Integrand size = 23, antiderivative size = 125 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^4}}{\sqrt {2}}}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]
2/3*(x^4-1)^(3/4)+arctan((x^4-1)^(1/4)/x)+1/4*arctan((-1/2*2^(1/2)+1/2*(x^ 4-1)^(1/2)*2^(1/2))/(x^4-1)^(1/4))*2^(1/2)-arctanh((x^4-1)^(1/4)/x)-1/4*ar ctanh(2^(1/2)*(x^4-1)^(1/4)/(1+(x^4-1)^(1/2)))*2^(1/2)
Time = 5.64 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]
(2*(-1 + x^4)^(3/4))/3 + ArcTan[(-1 + x^4)^(1/4)/x] - ArcTan[(Sqrt[2]*(-1 + x^4)^(1/4))/(-1 + Sqrt[-1 + x^4])]/(2*Sqrt[2]) - ArcTanh[(-1 + x^4)^(1/4 )/x] - ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(2*Sqrt[2] )
Time = 0.39 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2372, 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.
| \(\displaystyle \int \frac {2 x^4-2 x+1}{x \sqrt [4]{x^4-1}} \, dx\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \int \left (\frac {2 x^4+1}{x \sqrt [4]{x^4-1}}-\frac {2}{\sqrt [4]{x^4-1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {2}{3} \left (x^4-1\right )^{3/4}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}\) |
(2*(-1 + x^4)^(3/4))/3 - ArcTan[x/(-1 + x^4)^(1/4)] - ArcTan[1 - Sqrt[2]*( -1 + x^4)^(1/4)]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*(-1 + x^4)^(1/4)]/(2*Sqr t[2]) - ArcTanh[x/(-1 + x^4)^(1/4)] + Log[1 - Sqrt[2]*(-1 + x^4)^(1/4) + S qrt[-1 + x^4]]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*(-1 + x^4)^(1/4) + Sqrt[-1 + x^4]]/(4*Sqrt[2])
3.19.40.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 10.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14
| method | result | size |
| meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 1\right ], \left [2\right ], x^{4}\right )}{2 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {2 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(142\) |
| trager | \(\frac {2 \left (x^{4}-1\right )^{\frac {3}{4}}}{3}+\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-2 x^{4}+1\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}\) | \(345\) |
1/8/Pi*2^(1/2)*GAMMA(3/4)/signum(x^4-1)^(1/4)*(-signum(x^4-1))^(1/4)*(1/4* Pi*2^(1/2)/GAMMA(3/4)*x^4*hypergeom([1,1,5/4],[2,2],x^4)+(-3*ln(2)-1/2*Pi+ 4*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4))+1/2/signum(x^4-1)^(1/4)*(-signum(x^4- 1))^(1/4)*x^4*hypergeom([1/4,1],[2],x^4)-2/signum(x^4-1)^(1/4)*(-signum(x^ 4-1))^(1/4)*x*hypergeom([1/4,1/4],[5/4],x^4)
Result contains complex when optimal does not.
Time = 11.53 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.30 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=-\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - 2 i - 2\right )} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{4} - 1} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 4 i \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{4}}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + 2 i - 2\right )} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{4} - 1} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 4 i \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{4}}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - 2 i + 2\right )} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{4} - 1} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 4 i \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{4}}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + 2 i + 2\right )} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{4} - 1} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 4 i \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{4}}\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \frac {1}{2} \, \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{2} \, \log \left (-2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1\right ) \]
-(1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*((I + 1)*x^4 - 2*I - 2) - (2*I - 2)* sqrt(2)*sqrt(x^4 - 1) + 4*(x^4 - 1)^(3/4) - 4*I*(x^4 - 1)^(1/4))/x^4) + (1 /16*I - 1/16)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x^4 + 2*I - 2) + (2*I + 2)*sq rt(2)*sqrt(x^4 - 1) + 4*(x^4 - 1)^(3/4) + 4*I*(x^4 - 1)^(1/4))/x^4) - (1/1 6*I - 1/16)*sqrt(2)*log((sqrt(2)*((I - 1)*x^4 - 2*I + 2) - (2*I + 2)*sqrt( 2)*sqrt(x^4 - 1) + 4*(x^4 - 1)^(3/4) + 4*I*(x^4 - 1)^(1/4))/x^4) + (1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*(-(I + 1)*x^4 + 2*I + 2) + (2*I - 2)*sqrt(2) *sqrt(x^4 - 1) + 4*(x^4 - 1)^(3/4) - 4*I*(x^4 - 1)^(1/4))/x^4) + 2/3*(x^4 - 1)^(3/4) + 1/2*arctan(2*(x^4 - 1)^(1/4)*x^3 + 2*(x^4 - 1)^(3/4)*x) + 1/2 *log(-2*x^4 + 2*(x^4 - 1)^(1/4)*x^3 - 2*sqrt(x^4 - 1)*x^2 + 2*(x^4 - 1)^(3 /4)*x + 1)
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=- \frac {x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} + \frac {2 \left (x^{4} - 1\right )^{\frac {3}{4}}}{3} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \]
-x*exp(-I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), x**4)/(2*gamma(5/4)) + 2*(x**4 - 1)**(3/4)/3 - gamma(1/4)*hyper((1/4, 1/4), (5/4,), exp_polar(2 *I*pi)/x**4)/(4*x*gamma(5/4))
Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.24 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]
1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^4 - 1)^(1/4))) + 1/4*sqrt(2 )*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(x^4 - 1)^(1/4))) - 1/8*sqrt(2)*log(sqr t(2)*(x^4 - 1)^(1/4) + sqrt(x^4 - 1) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*(x^4 - 1)^(1/4) + sqrt(x^4 - 1) + 1) + 2/3*(x^4 - 1)^(3/4) + arctan((x^4 - 1)^( 1/4)/x) - 1/2*log((x^4 - 1)^(1/4)/x + 1) + 1/2*log((x^4 - 1)^(1/4)/x - 1)
\[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\int { \frac {2 \, x^{4} - 2 \, x + 1}{{\left (x^{4} - 1\right )}^{\frac {1}{4}} x} \,d x } \]
Time = 8.99 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2\,{\left (x^4-1\right )}^{3/4}}{3}-\frac {2\,x\,{\left (1-x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^4\right )}{{\left (x^4-1\right )}^{1/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]