Integrand size = 16, antiderivative size = 173 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \] Output:
-1/10*(3/2+1/2*5^(1/2))^(1/4)*arctan(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*5^(1 /2)+1/10*(3/2-1/2*5^(1/2))^(1/4)*arctan((3/2+1/2*5^(1/2))^(1/4)*x)*5^(1/2) -1/10*(3/2+1/2*5^(1/2))^(1/4)*arctanh(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*5^( 1/2)+1/10*(3/2-1/2*5^(1/2))^(1/4)*arctanh((3/2+1/2*5^(1/2))^(1/4)*x)*5^(1/ 2)
Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {\sqrt {-1+\sqrt {5}} \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {-1+\sqrt {5}} \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10}} \] Input:
Integrate[x^4/(1 - 3*x^4 + x^8),x]
Output:
(Sqrt[-1 + Sqrt[5]]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[1 + Sqrt[5]]*A rcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[-1 + Sqrt[5]]*ArcTanh[Sqrt[2/(-1 + S qrt[5])]*x] - Sqrt[1 + Sqrt[5]]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[ 10])
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1710, 756, 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.
| \(\displaystyle \int \frac {x^4}{x^8-3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (-3-\sqrt {5}\right )}dx+\frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{x^4+\frac {1}{2} \left (-3+\sqrt {5}\right )}dx\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}dx}{\sqrt {3-\sqrt {5}}}\right )+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}dx}{\sqrt {3+\sqrt {5}}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}\right )+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{10} \left (5+3 \sqrt {5}\right ) \left (-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}\right )+\frac {1}{10} \left (5-3 \sqrt {5}\right ) \left (-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}\right )\) |
Input:
Int[x^4/(1 - 3*x^4 + x^8),x]
Output:
((5 + 3*Sqrt[5])*(-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*(3 + Sqrt[5 ])^(3/4))) - ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*(3 + Sqrt[5])^(3/ 4))))/10 + ((5 - 3*Sqrt[5])*(-(ArcTan[((3 + Sqrt[5])/2)^(1/4)*x]/(2^(1/4)* (3 - Sqrt[5])^(3/4))) - ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x]/(2^(1/4)*(3 - S qrt[5])^(3/4))))/10
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) 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 = 60, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}\) | \(60\) |
| default | \(-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}\) | \(130\) |
Input:
int(x^4/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(10*_R^3+_R+x),_R=RootOf(25*_Z^4+5*_Z^2-1))+1/4*sum(_R*ln(-10 *_R^3+_R+x),_R=RootOf(25*_Z^4-5*_Z^2-1))
Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x - 5 \, x\right )} \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \log \left (x + \sqrt {5} \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}} \log \left (x - \sqrt {5} \sqrt {\frac {1}{10} \, \sqrt {5} + \frac {1}{10}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (x + \sqrt {5} \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}} \log \left (x - \sqrt {5} \sqrt {\frac {1}{10} \, \sqrt {5} - \frac {1}{10}}\right ) \] Input:
integrate(x^4/(x^8-3*x^4+1),x, algorithm="fricas")
Output:
