4)*5^(1/2)*x)/(2*(55*5^(1/2) - 123)^(1/4)))*(55*5^(1/2) - 123)^(1 /4))/20 + (2^(3/4)*5^(1/2)*atan((2^(1/4)*x*3i)/(2*(- 55*5^(1/2) - 123)^(1/ 4)) + (2^(1/4)*5^(1/2)*x*1i)/(2*(- 55*5^(1/2) - 123)^(1/4)))*(- 55*5^(1/2) - 123)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((2^(1/4)*x*3i)/(2*(55*5^(1/2) - 123)^(1/4)) - (2^(1/4)*5^(1/2)*x*1i)/(2*(55*5^(1/2) - 123)^(1/4)))*(55* 5^(1/2) - 123)^(1/4)*1i)/20
\[ \int \frac {x^8}{1+3 x^4+x^8} \, dx=-3 \left (\int \frac {x^{4}}{x^{8}+3 x^{4}+1}d x \right )-\left (\int \frac {1}{x^{8}+3 x^{4}+1}d x \right )+x \] Input:
int(x^8/(x^8+3*x^4+1),x)
Output:
- 3*int(x**4/(x**8 + 3*x**4 + 1),x) - int(1/(x**8 + 3*x**4 + 1),x) + x