Integrand size = 18, antiderivative size = 234 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \]
-1/28*arctan((-2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(a+b*2^(1/2) )*(-14+28*2^(1/2))^(1/2)+1/28*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/ 2))^(1/2))*(a+b*2^(1/2))*(-14+28*2^(1/2))^(1/2)-1/4*ln(x^2+2^(1/2)-x*(-1+2 *2^(1/2))^(1/2))*(a-b*2^(1/2))/(-2+4*2^(1/2))^(1/2)+1/4*ln(x^2+2^(1/2)+x*( -1+2*2^(1/2))^(1/2))*(a-b*2^(1/2))/(-2+4*2^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.47 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\frac {\left (-2 i a+\left (i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{\sqrt {14-14 i \sqrt {7}}}+\frac {\left (2 i a+\left (-i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{\sqrt {14+14 i \sqrt {7}}} \]
(((-2*I)*a + (I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 - I*Sqrt[7])/2]])/Sqrt[14 - (14*I)*Sqrt[7]] + (((2*I)*a + (-I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 + I*Sqrt [7])/2]])/Sqrt[14 + (14*I)*Sqrt[7]]
Time = 0.46 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1483, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {a+b x^2}{x^4+x^2+2} \, dx\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a-\left (a-\sqrt {2} b\right ) x}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a+\left (a-\sqrt {2} b\right ) x}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx-\frac {1}{2} \left (a-\sqrt {2} b\right ) \int -\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx+\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {2 x+\sqrt {-1+2 \sqrt {2}}}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx+\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}-2 x}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx+\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {2 x+\sqrt {-1+2 \sqrt {2}}}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}-2 x}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx-\sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{-\left (2 x-\sqrt {-1+2 \sqrt {2}}\right )^2-2 \sqrt {2}-1}d\left (2 x-\sqrt {-1+2 \sqrt {2}}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {2 x+\sqrt {-1+2 \sqrt {2}}}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx-\sqrt {2 \sqrt {2}-1} \left (a+\sqrt {2} b\right ) \int \frac {1}{-\left (2 x+\sqrt {-1+2 \sqrt {2}}\right )^2-2 \sqrt {2}-1}d\left (2 x+\sqrt {-1+2 \sqrt {2}}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}-2 x}{x^2-\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx+\sqrt {\frac {2 \sqrt {2}-1}{1+2 \sqrt {2}}} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {2 x-\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\frac {1}{2} \left (a-\sqrt {2} b\right ) \int \frac {2 x+\sqrt {-1+2 \sqrt {2}}}{x^2+\sqrt {-1+2 \sqrt {2}} x+\sqrt {2}}dx+\sqrt {\frac {2 \sqrt {2}-1}{1+2 \sqrt {2}}} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt {\frac {2 \sqrt {2}-1}{1+2 \sqrt {2}}} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {2 x-\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {1}{2} \left (a-\sqrt {2} b\right ) \log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\sqrt {\frac {2 \sqrt {2}-1}{1+2 \sqrt {2}}} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \left (a-\sqrt {2} b\right ) \log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{2 \sqrt {2 \left (2 \sqrt {2}-1\right )}}\) |
(Sqrt[(-1 + 2*Sqrt[2])/(1 + 2*Sqrt[2])]*(a + Sqrt[2]*b)*ArcTan[(-Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]] - ((a - Sqrt[2]*b)*Log[Sqrt[2] - S qrt[-1 + 2*Sqrt[2]]*x + x^2])/2)/(2*Sqrt[2*(-1 + 2*Sqrt[2])]) + (Sqrt[(-1 + 2*Sqrt[2])/(1 + 2*Sqrt[2])]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]] + ((a - Sqrt[2]*b)*Log[Sqrt[2] + Sqrt[-1 + 2* Sqrt[2]]*x + x^2])/2)/(2*Sqrt[2*(-1 + 2*Sqrt[2])])
3.1.100.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (\textit {\_R}^{2} b +a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+\textit {\_R}}\right )}{2}\) | \(38\) |
| default | \(\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}+x \sqrt {-1+2 \sqrt {2}}\right )}{56}+\frac {\left (7 \sqrt {2}\, a -\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}-\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}-x \sqrt {-1+2 \sqrt {2}}\right )}{56}-\frac {\left (-7 \sqrt {2}\, a +\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}\) | \(369\) |
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (167) = 334\).
