Integrand size = 18, antiderivative size = 157 \[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {-1+\sqrt {3}}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {-1+\sqrt {3}}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}} \]
1/4*arctan(2^(1/4)*x/(3^(1/2)-1)^(1/2))*2^(3/4)/(3^(1/2)-1)^(1/2)+1/4*arct anh(2^(1/4)*x/(3^(1/2)-1)^(1/2))*2^(3/4)/(3^(1/2)-1)^(1/2)-1/4*arctan(2^(1 /4)*x/(1+3^(1/2))^(1/2))*2^(3/4)/(1+3^(1/2))^(1/2)-1/4*arctanh(2^(1/4)*x/( 1+3^(1/2))^(1/2))*2^(3/4)/(1+3^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.34 \[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {x^4+1}{x^8-4 x^4+1} \, dx\) |
\(\Big \downarrow \) 1749 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4-\sqrt {6} x^2+1}dx+\frac {1}{2} \int \frac {1}{x^4+\sqrt {6} x^2+1}dx\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{x^2-\frac {1+\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}-\frac {\int \frac {1}{x^2+\frac {1-\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\int \frac {1}{x^2-\frac {1-\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}-\frac {\int \frac {1}{x^2+\frac {1+\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{x^2-\frac {1+\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}-\frac {\int \frac {1}{x^2+\frac {1-\sqrt {3}}{\sqrt {2}}}dx}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{\sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{\sqrt [4]{2} \sqrt {1+\sqrt {3}}}\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{\sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{\sqrt [4]{2} \sqrt {1+\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{\sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{\sqrt [4]{2} \sqrt {1+\sqrt {3}}}\right )\) |
(ArcTan[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2^(1/4)*Sqrt[-1 + Sqrt[3]]) - Arc Tan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2^(1/4)*Sqrt[1 + Sqrt[3]]))/2 + (ArcTa nh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTanh[ (2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2^(1/4)*Sqrt[1 + Sqrt[3]]))/2
3.1.17.3.1 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 = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.25
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-2 \textit {\_R}^{3}}\right )}{8}\) | \(40\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}-2 \textit {\_R}^{3}}\right )}{8}\) | \(40\) |
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (101) = 202\).
Time = 0.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.22 \[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {3} + 2}} \log \left ({\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {\sqrt {3} + 2}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {\sqrt {3} + 2}} \log \left (-{\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {\sqrt {3} + 2}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {3} + 2}} \log \left ({\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {-\sqrt {3} + 2}} + 2 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {-\sqrt {3} + 2}} \log \left (-{\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {-\sqrt {3} + 2}} + 2 \, x\right ) + \frac {1}{8} \, \sqrt {2} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left ({\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-{\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left ({\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) + \frac {1}{8} \, \sqrt {2} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-{\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) \]
1/8*sqrt(2)*sqrt(-sqrt(sqrt(3) + 2))*log((sqrt(3)*sqrt(2) - sqrt(2))*sqrt( -sqrt(sqrt(3) + 2)) + 2*x) - 1/8*sqrt(2)*sqrt(-sqrt(sqrt(3) + 2))*log(-(sq rt(3)*sqrt(2) - sqrt(2))*sqrt(-sqrt(sqrt(3) + 2)) + 2*x) - 1/8*sqrt(2)*sqr t(-sqrt(-sqrt(3) + 2))*log((sqrt(3)*sqrt(2) + sqrt(2))*sqrt(-sqrt(-sqrt(3) + 2)) + 2*x) + 1/8*sqrt(2)*sqrt(-sqrt(-sqrt(3) + 2))*log(-(sqrt(3)*sqrt(2 ) + sqrt(2))*sqrt(-sqrt(-sqrt(3) + 2)) + 2*x) + 1/8*sqrt(2)*(sqrt(3) + 2)^ (1/4)*log((sqrt(3)*sqrt(2) - sqrt(2))*(sqrt(3) + 2)^(1/4) + 2*x) - 1/8*sqr t(2)*(sqrt(3) + 2)^(1/4)*log(-(sqrt(3)*sqrt(2) - sqrt(2))*(sqrt(3) + 2)^(1 /4) + 2*x) - 1/8*sqrt(2)*(-sqrt(3) + 2)^(1/4)*log((sqrt(3)*sqrt(2) + sqrt( 2))*(-sqrt(3) + 2)^(1/4) + 2*x) + 1/8*sqrt(2)*(-sqrt(3) + 2)^(1/4)*log(-(s qrt(3)*sqrt(2) + sqrt(2))*(-sqrt(3) + 2)^(1/4) + 2*x)
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.15 \[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\operatorname {RootSum} {\left (1048576 t^{8} - 4096 t^{4} + 1, \left ( t \mapsto t \log {\left (4096 t^{5} - 12 t + x \right )} \right )\right )} \]
\[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\int { \frac {x^{4} + 1}{x^{8} - 4 \, x^{4} + 1} \,d x } \]
\[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\int { \frac {x^{4} + 1}{x^{8} - 4 \, x^{4} + 1} \,d x } \]
Time = 8.42 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.54 \[ \int \frac {1+x^4}{1-4 x^4+x^8} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {5184\,\sqrt {2}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {3024\,\sqrt {2}\,\sqrt {3}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,5184{}\mathrm {i}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}-\frac {\sqrt {2}\,\sqrt {3}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,3024{}\mathrm {i}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}\,1{}\mathrm {i}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {5184\,\sqrt {2}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}-\frac {3024\,\sqrt {2}\,\sqrt {3}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,5184{}\mathrm {i}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {\sqrt {2}\,\sqrt {3}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,3024{}\mathrm {i}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}\,1{}\mathrm {i}}{4} \]
(2^(1/2)*atan((2^(1/2)*x*(2 - 3^(1/2))^(1/4)*5184i)/(2160*3^(1/2)*(2 - 3^( 1/2))^(1/2) - 3888*(2 - 3^(1/2))^(1/2)) - (2^(1/2)*3^(1/2)*x*(2 - 3^(1/2)) ^(1/4)*3024i)/(2160*3^(1/2)*(2 - 3^(1/2))^(1/2) - 3888*(2 - 3^(1/2))^(1/2) ))*(2 - 3^(1/2))^(1/4)*1i)/4 - (2^(1/2)*atan((5184*2^(1/2)*x*(2 - 3^(1/2)) ^(1/4))/(2160*3^(1/2)*(2 - 3^(1/2))^(1/2) - 3888*(2 - 3^(1/2))^(1/2)) - (3 024*2^(1/2)*3^(1/2)*x*(2 - 3^(1/2))^(1/4))/(2160*3^(1/2)*(2 - 3^(1/2))^(1/ 2) - 3888*(2 - 3^(1/2))^(1/2)))*(2 - 3^(1/2))^(1/4))/4 + (2^(1/2)*atan((51 84*2^(1/2)*x*(3^(1/2) + 2)^(1/4))/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2) *(3^(1/2) + 2)^(1/2)) + (3024*2^(1/2)*3^(1/2)*x*(3^(1/2) + 2)^(1/4))/(3888 *(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)))*(3^(1/2) + 2)^(1 /4))/4 - (2^(1/2)*atan((2^(1/2)*x*(3^(1/2) + 2)^(1/4)*5184i)/(3888*(3^(1/2 ) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)) + (2^(1/2)*3^(1/2)*x*(3^( 1/2) + 2)^(1/4)*3024i)/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)))*(3^(1/2) + 2)^(1/4)*1i)/4