1/2*sqrt(1/10*sqrt(5) + 1/10)*arctan(1/2*(sqrt(5)*x - 5*x)*sqrt(1/10*sqrt( 5) + 1/10)) + 1/2*sqrt(1/10*sqrt(5) - 1/10)*arctan(1/2*(sqrt(5)*x + 5*x)*s qrt(1/10*sqrt(5) - 1/10)) - 1/4*sqrt(1/10*sqrt(5) + 1/10)*log(x + sqrt(5)* sqrt(1/10*sqrt(5) + 1/10)) + 1/4*sqrt(1/10*sqrt(5) + 1/10)*log(x - sqrt(5) *sqrt(1/10*sqrt(5) + 1/10)) + 1/4*sqrt(1/10*sqrt(5) - 1/10)*log(x + sqrt(5 )*sqrt(1/10*sqrt(5) - 1/10)) - 1/4*sqrt(1/10*sqrt(5) - 1/10)*log(x - sqrt( 5)*sqrt(1/10*sqrt(5) - 1/10))
Time = 0.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.28 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} \] Input:
integrate(x**4/(x**8-3*x**4+1),x)
Output:
RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x))) + RootSum(6400*_t**4 + 80*_t**2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x)))
\[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\int { \frac {x^{4}}{x^{8} - 3 \, x^{4} + 1} \,d x } \] Input:
integrate(x^4/(x^8-3*x^4+1),x, algorithm="maxima")
Output:
integrate(x^4/(x^8 - 3*x^4 + 1), x)
Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \] Input:
integrate(x^4/(x^8-3*x^4+1),x, algorithm="giac")
Output:
-1/20*sqrt(10*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt( 10*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(10*sqrt(5) + 10)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)*l og(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) - 10)*log(abs( x + sqrt(1/2*sqrt(5) - 1/2))) - 1/40*sqrt(10*sqrt(5) - 10)*log(abs(x - sqr t(1/2*sqrt(5) - 1/2)))
Time = 19.51 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.55 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{20} \] Input:
int(x^4/(x^8 - 3*x^4 + 1),x)
Output:
(10^(1/2)*atan((10^(1/2)*x*(- 5^(1/2) - 1)^(1/2)*1i)/(2*(5^(1/2) - 1)) - ( 5^(1/2)*10^(1/2)*x*(- 5^(1/2) - 1)^(1/2)*3i)/(10*(5^(1/2) - 1)))*(- 5^(1/2 ) - 1)^(1/2)*1i)/20 + (10^(1/2)*atan((10^(1/2)*x*(1 - 5^(1/2))^(1/2)*1i)/( 2*(5^(1/2) + 1)) + (5^(1/2)*10^(1/2)*x*(1 - 5^(1/2))^(1/2)*3i)/(10*(5^(1/2 ) + 1)))*(1 - 5^(1/2))^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(5^(1/2) + 1)^(1/2)*1i)/(2*(5^(1/2) - 1)) - (5^(1/2)*10^(1/2)*x*(5^(1/2) + 1)^(1/2) *3i)/(10*(5^(1/2) - 1)))*(5^(1/2) + 1)^(1/2)*1i)/20 - (10^(1/2)*atan((10^( 1/2)*x*(5^(1/2) - 1)^(1/2)*1i)/(2*(5^(1/2) + 1)) + (5^(1/2)*10^(1/2)*x*(5^ (1/2) - 1)^(1/2)*3i)/(10*(5^(1/2) + 1)))*(5^(1/2) - 1)^(1/2)*1i)/20
\[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {\sqrt {\sqrt {5}+1}\, \sqrt {10}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {5}+1}\, \sqrt {2}}\right )}{20}-\frac {\sqrt {\sqrt {5}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {5}+1}\, \sqrt {2}}\right )}{4}+\frac {\sqrt {\sqrt {5}-1}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{40}-\frac {\sqrt {\sqrt {5}-1}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{40}+\frac {\sqrt {\sqrt {5}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{8}-\frac {\sqrt {\sqrt {5}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {5}-1}+\sqrt {2}\, x \right )}{8}+\int \frac {x^{2}}{x^{8}-3 x^{4}+1}d x +\int \frac {1}{x^{8}-3 x^{4}+1}d x \] Input:
int(x^4/(x^8-3*x^4+1),x)
Output:
(2*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))) + sqrt( sqrt(5) - 1)*sqrt(10)*log( - sqrt(sqrt(5) - 1) + sqrt(2)*x) - sqrt(sqrt(5) - 1)*sqrt(10)*log(sqrt(sqrt(5) - 1) + sqrt(2)*x) + 5*sqrt(sqrt(5) - 1)*sq rt(2)*log( - sqrt(sqrt(5) - 1) + sqrt(2)*x) - 5*sqrt(sqrt(5) - 1)*sqrt(2)* log(sqrt(sqrt(5) - 1) + sqrt(2)*x) + 40*int(x**2/(x**8 - 3*x**4 + 1),x) + 40*int(1/(x**8 - 3*x**4 + 1),x))/40