Integrand size = 27, antiderivative size = 139 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {\left (-\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{-1+2 x-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\left (-\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1-2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
1/2*arctan(x/(x^4+1)^(1/4))-1/2*arctan((-2^(1/2)+x*2^(1/2))*(x^4+1)^(1/4)/ (-1+2*x-x^2+(x^4+1)^(1/2)))*2^(1/2)+1/2*arctanh(x/(x^4+1)^(1/4))-1/2*arcta nh((-2^(1/2)+x*2^(1/2))*(x^4+1)^(1/4)/(1-2*x+x^2+(x^4+1)^(1/2)))*2^(1/2)
Time = 8.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {1}{2} \left (\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} (-1+x) \sqrt [4]{1+x^4}}{-1+2 x-x^2+\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (-1+x) \sqrt [4]{1+x^4}}{1-2 x+x^2+\sqrt {1+x^4}}\right )\right ) \]
(ArcTan[x/(1 + x^4)^(1/4)] - Sqrt[2]*ArcTan[(Sqrt[2]*(-1 + x)*(1 + x^4)^(1 /4))/(-1 + 2*x - x^2 + Sqrt[1 + x^4])] + ArcTanh[x/(1 + x^4)^(1/4)] - Sqrt [2]*ArcTanh[(Sqrt[2]*(-1 + x)*(1 + x^4)^(1/4))/(1 - 2*x + x^2 + Sqrt[1 + x ^4])])/2
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.88 (sec) , antiderivative size = 851, normalized size of antiderivative = 6.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2027, 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.
| \(\displaystyle \int \frac {x^2-2 x}{\left (x^2-x+1\right ) \sqrt [4]{x^4+1}} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {(x-2) x}{\left (x^2-x+1\right ) \sqrt [4]{x^4+1}}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\sqrt [4]{x^4+1}}-\frac {x+1}{\left (x^2-x+1\right ) \sqrt [4]{x^4+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} \left (1+i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right ) x^3-\frac {1}{6} \left (1-i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right ) x^3+\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}\) |
-1/6*((1 + I*Sqrt[3])*x^3*AppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^4)/(1 - I *Sqrt[3])]) - ((1 - I*Sqrt[3])*x^3*AppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^ 4)/(1 + I*Sqrt[3])])/6 + ArcTan[x/(1 + x^4)^(1/4)]/2 - ((1 + I*Sqrt[3])*(- ((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*ArcTan[x/((-((I - Sqrt[3])/(I + Sqrt[ 3])))^(1/4)*(1 + x^4)^(1/4))])/4 - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4) *ArcTan[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1/4)])/2 - ( ((1 - I*Sqrt[3])/2)^(3/4)*ArcTan[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sqrt[3]) ^(1/4)])/2 - (((1 + I*Sqrt[3])/2)^(3/4)*ArcTan[(2^(1/4)*(1 + x^4)^(1/4))/( 1 + I*Sqrt[3])^(1/4)])/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 - ((1 + I*Sqrt[3]) *(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*ArcTanh[x/((-((I - Sqrt[3])/(I + S qrt[3])))^(1/4)*(1 + x^4)^(1/4))])/4 - ((-((I - Sqrt[3])/(I + Sqrt[3])))^( 3/4)*ArcTanh[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1/4)])/ 2 + (((1 - I*Sqrt[3])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sq rt[3])^(1/4)])/2 + (((1 + I*Sqrt[3])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^( 1/4))/(1 + I*Sqrt[3])^(1/4)])/2 - ((I/2)*Sqrt[-x^4]*EllipticPi[(-I - Sqrt[ 3])/2, ArcSin[(1 + x^4)^(1/4)], -1])/x^2 + ((I/2)*Sqrt[-x^4]*EllipticPi[(I - Sqrt[3])/2, ArcSin[(1 + x^4)^(1/4)], -1])/x^2 + ((I/2)*Sqrt[-x^4]*Ellip ticPi[1/Sqrt[(1 - I*Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)], -1])/x^2 - ((I/2 )*Sqrt[-x^4]*EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)] , -1])/x^2
3.20.65.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.81 (sec) , antiderivative size = 593, normalized size of antiderivative = 4.27
