Integrand size = 12, antiderivative size = 304 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=-\frac {\sqrt [4]{9+4 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} x}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}\right )}{2 \sqrt {10}}-\frac {\text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} x}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \] Output:
1/20*(9+4*5^(1/2))^(1/4)*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*10^(1/2)+1 /20*(9+4*5^(1/2))^(1/4)*arctan(1+2^(3/4)*x/(3-5^(1/2))^(1/4))*10^(1/2)-1/5 *arctan(-1+2^(3/4)*x/(3+5^(1/2))^(1/4))*5^(1/2)/(6+2*5^(1/2))^(3/4)-1/5*ar ctan(1+2^(3/4)*x/(3+5^(1/2))^(1/4))*5^(1/2)/(6+2*5^(1/2))^(3/4)+1/20*(9+4* 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)))*10^(1/2)-1/5*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)))*5^(1/2)/(6+2*5^(1/2))^(3/4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.14 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(1 + 3*x^4 + x^8)^(-1),x]
Output:
RootSum[1 + 3*#1^4 + #1^8 & , Log[x - #1]/(3*#1^3 + 2*#1^7) & ]/4
Time = 0.68 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.53, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {1685, 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.
| \(\displaystyle \int \frac {1}{x^8+3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {\int \frac {1}{x^4+\frac {1}{2} \left (3-\sqrt {5}\right )}dx}{\sqrt {5}}-\frac {\int \frac {1}{x^4+\frac {1}{2} \left (3+\sqrt {5}\right )}dx}{\sqrt {5}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3-\sqrt {5}}\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {2 \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3+\sqrt {5}}\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}+\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
Input:
Int[(1 + 3*x^4 + x^8)^(-1),x]
Output:
((-(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)/S qrt[3 - Sqrt[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]] + (-1/2*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]])/Sqrt[5]
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[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(37\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(37\) |
Input:
int(1/(x^8+3*x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(1/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.15 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=-\frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {5} + 5\right )} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} {\left (3 \, \sqrt {5} + 5\right )} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5}} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (3 \, \sqrt {5} - 5\right )} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} + 2 \, x\right ) \] Input:
integrate(1/(x^8+3*x^4+1),x, algorithm="fricas")
Output:
-1/4*sqrt(1/5)*sqrt(-sqrt(4*sqrt(5) - 9))*log(sqrt(1/5)*(3*sqrt(5) + 5)*sq rt(-sqrt(4*sqrt(5) - 9)) + 2*x) + 1/4*sqrt(1/5)*sqrt(-sqrt(4*sqrt(5) - 9)) *log(-sqrt(1/5)*(3*sqrt(5) + 5)*sqrt(-sqrt(4*sqrt(5) - 9)) + 2*x) + 1/4*sq rt(1/5)*sqrt(-sqrt(-4*sqrt(5) - 9))*log(sqrt(1/5)*(3*sqrt(5) - 5)*sqrt(-sq rt(-4*sqrt(5) - 9)) + 2*x) - 1/4*sqrt(1/5)*sqrt(-sqrt(-4*sqrt(5) - 9))*log (-sqrt(1/5)*(3*sqrt(5) - 5)*sqrt(-sqrt(-4*sqrt(5) - 9)) + 2*x) - 1/4*sqrt( 1/5)*(4*sqrt(5) - 9)^(1/4)*log(sqrt(1/5)*(4*sqrt(5) - 9)^(1/4)*(3*sqrt(5) + 5) + 2*x) + 1/4*sqrt(1/5)*(4*sqrt(5) - 9)^(1/4)*log(-sqrt(1/5)*(4*sqrt(5 ) - 9)^(1/4)*(3*sqrt(5) + 5) + 2*x) + 1/4*sqrt(1/5)*(-4*sqrt(5) - 9)^(1/4) *log(sqrt(1/5)*(3*sqrt(5) - 5)*(-4*sqrt(5) - 9)^(1/4) + 2*x) - 1/4*sqrt(1/ 5)*(-4*sqrt(5) - 9)^(1/4)*log(-sqrt(1/5)*(3*sqrt(5) - 5)*(-4*sqrt(5) - 9)^ (1/4) + 2*x)
Time = 1.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 115200 t^{4} + 1, \left ( t \mapsto t \log {\left (- 9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \] Input:
integrate(1/(x**8+3*x**4+1),x)
Output:
RootSum(40960000*_t**8 + 115200*_t**4 + 1, Lambda(_t, _t*log(-9600*_t**5 - 47*_t/2 + x)))
\[ \int \frac {1}{1+3 x^4+x^8} \, dx=\int { \frac {1}{x^{8} + 3 \, x^{4} + 1} \,d x } \] Input:
integrate(1/(x^8+3*x^4+1),x, algorithm="maxima")
Output:
integrate(1/(x^8 + 3*x^4 + 1), x)
Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) \] Input:
integrate(1/(x^8+3*x^4+1),x, algorithm="giac")
Output:
1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(10*sqrt(5) + 20) - 1/80 *(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) + 1))*sqrt(10*sqrt(5) + 20) - 1/80*(p i + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) + 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) - 1/40*sqrt(10*s qrt(5) - 20)*log(10000*(x + sqrt(sqrt(5) + 1))^2 + 10000*x^2) + 1/40*sqrt( 10*sqrt(5) - 20)*log(10000*(x - sqrt(sqrt(5) + 1))^2 + 10000*x^2) + 1/40*s qrt(10*sqrt(5) + 20)*log(400*(x + sqrt(sqrt(5) - 1))^2 + 400*x^2) - 1/40*s qrt(10*sqrt(5) + 20)*log(400*(x - sqrt(sqrt(5) - 1))^2 + 400*x^2)
Time = 0.08 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.33 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {64\,\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {64\,\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10} \] Input:
int(1/(3*x^4 + x^8 + 1),x)
Output:
(5^(1/2)*atan((144*x*(- 4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(- 4*5^(1/2) - 9 )^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)) + (64*5^(1/2)*x*(- 4*5^(1/2) - 9)^(1 /4))/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*(- 4*5^(1/2) - 9)^(1/4))/10 + (5^(1/2)*atan((144*x*(4*5^(1/2) - 9)^(1/4))/(2 4*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)) - (64*5^(1/2)* x*(4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4))/10 - (5^(1/2)*atan((x*(- 4*5^(1/2) - 9)^(1/4)*144i)/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^ (1/2)) + (5^(1/2)*x*(- 4*5^(1/2) - 9)^(1/4)*64i)/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*(- 4*5^(1/2) - 9)^(1/4)*1i)/10 - (5^(1/2)*atan((x*(4*5^(1/2) - 9)^(1/4)*144i)/(24*5^(1/2)*(4*5^(1/2) - 9)^ (1/2) - 56*(4*5^(1/2) - 9)^(1/2)) - (5^(1/2)*x*(4*5^(1/2) - 9)^(1/4)*64i)/ (24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4)*1i)/10
\[ \int \frac {1}{1+3 x^4+x^8} \, dx=\int \frac {1}{x^{8}+3 x^{4}+1}d x \] Input:
int(1/(x^8+3*x^4+1),x)
Output:
int(1/(x**8 + 3*x**4 + 1),x)