Integrand size = 16, antiderivative size = 167 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{144-64 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{144-64 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \] Output:
1/20*(3+5^(1/2))^(3/4)*arctan(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*2^(1/4)*5^( 1/2)-1/20*(144-64*5^(1/2))^(1/4)*arctan((3/2+1/2*5^(1/2))^(1/4)*x)*5^(1/2) -1/20*(3+5^(1/2))^(3/4)*arctanh(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*2^(1/4)*5 ^(1/2)+1/20*(144-64*5^(1/2))^(1/4)*arctanh((3/2+1/2*5^(1/2))^(1/4)*x)*5^(1 /2)
Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\frac {\left (-3+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}+\frac {\left (3+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}-\frac {\left (-3+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \] Input:
Integrate[x^6/(1 - 3*x^4 + x^8),x]
Output:
(((-3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] + (( 3 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] - ((-3 + S qrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((3 + Sqrt [5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])
Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1710, 27, 827, 216, 219}
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.
| \(\displaystyle \int \frac {x^6}{x^8-3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {1}{10} \left (5-3 \sqrt {5}\right ) \int -\frac {2 x^2}{-2 x^4-\sqrt {5}+3}dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int -\frac {2 x^2}{-2 x^4+\sqrt {5}+3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {x^2}{-2 x^4-\sqrt {5}+3}dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{-2 x^4+\sqrt {5}+3}dx\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{2 \sqrt {2}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}dx}{2 \sqrt {2}}\right )-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{2 \sqrt {2}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}dx}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{2 \sqrt {2}}-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt [4]{3-\sqrt {5}}}\right )-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{2 \sqrt {2}}-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {1}{5} \left (5+3 \sqrt {5}\right ) \left (\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \left (\frac {\text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt [4]{3-\sqrt {5}}}\right )\) |
Input:
Int[x^6/(1 - 3*x^4 + x^8),x]
Output:
-1/5*((5 + 3*Sqrt[5])*(-1/2*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(3/4)*(3 + Sqrt[5])^(1/4)) + ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2*2^(3/4)*(3 + Sqr t[5])^(1/4)))) - ((5 - 3*Sqrt[5])*(-1/2*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x]/ (2^(3/4)*(3 - Sqrt[5])^(1/4)) + ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x]/(2*2^(3 /4)*(3 - Sqrt[5])^(1/4))))/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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^n/2)*(b/q + 1) Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Simp[(d^n/2)*(b/q - 1) Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] & & NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}-7 \textit {\_R} +2 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}+7 \textit {\_R} +2 x \right )\right )}{4}\) | \(68\) |
| default | \(-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}\) | \(130\) |
Input:
int(x^6/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(5*_R^3-7*_R+2*x),_R=RootOf(25*_Z^4-20*_Z^2-1))+1/4*sum(_R*ln (5*_R^3+7*_R+2*x),_R=RootOf(25*_Z^4+20*_Z^2-1))
Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \arctan \left (\frac {1}{2} \, {\left (3 \, \sqrt {5} x - 5 \, x\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \arctan \left (\frac {1}{2} \, {\left (3 \, \sqrt {5} x + 5 \, x\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \log \left ({\left (\sqrt {5} - 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \log \left (-{\left (\sqrt {5} - 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \log \left ({\left (\sqrt {5} + 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \log \left (-{\left (\sqrt {5} + 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} + 2 \, x\right ) \] Input:
