Integrand size = 16, antiderivative size = 348 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}+\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{843+377 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} x}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{843-377 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} x}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}\right )}{2\ 2^{3/4} \sqrt {5}} \] Output:
-1/3/x^3-1/20*(843+377*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*(843+377*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*(843-377*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*(843-377*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*(843+377*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*(843-377*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 = 65, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[1/(x^4*(1 + 3*x^4 + x^8)),x]
Output:
-1/3*1/x^3 - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1 ^4)/(3*#1^3 + 2*#1^7) & ]/4
Time = 0.72 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.40, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1704, 27, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (x^8+3 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \frac {1}{3} \int -\frac {3 \left (x^4+3\right )}{x^8+3 x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {x^4+3}{x^8+3 x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3-\sqrt {5}\right )}dx-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (3+\sqrt {5}\right )}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3-\sqrt {5}}\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {2 \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3+\sqrt {5}}\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}+\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{\sqrt {3-\sqrt {5}}}\right )-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}\right )-\frac {1}{3 x^3}\) |
Input:
Int[1/(x^4*(1 + 3*x^4 + x^8)),x]
Output:
-1/3*1/x^3 - ((5 + 3*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 + Sqr t[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 - Sq rt[5]))^(1/4)*x + 2*x^2])/4)/Sqrt[3 - Sqrt[5]]))/10 - ((5 - 3*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]]))/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*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) )), x] - Simp[1/(a*d^n*(m + 1)) Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.11
| method | result | size |
| risch | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (625 \textit {\_Z}^{8}+21075 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (175 \textit {\_R}^{5}+5778 \textit {\_R} +377 x \right )\right )}{4}\) | \(40\) |
| default | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}-3\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(50\) |
Input:
int(1/x^4/(x^8+3*x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/3/x^3+1/4*sum(_R*ln(175*_R^5+5778*_R+377*x),_R=RootOf(625*_Z^8+21075*_Z ^4+1))
Time = 0.07 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {3 \, \sqrt {\frac {1}{5}} x^{3} \sqrt {-\sqrt {\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} \log \left (\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} + 15\right )} \sqrt {-\sqrt {\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} + 2 \, x\right ) - 3 \, \sqrt {\frac {1}{5}} x^{3} \sqrt {-\sqrt {\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} \log \left (-\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} + 15\right )} \sqrt {-\sqrt {\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} + 2 \, x\right ) - 3 \, \sqrt {\frac {1}{5}} x^{3} \sqrt {-\sqrt {-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} \log \left (\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} - 15\right )} \sqrt {-\sqrt {-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} + 2 \, x\right ) + 3 \, \sqrt {\frac {1}{5}} x^{3} \sqrt {-\sqrt {-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} \log \left (-\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} - 15\right )} \sqrt {-\sqrt {-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}}} + 2 \, x\right ) + 3 \, \sqrt {\frac {1}{5}} x^{3} {\left (\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} {\left (7 \, \sqrt {5} + 15\right )} + 2 \, x\right ) - 3 \, \sqrt {\frac {1}{5}} x^{3} {\left (\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} {\left (7 \, \sqrt {5} + 15\right )} + 2 \, x\right ) - 3 \, \sqrt {\frac {1}{5}} x^{3} {\left (-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} \log \left (\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} - 15\right )} {\left (-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} + 2 \, x\right ) + 3 \, \sqrt {\frac {1}{5}} x^{3} {\left (-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} \log \left (-\sqrt {\frac {1}{5}} {\left (7 \, \sqrt {5} - 15\right )} {\left (-\frac {377}{2} \, \sqrt {5} - \frac {843}{2}\right )}^{\frac {1}{4}} + 2 \, x\right ) - 4}{12 \, x^{3}} \] Input:
integrate(1/x^4/(x^8+3*x^4+1),x, algorithm="fricas")
Output:
1/12*(3*sqrt(1/5)*x^3*sqrt(-sqrt(377/2*sqrt(5) - 843/2))*log(sqrt(1/5)*(7* sqrt(5) + 15)*sqrt(-sqrt(377/2*sqrt(5) - 843/2)) + 2*x) - 3*sqrt(1/5)*x^3* sqrt(-sqrt(377/2*sqrt(5) - 843/2))*log(-sqrt(1/5)*(7*sqrt(5) + 15)*sqrt(-s qrt(377/2*sqrt(5) - 843/2)) + 2*x) - 3*sqrt(1/5)*x^3*sqrt(-sqrt(-377/2*sqr t(5) - 843/2))*log(sqrt(1/5)*(7*sqrt(5) - 15)*sqrt(-sqrt(-377/2*sqrt(5) - 843/2)) + 2*x) + 3*sqrt(1/5)*x^3*sqrt(-sqrt(-377/2*sqrt(5) - 843/2))*log(- sqrt(1/5)*(7*sqrt(5) - 15)*sqrt(-sqrt(-377/2*sqrt(5) - 843/2)) + 2*x) + 3* sqrt(1/5)*x^3*(377/2*sqrt(5) - 843/2)^(1/4)*log(sqrt(1/5)*(377/2*sqrt(5) - 843/2)^(1/4)*(7*sqrt(5) + 15) + 2*x) - 3*sqrt(1/5)*x^3*(377/2*sqrt(5) - 8 43/2)^(1/4)*log(-sqrt(1/5)*(377/2*sqrt(5) - 843/2)^(1/4)*(7*sqrt(5) + 15) + 2*x) - 3*sqrt(1/5)*x^3*(-377/2*sqrt(5) - 843/2)^(1/4)*log(sqrt(1/5)*(7*s qrt(5) - 15)*(-377/2*sqrt(5) - 843/2)^(1/4) + 2*x) + 3*sqrt(1/5)*x^3*(-377 /2*sqrt(5) - 843/2)^(1/4)*log(-sqrt(1/5)*(7*sqrt(5) - 15)*(-377/2*sqrt(5) - 843/2)^(1/4) + 2*x) - 4)/x^3
Time = 1.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 5395200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {179200 t^{5}}{377} + \frac {23112 t}{377} + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \] Input:
integrate(1/x**4/(x**8+3*x**4+1),x)
Output:
RootSum(40960000*_t**8 + 5395200*_t**4 + 1, Lambda(_t, _t*log(179200*_t**5 /377 + 23112*_t/377 + x))) - 1/(3*x**3)
\[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + 3 \, x^{4} + 1\right )} x^{4}} \,d x } \] Input:
integrate(1/x^4/(x^8+3*x^4+1),x, algorithm="maxima")
Output:
-1/3/x^3 - integrate((x^4 + 3)/(x^8 + 3*x^4 + 1), x)
Time = 0.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {65 \, \sqrt {5} + 145} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {65 \, \sqrt {5} + 145} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {65 \, \sqrt {5} - 145} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {65 \, \sqrt {5} - 145} + \frac {1}{40} \, \sqrt {65 \, \sqrt {5} - 145} \log \left (93122500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 93122500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {65 \, \sqrt {5} - 145} \log \left (93122500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 93122500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {65 \, \sqrt {5} + 145} \log \left (53728900 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 53728900 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {65 \, \sqrt {5} + 145} \log \left (53728900 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 53728900 \, x^{2}\right ) - \frac {1}{3 \, x^{3}} \] Input:
integrate(1/x^4/(x^8+3*x^4+1),x, algorithm="giac")
Output:
-1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(65*sqrt(5) + 145) + 1/ 80*(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) + 1))*sqrt(65*sqrt(5) + 145) + 1/80 *(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(65*sqrt(5) - 145) - 1/80*(p i + 4*arctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(65*sqrt(5) - 145) + 1/40*sqrt (65*sqrt(5) - 145)*log(93122500*(x + sqrt(sqrt(5) + 1))^2 + 93122500*x^2) - 1/40*sqrt(65*sqrt(5) - 145)*log(93122500*(x - sqrt(sqrt(5) + 1))^2 + 931 22500*x^2) - 1/40*sqrt(65*sqrt(5) + 145)*log(53728900*(x + sqrt(sqrt(5) - 1))^2 + 53728900*x^2) + 1/40*sqrt(65*sqrt(5) + 145)*log(53728900*(x - sqrt (sqrt(5) - 1))^2 + 53728900*x^2) - 1/3/x^3
Time = 20.26 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {46371\,2^{3/4}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}-\frac {20735\,2^{3/4}\,\sqrt {5}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}\right )\,{\left (377\,\sqrt {5}-843\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {46371\,2^{3/4}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}+\frac {20735\,2^{3/4}\,\sqrt {5}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}\right )\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}}{20}-\frac {1}{3\,x^3}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,46371{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,20735{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {-377\,\sqrt {5}-843}+1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {-377\,\sqrt {5}-843}\right )}\right )\,{\left (-377\,\sqrt {5}-843\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,46371{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,20735{}\mathrm {i}}{2\,\left (3393\,\sqrt {2}\,\sqrt {377\,\sqrt {5}-843}-1508\,\sqrt {2}\,\sqrt {5}\,\sqrt {377\,\sqrt {5}-843}\right )}\right )\,{\left (377\,\sqrt {5}-843\right )}^{1/4}\,1{}\mathrm {i}}{20} \] Input:
int(1/(x^4*(3*x^4 + x^8 + 1)),x)
Output:
(2^(3/4)*5^(1/2)*atan((46371*2^(3/4)*x*(377*5^(1/2) - 843)^(1/4))/(2*(3393 *2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 1508*2^(1/2)*5^(1/2)*(377*5^(1/2) - 8 43)^(1/2))) - (20735*2^(3/4)*5^(1/2)*x*(377*5^(1/2) - 843)^(1/4))/(2*(3393 *2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 1508*2^(1/2)*5^(1/2)*(377*5^(1/2) - 8 43)^(1/2))))*(377*5^(1/2) - 843)^(1/4))/20 - (2^(3/4)*5^(1/2)*atan((46371* 2^(3/4)*x*(- 377*5^(1/2) - 843)^(1/4))/(2*(3393*2^(1/2)*(- 377*5^(1/2) - 8 43)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))) + (20735*2^ (3/4)*5^(1/2)*x*(- 377*5^(1/2) - 843)^(1/4))/(2*(3393*2^(1/2)*(- 377*5^(1/ 2) - 843)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))))*(- 3 77*5^(1/2) - 843)^(1/4))/20 - 1/(3*x^3) + (2^(3/4)*5^(1/2)*atan((2^(3/4)*x *(- 377*5^(1/2) - 843)^(1/4)*46371i)/(2*(3393*2^(1/2)*(- 377*5^(1/2) - 843 )^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))) + (2^(3/4)*5^ (1/2)*x*(- 377*5^(1/2) - 843)^(1/4)*20735i)/(2*(3393*2^(1/2)*(- 377*5^(1/2 ) - 843)^(1/2) + 1508*2^(1/2)*5^(1/2)*(- 377*5^(1/2) - 843)^(1/2))))*(- 37 7*5^(1/2) - 843)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((2^(3/4)*x*(377*5^(1 /2) - 843)^(1/4)*46371i)/(2*(3393*2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 1508 *2^(1/2)*5^(1/2)*(377*5^(1/2) - 843)^(1/2))) - (2^(3/4)*5^(1/2)*x*(377*5^( 1/2) - 843)^(1/4)*20735i)/(2*(3393*2^(1/2)*(377*5^(1/2) - 843)^(1/2) - 150 8*2^(1/2)*5^(1/2)*(377*5^(1/2) - 843)^(1/2))))*(377*5^(1/2) - 843)^(1/4)*1 i)/20
\[ \int \frac {1}{x^4 \left (1+3 x^4+x^8\right )} \, dx=\frac {-3 \left (\int \frac {x^{4}}{x^{8}+3 x^{4}+1}d x \right ) x^{3}-9 \left (\int \frac {1}{x^{8}+3 x^{4}+1}d x \right ) x^{3}-1}{3 x^{3}} \] Input:
int(1/x^4/(x^8+3*x^4+1),x)
Output:
( - 3*int(x**4/(x**8 + 3*x**4 + 1),x)*x**3 - 9*int(1/(x**8 + 3*x**4 + 1),x )*x**3 - 1)/(3*x**3)