1/4*RootOf(_Z^2+1)*ln(-2*RootOf(_Z^2+1)*(x^4+1)^(1/2)*x^2+2*RootOf(_Z^2+1) *x^4+2*(x^4+1)^(3/4)*x-2*x^3*(x^4+1)^(1/4)+RootOf(_Z^2+1))-1/4*ln(2*(x^4+1 )^(3/4)*x-2*(x^4+1)^(1/2)*x^2+2*x^3*(x^4+1)^(1/4)-2*x^4-1)+1/2*RootOf(_Z^2 +RootOf(_Z^2+1))*ln((-(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^ 2+1)*x^2+2*(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+Root Of(_Z^2+1)*(x^4+1)^(1/4)*x^3+(x^4+1)^(3/4)*x-(x^4+1)^(1/2)*RootOf(_Z^2+Roo tOf(_Z^2+1))*RootOf(_Z^2+1)-3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x^2-2*RootOf(_Z ^2+RootOf(_Z^2+1))*x^3-(x^4+1)^(3/4)+3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x+3*Ro otOf(_Z^2+RootOf(_Z^2+1))*x^2-RootOf(_Z^2+1)*(x^4+1)^(1/4)-2*RootOf(_Z^2+R ootOf(_Z^2+1))*x)/(x^2-x+1)^2)-1/2*RootOf(_Z^2+1)*RootOf(_Z^2+RootOf(_Z^2+ 1))*ln(((x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*x^2-RootOf(_Z^2+1)*(x^4+ 1)^(1/4)*x^3+2*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^3+(x^4+1)^(3/4 )*x-2*(x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))*x+3*RootOf(_Z^2+1)*(x^4+1) ^(1/4)*x^2-3*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^2-(x^4+1)^(3/4)+ (x^4+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^2+1))-3*RootOf(_Z^2+1)*(x^4+1)^(1/4)*x +2*RootOf(_Z^2+RootOf(_Z^2+1))*RootOf(_Z^2+1)*x+RootOf(_Z^2+1)*(x^4+1)^(1/ 4))/(x^2-x+1)^2)
Result contains complex when optimal does not.
Time = 7.26 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.47 \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i + 1\right ) \, x^{2} - \left (2 i + 2\right ) \, x + i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{3} + \left (3 i - 3\right ) \, x^{2} - \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} + 3 i \, x^{2} - 3 i \, x + i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i - 1\right ) \, x^{2} + \left (2 i - 2\right ) \, x - i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{3} - \left (3 i + 3\right ) \, x^{2} + \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} - 3 i \, x^{2} + 3 i \, x - i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i - 1\right ) \, x^{2} - \left (2 i - 2\right ) \, x + i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{3} + \left (3 i + 3\right ) \, x^{2} - \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} - 3 i \, x^{2} + 3 i \, x - i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i + 1\right ) \, x^{2} + \left (2 i + 2\right ) \, x - i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} + \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{3} - \left (3 i - 3\right ) \, x^{2} + \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} + 3 i \, x^{2} - 3 i \, x + i\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{4} \, \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}{4} \, \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 ) \]
(1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*((I + 1)*x^2 - (2*I + 2)* x + I + 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqrt(2)*(-(2*I - 2)*x^3 + (3*I - 3)*x^2 - (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I*x^3 + 3*I*x^2 - 3*I*x + I))/ (x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) - (1/8*I + 1/8)*sqrt(2)*log((sqrt(2)*sqrt (x^4 + 1)*(-(I - 1)*x^2 + (2*I - 2)*x - I + 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqrt(2)*((2*I + 2)*x^3 - (3*I + 3)*x^2 + (2*I + 2)*x) - 2*(x^4 + 1)^(1/ 4)*(I*x^3 - 3*I*x^2 + 3*I*x - I))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + (1/8* I + 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*((I - 1)*x^2 - (2*I - 2)*x + I - 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqrt(2)*(-(2*I + 2)*x^3 + (3*I + 3)*x^ 2 - (2*I + 2)*x) - 2*(x^4 + 1)^(1/4)*(I*x^3 - 3*I*x^2 + 3*I*x - I))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) - (1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*(-(I + 1)*x^2 + (2*I + 2)*x - I - 1) + 2*(x^4 + 1)^(3/4)*(x - 1) + sqr t(2)*((2*I - 2)*x^3 - (3*I - 3)*x^2 + (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I *x^3 + 3*I*x^2 - 3*I*x + I))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/4*arctan (2*(x^4 + 1)^(1/4)*x^3 + 2*(x^4 + 1)^(3/4)*x) + 1/4*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)
\[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x \left (x - 2\right )}{\sqrt [4]{x^{4} + 1} \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x^{2} - 2 \, x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}} \,d x } \]
\[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x^{2} - 2 \, x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-2 x+x^2}{\left (1-x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int -\frac {2\,x-x^2}{{\left (x^4+1\right )}^{1/4}\,\left (x^2-x+1\right )} \,d x \]