Integrand size = 18, antiderivative size = 131 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \]
arctan(x*2^(1/2)/(5^(1/2)-1)^(1/2))/(-2+2*5^(1/2))^(1/2)+arctanh(x*2^(1/2) /(5^(1/2)-1)^(1/2))/(-2+2*5^(1/2))^(1/2)-arctan(x*2^(1/2)/(5^(1/2)+1)^(1/2 ))/(2+2*5^(1/2))^(1/2)-arctanh(x*2^(1/2)/(5^(1/2)+1)^(1/2))/(2+2*5^(1/2))^ (1/2)
Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \]
ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/ Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt [5])]
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1749, 1406, 216, 220}
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.
\(\displaystyle \int \frac {x^4+1}{x^8-3 x^4+1} \, dx\) |
\(\Big \downarrow \) 1749 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4-\sqrt {5} x^2+1}dx+\frac {1}{2} \int \frac {1}{x^4+\sqrt {5} x^2+1}dx\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^2+\frac {1}{2} \left (-1-\sqrt {5}\right )}dx-\int \frac {1}{x^2+\frac {1}{2} \left (1-\sqrt {5}\right )}dx\right )+\frac {1}{2} \left (\int \frac {1}{x^2+\frac {1}{2} \left (-1+\sqrt {5}\right )}dx-\int \frac {1}{x^2+\frac {1}{2} \left (1+\sqrt {5}\right )}dx\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^2+\frac {1}{2} \left (-1-\sqrt {5}\right )}dx-\int \frac {1}{x^2+\frac {1}{2} \left (1-\sqrt {5}\right )}dx\right )+\frac {1}{2} \left (\sqrt {\frac {2}{\sqrt {5}-1}} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-\sqrt {\frac {2}{1+\sqrt {5}}} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{2} \left (\sqrt {\frac {2}{\sqrt {5}-1}} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-\sqrt {\frac {2}{1+\sqrt {5}}} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )\right )+\frac {1}{2} \left (\sqrt {\frac {2}{\sqrt {5}-1}} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-\sqrt {\frac {2}{1+\sqrt {5}}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )\right )\) |
(Sqrt[2/(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[2/(1 + Sqr t[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/2 + (Sqrt[2/(-1 + Sqrt[5])]*ArcTan h[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[2/(1 + Sqrt[5])]*ArcTanh[Sqrt[2/(1 + Sq rt[5])]*x])/2
3.1.16.3.1 Defintions of rubi rules used
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[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}\) | \(56\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{\sqrt {2 \sqrt {5}+2}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{\sqrt {2 \sqrt {5}-2}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{\sqrt {2 \sqrt {5}+2}}+\frac {\operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{\sqrt {2 \sqrt {5}-2}}\) | \(96\) |
1/4*sum(_R*ln(_R^3-_R+x),_R=RootOf(_Z^4-_Z^2-1))+1/4*sum(_R*ln(-_R^3-_R+x) ,_R=RootOf(_Z^4+_Z^2-1))
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (95) = 190\).
Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.42 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} + 1} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} + 1} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, x\right ) \]
1/8*sqrt(2)*sqrt(sqrt(5) + 1)*log((sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 1) + 4*x) - 1/8*sqrt(2)*sqrt(sqrt(5) + 1)*log(-(sqrt(5)*sqrt(2) - sqrt( 2))*sqrt(sqrt(5) + 1) + 4*x) - 1/8*sqrt(2)*sqrt(sqrt(5) - 1)*log((sqrt(5)* sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*x) + 1/8*sqrt(2)*sqrt(sqrt(5) - 1 )*log(-(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*x) - 1/8*sqrt(2)* sqrt(-sqrt(5) + 1)*log((sqrt(5)*sqrt(2) + sqrt(2))*sqrt(-sqrt(5) + 1) + 4* x) + 1/8*sqrt(2)*sqrt(-sqrt(5) + 1)*log(-(sqrt(5)*sqrt(2) + sqrt(2))*sqrt( -sqrt(5) + 1) + 4*x) + 1/8*sqrt(2)*sqrt(-sqrt(5) - 1)*log((sqrt(5)*sqrt(2) - sqrt(2))*sqrt(-sqrt(5) - 1) + 4*x) - 1/8*sqrt(2)*sqrt(-sqrt(5) - 1)*log (-(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(-sqrt(5) - 1) + 4*x)
Time = 0.66 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.37 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (256 t^{4} - 16 t^{2} - 1, \left ( t \mapsto t \log {\left (1024 t^{5} - 8 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (256 t^{4} + 16 t^{2} - 1, \left ( t \mapsto t \log {\left (1024 t^{5} - 8 t + x \right )} \right )\right )} \]
RootSum(256*_t**4 - 16*_t**2 - 1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x) )) + RootSum(256*_t**4 + 16*_t**2 - 1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x)))
\[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\int { \frac {x^{4} + 1}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=-\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
-1/4*sqrt(2*sqrt(5) - 2)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/4*sqrt(2*sq rt(5) + 2)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/8*sqrt(2*sqrt(5) - 2)*log (abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/8*sqrt(2*sqrt(5) - 2)*log(abs(x - s qrt(1/2*sqrt(5) + 1/2))) + 1/8*sqrt(2*sqrt(5) + 2)*log(abs(x + sqrt(1/2*sq rt(5) - 1/2))) - 1/8*sqrt(2*sqrt(5) + 2)*log(abs(x - sqrt(1/2*sqrt(5) - 1/ 2)))
Time = 0.12 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.05 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}-\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}-1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}+1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}+\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}+1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {5}}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}-\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {1-\sqrt {5}}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}+\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {-\sqrt {5}-1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{4} \]
(2^(1/2)*atan((2^(1/2)*x*(1 - 5^(1/2))^(1/2)*1875i)/(2*(875*5^(1/2) - 1875 )) - (2^(1/2)*5^(1/2)*x*(1 - 5^(1/2))^(1/2)*875i)/(2*(875*5^(1/2) - 1875)) )*(1 - 5^(1/2))^(1/2)*1i)/4 - (2^(1/2)*atan((2^(1/2)*x*(5^(1/2) + 1)^(1/2) *1875i)/(2*(875*5^(1/2) + 1875)) + (2^(1/2)*5^(1/2)*x*(5^(1/2) + 1)^(1/2)* 875i)/(2*(875*5^(1/2) + 1875)))*(5^(1/2) + 1)^(1/2)*1i)/4 - (2^(1/2)*atan( (2^(1/2)*x*(5^(1/2) - 1)^(1/2)*1875i)/(2*(875*5^(1/2) - 1875)) - (2^(1/2)* 5^(1/2)*x*(5^(1/2) - 1)^(1/2)*875i)/(2*(875*5^(1/2) - 1875)))*(5^(1/2) - 1 )^(1/2)*1i)/4 + (2^(1/2)*atan((2^(1/2)*x*(- 5^(1/2) - 1)^(1/2)*1875i)/(2*( 875*5^(1/2) + 1875)) + (2^(1/2)*5^(1/2)*x*(- 5^(1/2) - 1)^(1/2)*875i)/(2*( 875*5^(1/2) + 1875)))*(- 5^(1/2) - 1)^(1/2)*1i)/4