Time = 0.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.67 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x + \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} + \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) + \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x - \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} + \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) - \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x + \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} - \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) + \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x - \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} - \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) \]
-1/28*sqrt(7)*sqrt(a^2 - 8*a*b + 2*b^2 + sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4 *b^4))*log(-4*(a^4 - a^3*b + 2*a*b^3 - 4*b^4)*x + sqrt(a^2 - 8*a*b + 2*b^2 + sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4*b^4))*(sqrt(7)*(a^3 - 2*a*b^2) + sqrt (-a^4 + 4*a^2*b^2 - 4*b^4)*(a - 4*b))) + 1/28*sqrt(7)*sqrt(a^2 - 8*a*b + 2 *b^2 + sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4*b^4))*log(-4*(a^4 - a^3*b + 2*a*b ^3 - 4*b^4)*x - sqrt(a^2 - 8*a*b + 2*b^2 + sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4*b^4))*(sqrt(7)*(a^3 - 2*a*b^2) + sqrt(-a^4 + 4*a^2*b^2 - 4*b^4)*(a - 4* b))) - 1/28*sqrt(7)*sqrt(a^2 - 8*a*b + 2*b^2 - sqrt(7)*sqrt(-a^4 + 4*a^2*b ^2 - 4*b^4))*log(-4*(a^4 - a^3*b + 2*a*b^3 - 4*b^4)*x + sqrt(a^2 - 8*a*b + 2*b^2 - sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4*b^4))*(sqrt(7)*(a^3 - 2*a*b^2) - sqrt(-a^4 + 4*a^2*b^2 - 4*b^4)*(a - 4*b))) + 1/28*sqrt(7)*sqrt(a^2 - 8*a *b + 2*b^2 - sqrt(7)*sqrt(-a^4 + 4*a^2*b^2 - 4*b^4))*log(-4*(a^4 - a^3*b + 2*a*b^3 - 4*b^4)*x - sqrt(a^2 - 8*a*b + 2*b^2 - sqrt(7)*sqrt(-a^4 + 4*a^2 *b^2 - 4*b^4))*(sqrt(7)*(a^3 - 2*a*b^2) - sqrt(-a^4 + 4*a^2*b^2 - 4*b^4)*( a - 4*b)))
Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.52 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\operatorname {RootSum} {\left (1568 t^{4} + t^{2} \left (- 28 a^{2} + 224 a b - 56 b^{2}\right ) + a^{4} - 2 a^{3} b + 5 a^{2} b^{2} - 4 a b^{3} + 4 b^{4}, \left ( t \mapsto t \log {\left (x + \frac {112 t^{3} a - 448 t^{3} b + 6 t a^{3} + 12 t a^{2} b - 48 t a b^{2} + 8 t b^{3}}{a^{4} - a^{3} b + 2 a b^{3} - 4 b^{4}} \right )} \right )\right )} \]
RootSum(1568*_t**4 + _t**2*(-28*a**2 + 224*a*b - 56*b**2) + a**4 - 2*a**3* b + 5*a**2*b**2 - 4*a*b**3 + 4*b**4, Lambda(_t, _t*log(x + (112*_t**3*a - 448*_t**3*b + 6*_t*a**3 + 12*_t*a**2*b - 48*_t*a*b**2 + 8*_t*b**3)/(a**4 - a**3*b + 2*a*b**3 - 4*b**4))))
\[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\int { \frac {b x^{2} + a}{x^{4} + x^{2} + 2} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (167) = 334\).
