Integrand size = 16, antiderivative size = 340 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{x}+\frac {\left (3+\sqrt {5}\right )^{5/4} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{123-55 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} x}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{123-55 \sqrt {5}} \text {arctanh}\left (\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} x}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}\right )}{2\ 2^{3/4} \sqrt {5}} \] Output:
-1/x-1/40*(3+5^(1/2))^(5/4)*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*2^(1/4) *5^(1/2)-1/40*(3+5^(1/2))^(5/4)*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)+1/40*(3+5^(1/2))^(5/4)*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 = 61, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{x}-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[1/(x^2*(1 + 3*x^4 + x^8)),x]
Output:
-x^(-1) - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4) /(3*#1 + 2*#1^5) & ]/4
Time = 0.69 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.38, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1704, 25, 1834, 27, 826, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (x^8+3 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \int -\frac {x^2 \left (x^4+3\right )}{x^8+3 x^4+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {x^2 \left (x^4+3\right )}{x^8+3 x^4+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle -\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {2 x^2}{2 x^4-\sqrt {5}+3}dx-\frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {2 x^2}{2 x^4+\sqrt {5}+3}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{2 x^4-\sqrt {5}+3}dx-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {x^2}{2 x^4+\sqrt {5}+3}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )-\frac {1}{x}\) |
Input:
Int[1/(x^2*(1 + 3*x^4 + x^8)),x]
Output:
-x^(-1) - ((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)))/(2*Sqrt[2]) - (-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)/(2*Sqrt[2])))/5 - ((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)))/(2*Sqrt[2]) - (-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)))/(2*Sqrt[2])))/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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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[(((f_.)*(x_))^(m_.)*((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[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ [{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12
| method | result | size |
| risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (625 \textit {\_Z}^{8}+3075 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (1175 \textit {\_R}^{7}+5778 \textit {\_R}^{3}+11 x \right )\right )}{4}\) | \(42\) |
| default | \(-\frac {1}{x}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(52\) |
Input:
int(1/x^2/(x^8+3*x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/x+1/4*sum(_R*ln(1175*_R^7+5778*_R^3+11*x),_R=RootOf(625*_Z^8+3075*_Z^4+ 1))
Time = 0.07 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(x^8+3*x^4+1),x, algorithm="fricas")
Output:
