Integrand size = 37, antiderivative size = 94 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \]
-(x^4+1)^(1/4)/x-1/3*arctan(3^(1/2)*x*(x^4+1)^(1/4)/(-x^2+(x^4+1)^(1/2)))* 3^(1/2)+1/3*arctanh(3^(1/2)*x*(x^4+1)^(1/4)/(x^2+(x^4+1)^(1/2)))*3^(1/2)
Time = 0.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt [4]{1+x^4}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \]
-((1 + x^4)^(1/4)/x) + ArcTan[(Sqrt[3]*x)/(x - 2*(1 + x^4)^(1/4))]/Sqrt[3] - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/4))]/Sqrt[3] + ArcTanh[(Sqrt[3]* x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])]/Sqrt[3]
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {x^8+3 x^4+1}{x^2 \left (x^4+1\right )^{3/4} \left (3 x^8+3 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{x^2 \left (x^4+1\right )^{3/4}}-\frac {2 x^6}{\left (x^4+1\right )^{3/4} \left (3 x^8+3 x^4+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \arctan \left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}+\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )+\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}-\frac {\sqrt [4]{x^4+1}}{x}\) |
-((1 + x^4)^(1/4)/x) - ((3 + I*Sqrt[3])*(-((3*I - Sqrt[3])/(3*I + Sqrt[3]) ))^(1/4)*ArcTan[x/((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*(1 + x^4)^(1 /4))])/6 - ((3 - I*Sqrt[3])*ArcTan[((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^( 1/4)*x)/(1 + x^4)^(1/4)])/(6*(-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)) + ((3 + I*Sqrt[3])*(-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*ArcTanh[x/((- ((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))])/6 + ((3 - I*Sq rt[3])*ArcTanh[((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1 /4)])/(6*(-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4))
3.14.11.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 49.72 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {3}\, \ln \left (\frac {\sqrt {3}\, \left (x^{4}+1\right )^{\frac {1}{4}} x +x^{2}+\sqrt {x^{4}+1}}{x^{2}}\right ) x -\sqrt {3}\, \ln \left (\frac {-\sqrt {3}\, \left (x^{4}+1\right )^{\frac {1}{4}} x +x^{2}+\sqrt {x^{4}+1}}{x^{2}}\right ) x +2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{4}}+x \right ) \sqrt {3}}{3 x}\right ) x -2 \sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{4}+1\right )^{\frac {1}{4}}+x \right ) \sqrt {3}}{3 x}\right ) x -6 \left (x^{4}+1\right )^{\frac {1}{4}}}{6 x}\) | \(136\) |
| trager | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{6}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{8}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{6}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{8}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}\) | \(287\) |
| risch | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{16}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{10}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{12}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}-42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{6}+28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{8}-12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{4}-6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{16}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{10}-27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{12}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}+42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{8}-12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}+30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}+6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}\right ) {\left (\left (x^{4}+1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}+1\right )^{\frac {3}{4}}}\) | \(628\) |
1/6*(3^(1/2)*ln((3^(1/2)*(x^4+1)^(1/4)*x+x^2+(x^4+1)^(1/2))/x^2)*x-3^(1/2) *ln((-3^(1/2)*(x^4+1)^(1/4)*x+x^2+(x^4+1)^(1/2))/x^2)*x+2*3^(1/2)*arctan(1 /3*(2*(x^4+1)^(1/4)+x)*3^(1/2)/x)*x-2*3^(1/2)*arctan(1/3*(-2*(x^4+1)^(1/4) +x)*3^(1/2)/x)*x-6*(x^4+1)^(1/4))/x
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (78) = 156\).
Time = 11.62 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.41 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, {\left (\sqrt {3} {\left (3 \, x^{5} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \sqrt {3} {\left (3 \, x^{7} + 4 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )}}{21 \, x^{8} + 21 \, x^{4} - 1}\right ) - \sqrt {3} x \log \left (-\frac {441 \, x^{16} + 882 \, x^{12} + 543 \, x^{8} + 102 \, x^{4} + 4 \, \sqrt {3} {\left (63 \, x^{13} + 78 \, x^{9} + 24 \, x^{5} + x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, \sqrt {3} {\left (63 \, x^{15} + 111 \, x^{11} + 57 \, x^{7} + 8 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 24 \, {\left (18 \, x^{14} + 27 \, x^{10} + 11 \, x^{6} + x^{2}\right )} \sqrt {x^{4} + 1} + 1}{9 \, x^{16} + 18 \, x^{12} + 15 \, x^{8} + 6 \, x^{4} + 1}\right ) + 12 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x} \]
-1/12*(2*sqrt(3)*x*arctan(2*(sqrt(3)*(3*x^5 - x)*(x^4 + 1)^(3/4) - sqrt(3) *(3*x^7 + 4*x^3)*(x^4 + 1)^(1/4))/(21*x^8 + 21*x^4 - 1)) - sqrt(3)*x*log(- (441*x^16 + 882*x^12 + 543*x^8 + 102*x^4 + 4*sqrt(3)*(63*x^13 + 78*x^9 + 2 4*x^5 + x)*(x^4 + 1)^(3/4) + 4*sqrt(3)*(63*x^15 + 111*x^11 + 57*x^7 + 8*x^ 3)*(x^4 + 1)^(1/4) + 24*(18*x^14 + 27*x^10 + 11*x^6 + x^2)*sqrt(x^4 + 1) + 1)/(9*x^16 + 18*x^12 + 15*x^8 + 6*x^4 + 1)) + 12*(x^4 + 1)^(1/4))/x
Timed out. \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int { \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int { \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int \frac {x^8+3\,x^4+1}{x^2\,{\left (x^4+1\right )}^{3/4}\,\left (3\,x^8+3\,x^4+1\right )} \,d x \]