Time = 0.54 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.58 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{896} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} + 4\right )} + 3 \, \sqrt {7} 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} - 4\right )} - 3 \cdot 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-2 \, \sqrt {2} + 8} - 2^{\frac {3}{4}} b {\left (\sqrt {2} - 4\right )} \sqrt {-2 \, \sqrt {2} + 8} - 8 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {2 \, \sqrt {2} + 8} + 8 \cdot 2^{\frac {1}{4}} a \sqrt {-2 \, \sqrt {2} + 8}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x + 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) - \frac {1}{896} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} + 4\right )} + 3 \, \sqrt {7} 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} - 4\right )} - 3 \cdot 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-2 \, \sqrt {2} + 8} - 2^{\frac {3}{4}} b {\left (\sqrt {2} - 4\right )} \sqrt {-2 \, \sqrt {2} + 8} - 8 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {2 \, \sqrt {2} + 8} + 8 \cdot 2^{\frac {1}{4}} a \sqrt {-2 \, \sqrt {2} + 8}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x - 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) - \frac {1}{1792} \, \sqrt {7} {\left (3 \, \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-2 \, \sqrt {2} + 8} + \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} - 4\right )} \sqrt {-2 \, \sqrt {2} + 8} + 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} + 4\right )} + 3 \cdot 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} - 4\right )} - 8 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {-2 \, \sqrt {2} + 8} - 8 \cdot 2^{\frac {1}{4}} a \sqrt {2 \, \sqrt {2} + 8}\right )} \log \left (x^{2} + 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) + \frac {1}{1792} \, \sqrt {7} {\left (3 \, \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} + 4\right )} \sqrt {-2 \, \sqrt {2} + 8} + \sqrt {7} 2^{\frac {3}{4}} b {\left (\sqrt {2} - 4\right )} \sqrt {-2 \, \sqrt {2} + 8} + 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} + 4\right )} + 3 \cdot 2^{\frac {3}{4}} b \sqrt {2 \, \sqrt {2} + 8} {\left (\sqrt {2} - 4\right )} - 8 \, \sqrt {7} 2^{\frac {1}{4}} a \sqrt {-2 \, \sqrt {2} + 8} - 8 \cdot 2^{\frac {1}{4}} a \sqrt {2 \, \sqrt {2} + 8}\right )} \log \left (x^{2} - 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) \]
-1/896*sqrt(7)*(sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*sq rt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 3*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) - 2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) - 8*sqrt(7)*2^(1/4)*a*sqrt(2*sqrt(2) + 8) + 8*2^(1/4)*a*sqrt(-2*sqrt(2) + 8) )*arctan(2*2^(3/4)*sqrt(1/2)*(x + 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sqrt(s qrt(2) + 4)) - 1/896*sqrt(7)*(sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt( 2) + 4) + 3*sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 3*2^(3/4 )*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) - 2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*s qrt(2) + 8) - 8*sqrt(7)*2^(1/4)*a*sqrt(2*sqrt(2) + 8) + 8*2^(1/4)*a*sqrt(- 2*sqrt(2) + 8))*arctan(2*2^(3/4)*sqrt(1/2)*(x - 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sqrt(sqrt(2) + 4)) - 1/1792*sqrt(7)*(3*sqrt(7)*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) + sqrt(7)*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2 ) + 8) + 2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*2^(3/4)*b*sqrt(2* sqrt(2) + 8)*(sqrt(2) - 4) - 8*sqrt(7)*2^(1/4)*a*sqrt(-2*sqrt(2) + 8) - 8* 2^(1/4)*a*sqrt(2*sqrt(2) + 8))*log(x^2 + 2*2^(1/4)*x*sqrt(-1/8*sqrt(2) + 1 /2) + sqrt(2)) + 1/1792*sqrt(7)*(3*sqrt(7)*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2 *sqrt(2) + 8) + sqrt(7)*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 2^( 3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*2^(3/4)*b*sqrt(2*sqrt(2) + 8) *(sqrt(2) - 4) - 8*sqrt(7)*2^(1/4)*a*sqrt(-2*sqrt(2) + 8) - 8*2^(1/4)*a*sq rt(2*sqrt(2) + 8))*log(x^2 - 2*2^(1/4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqr...