-1/4*(sqrt(1/5)*x*sqrt(-sqrt(55/2*sqrt(5) - 123/2))*log(sqrt(1/5)*(47*sqrt (5) + 105)*sqrt(55/2*sqrt(5) - 123/2)*sqrt(-sqrt(55/2*sqrt(5) - 123/2)) + 2*x) - sqrt(1/5)*x*sqrt(-sqrt(55/2*sqrt(5) - 123/2))*log(-sqrt(1/5)*(47*sq rt(5) + 105)*sqrt(55/2*sqrt(5) - 123/2)*sqrt(-sqrt(55/2*sqrt(5) - 123/2)) + 2*x) - sqrt(1/5)*x*sqrt(-sqrt(-55/2*sqrt(5) - 123/2))*log(sqrt(1/5)*(47* sqrt(5) - 105)*sqrt(-55/2*sqrt(5) - 123/2)*sqrt(-sqrt(-55/2*sqrt(5) - 123/ 2)) + 2*x) + sqrt(1/5)*x*sqrt(-sqrt(-55/2*sqrt(5) - 123/2))*log(-sqrt(1/5) *(47*sqrt(5) - 105)*sqrt(-55/2*sqrt(5) - 123/2)*sqrt(-sqrt(-55/2*sqrt(5) - 123/2)) + 2*x) - sqrt(1/5)*x*(55/2*sqrt(5) - 123/2)^(1/4)*log(sqrt(1/5)*( 47*sqrt(5) + 105)*(55/2*sqrt(5) - 123/2)^(3/4) + 2*x) + sqrt(1/5)*x*(55/2* sqrt(5) - 123/2)^(1/4)*log(-sqrt(1/5)*(47*sqrt(5) + 105)*(55/2*sqrt(5) - 1 23/2)^(3/4) + 2*x) + sqrt(1/5)*x*(-55/2*sqrt(5) - 123/2)^(1/4)*log(sqrt(1/ 5)*(47*sqrt(5) - 105)*(-55/2*sqrt(5) - 123/2)^(3/4) + 2*x) - sqrt(1/5)*x*( -55/2*sqrt(5) - 123/2)^(1/4)*log(-sqrt(1/5)*(47*sqrt(5) - 105)*(-55/2*sqrt (5) - 123/2)^(3/4) + 2*x) + 4)/x
Time = 1.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log {\left (\frac {19251200 t^{7}}{11} + \frac {369792 t^{3}}{11} + x \right )} \right )\right )} - \frac {1}{x} \] Input:
integrate(1/x**2/(x**8+3*x**4+1),x)
Output:
RootSum(40960000*_t**8 + 787200*_t**4 + 1, Lambda(_t, _t*log(19251200*_t** 7/11 + 369792*_t**3/11 + x))) - 1/x
\[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + 3 \, x^{4} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8+3*x^4+1),x, algorithm="maxima")
Output:
-1/x - integrate((x^6 + 3*x^2)/(x^8 + 3*x^4 + 1), x)
Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {25 \, \sqrt {5} + 55} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {25 \, \sqrt {5} - 55} - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} - 55} \log \left (748225 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 748225 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {25 \, \sqrt {5} + 55} \log \left (180625 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 180625 \, x^{2}\right ) - \frac {1}{x} \] Input:
integrate(1/x^2/(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(748225*(x + sqrt(sqrt(5) + 1))^2 + 748225*x^2) + 1/40*sq rt(25*sqrt(5) - 55)*log(748225*(x - sqrt(sqrt(5) + 1))^2 + 748225*x^2) + 1 /40*sqrt(25*sqrt(5) + 55)*log(180625*(x + sqrt(sqrt(5) - 1))^2 + 180625*x^ 2) - 1/40*sqrt(25*sqrt(5) + 55)*log(180625*(x - sqrt(sqrt(5) - 1))^2 + 180 625*x^2) - 1/x
Time = 20.14 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=-\frac {1}{x}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2585\,2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {1155\,2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,{\left (-55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,2585{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,{\left (55\,\sqrt {5}-123\right )}^{1/4}\,1{}\mathrm {i}}{20} \] Input:
int(1/(x^2*(3*x^4 + x^8 + 1)),x)
Output:
- 1/x - (2^(3/4)*5^(1/2)*atan((2585*2^(3/4)*x*(- 55*5^(1/2) - 123)^(1/4))/ (2*(3025*5^(1/2) + 6765)) + (1155*2^(3/4)*5^(1/2)*x*(- 55*5^(1/2) - 123)^( 1/4))/(2*(3025*5^(1/2) + 6765)))*(- 55*5^(1/2) - 123)^(1/4))/20 - (2^(3/4) *5^(1/2)*atan((2585*2^(3/4)*x*(55*5^(1/2) - 123)^(1/4))/(2*(3025*5^(1/2) - 6765)) - (1155*2^(3/4)*5^(1/2)*x*(55*5^(1/2) - 123)^(1/4))/(2*(3025*5^(1/ 2) - 6765)))*(55*5^(1/2) - 123)^(1/4))/20 - (2^(3/4)*5^(1/2)*atan((2^(3/4) *x*(- 55*5^(1/2) - 123)^(1/4)*2585i)/(2*(3025*5^(1/2) + 6765)) + (2^(3/4)* 5^(1/2)*x*(- 55*5^(1/2) - 123)^(1/4)*1155i)/(2*(3025*5^(1/2) + 6765)))*(- 55*5^(1/2) - 123)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((2^(3/4)*x*(55*5^(1 /2) - 123)^(1/4)*2585i)/(2*(3025*5^(1/2) - 6765)) - (2^(3/4)*5^(1/2)*x*(55 *5^(1/2) - 123)^(1/4)*1155i)/(2*(3025*5^(1/2) - 6765)))*(55*5^(1/2) - 123) ^(1/4)*1i)/20
\[ \int \frac {1}{x^2 \left (1+3 x^4+x^8\right )} \, dx=\frac {-\left (\int \frac {x^{6}}{x^{8}+3 x^{4}+1}d x \right ) x -3 \left (\int \frac {x^{2}}{x^{8}+3 x^{4}+1}d x \right ) x -1}{x} \] Input:
int(1/x^2/(x^8+3*x^4+1),x)
Output:
( - int(x**6/(x**8 + 3*x**4 + 1),x)*x - 3*int(x**2/(x**8 + 3*x**4 + 1),x)* x - 1)/x