integrate(x^6/(x^8-3*x^4+1),x, algorithm="fricas")
Output:
1/2*sqrt(1/5*sqrt(5) + 2/5)*arctan(1/2*(3*sqrt(5)*x - 5*x)*sqrt(1/5*sqrt(5 ) + 2/5)) - 1/2*sqrt(1/5*sqrt(5) - 2/5)*arctan(1/2*(3*sqrt(5)*x + 5*x)*sqr t(1/5*sqrt(5) - 2/5)) + 1/4*sqrt(1/5*sqrt(5) + 2/5)*log((sqrt(5) - 5)*sqrt (1/5*sqrt(5) + 2/5) + 2*x) - 1/4*sqrt(1/5*sqrt(5) + 2/5)*log(-(sqrt(5) - 5 )*sqrt(1/5*sqrt(5) + 2/5) + 2*x) + 1/4*sqrt(1/5*sqrt(5) - 2/5)*log((sqrt(5 ) + 5)*sqrt(1/5*sqrt(5) - 2/5) + 2*x) - 1/4*sqrt(1/5*sqrt(5) - 2/5)*log(-( sqrt(5) + 5)*sqrt(1/5*sqrt(5) - 2/5) + 2*x)
Time = 0.82 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.32 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \] Input:
integrate(x**6/(x**8-3*x**4+1),x)
Output:
RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(-1792000*_t**7 + 492 0*_t**3 + x))) + RootSum(6400*_t**4 + 320*_t**2 - 1, Lambda(_t, _t*log(-17 92000*_t**7 + 4920*_t**3 + x)))
\[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\int { \frac {x^{6}}{x^{8} - 3 \, x^{4} + 1} \,d x } \] Input:
integrate(x^6/(x^8-3*x^4+1),x, algorithm="maxima")
Output:
integrate(x^6/(x^8 - 3*x^4 + 1), x)
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \] Input:
integrate(x^6/(x^8-3*x^4+1),x, algorithm="giac")
Output:
1/10*sqrt(5*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt(5* sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) + 10 )*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*log(ab s(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*log(abs(x + sq rt(1/2*sqrt(5) - 1/2))) - 1/20*sqrt(5*sqrt(5) - 10)*log(abs(x - sqrt(1/2*s qrt(5) - 1/2)))
Time = 19.53 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {2-\sqrt {5}}}{8\,\sqrt {5}-24}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {-\sqrt {5}-2}}{8\,\sqrt {5}+24}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,16{}\mathrm {i}}{8\,\sqrt {5}-24}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,16{}\mathrm {i}}{8\,\sqrt {5}+24}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10} \] Input:
int(x^6/(x^8 - 3*x^4 + 1),x)
Output:
(5^(1/2)*atan((16*x*(2 - 5^(1/2))^(1/2))/(8*5^(1/2) - 24))*(5^(1/2) - 2)^( 1/2)*1i)/10 + (5^(1/2)*atan((16*x*(- 5^(1/2) - 2)^(1/2))/(8*5^(1/2) + 24)) *(5^(1/2) + 2)^(1/2)*1i)/10 + (5^(1/2)*atan((x*(2 - 5^(1/2))^(1/2)*16i)/(8 *5^(1/2) - 24))*(2 - 5^(1/2))^(1/2)*1i)/10 + (5^(1/2)*atan((x*(- 5^(1/2) - 2)^(1/2)*16i)/(8*5^(1/2) + 24))*(- 5^(1/2) - 2)^(1/2)*1i)/10
\[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {3 \sqrt {\sqrt {5}+1}\, \sqrt {10}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {5}+1}\, \sqrt {2}}\right )}{20}-\frac {\sqrt {\sqrt {5}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {5}+1}\, \sqrt {2}}\right )}{4}+\frac {3 \sqrt {\sqrt {5}-1}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{40}-\frac {3 \sqrt {\sqrt {5}-1}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{40}+\frac {\sqrt {\sqrt {5}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{8}-\frac {\sqrt {\sqrt {5}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{8}+2 \left (\int \frac {x^{2}}{x^{8}-3 x^{4}+1}d x \right )+\int \frac {1}{x^{8}-3 x^{4}+1}d x \] Input:
int(x^6/(x^8-3*x^4+1),x)
Output:
(6*sqrt(sqrt(5) + 1)*sqrt(10)*atan((2*x)/(sqrt(sqrt(5) + 1)*sqrt(2))) - 10 *sqrt(sqrt(5) + 1)*sqrt(2)*atan((2*x)/(sqrt(sqrt(5) + 1)*sqrt(2))) + 3*sqr t(sqrt(5) - 1)*sqrt(10)*log( - sqrt(sqrt(5) - 1) + sqrt(2)*x) - 3*sqrt(sqr t(5) - 1)*sqrt(10)*log(sqrt(sqrt(5) - 1) + sqrt(2)*x) + 5*sqrt(sqrt(5) - 1 )*sqrt(2)*log( - sqrt(sqrt(5) - 1) + sqrt(2)*x) - 5*sqrt(sqrt(5) - 1)*sqrt (2)*log(sqrt(sqrt(5) - 1) + sqrt(2)*x) + 80*int(x**2/(x**8 - 3*x**4 + 1),x ) + 40*int(1/(x**8 - 3*x**4 + 1),x))/40