Time = 13.52 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.29 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\mathrm {atan}\left (\frac {a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,7{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,14{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}+\frac {\sqrt {7}\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {2\,\sqrt {7}\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}-a\,b^2-2\,a^2\,b+\frac {a^3}{2}+4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}\right )\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}+\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}-\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,2{}\mathrm {i}-2\,\mathrm {atanh}\left (\frac {7\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {14\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}+\frac {\sqrt {7}\,a^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,1{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}-\frac {\sqrt {7}\,b^2\,x\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}}\,2{}\mathrm {i}}{\frac {\sqrt {7}\,a^3\,1{}\mathrm {i}}{2}+a\,b^2+2\,a^2\,b-\frac {a^3}{2}-4\,b^3-\sqrt {7}\,a\,b^2\,1{}\mathrm {i}}\right )\,\sqrt {\frac {a^2}{112}-\frac {a\,b}{14}+\frac {b^2}{56}-\frac {\sqrt {7}\,a^2\,1{}\mathrm {i}}{112}+\frac {\sqrt {7}\,b^2\,1{}\mathrm {i}}{56}} \]
- atan((a^2*x*((7^(1/2)*a^2*1i)/112 - (a*b)/14 - (7^(1/2)*b^2*1i)/56 + a^2 /112 + b^2/56)^(1/2)*7i)/((7^(1/2)*a^3*1i)/2 - a*b^2 - 2*a^2*b + a^3/2 + 4 *b^3 - 7^(1/2)*a*b^2*1i) - (b^2*x*((7^(1/2)*a^2*1i)/112 - (a*b)/14 - (7^(1 /2)*b^2*1i)/56 + a^2/112 + b^2/56)^(1/2)*14i)/((7^(1/2)*a^3*1i)/2 - a*b^2 - 2*a^2*b + a^3/2 + 4*b^3 - 7^(1/2)*a*b^2*1i) + (7^(1/2)*a^2*x*((7^(1/2)*a ^2*1i)/112 - (a*b)/14 - (7^(1/2)*b^2*1i)/56 + a^2/112 + b^2/56)^(1/2))/((7 ^(1/2)*a^3*1i)/2 - a*b^2 - 2*a^2*b + a^3/2 + 4*b^3 - 7^(1/2)*a*b^2*1i) - ( 2*7^(1/2)*b^2*x*((7^(1/2)*a^2*1i)/112 - (a*b)/14 - (7^(1/2)*b^2*1i)/56 + a ^2/112 + b^2/56)^(1/2))/((7^(1/2)*a^3*1i)/2 - a*b^2 - 2*a^2*b + a^3/2 + 4* b^3 - 7^(1/2)*a*b^2*1i))*((7^(1/2)*a^2*1i)/112 - (a*b)/14 - (7^(1/2)*b^2*1 i)/56 + a^2/112 + b^2/56)^(1/2)*2i - 2*atanh((7*a^2*x*((7^(1/2)*b^2*1i)/56 - (7^(1/2)*a^2*1i)/112 - (a*b)/14 + a^2/112 + b^2/56)^(1/2))/((7^(1/2)*a^ 3*1i)/2 + a*b^2 + 2*a^2*b - a^3/2 - 4*b^3 - 7^(1/2)*a*b^2*1i) - (14*b^2*x* ((7^(1/2)*b^2*1i)/56 - (7^(1/2)*a^2*1i)/112 - (a*b)/14 + a^2/112 + b^2/56) ^(1/2))/((7^(1/2)*a^3*1i)/2 + a*b^2 + 2*a^2*b - a^3/2 - 4*b^3 - 7^(1/2)*a* b^2*1i) + (7^(1/2)*a^2*x*((7^(1/2)*b^2*1i)/56 - (7^(1/2)*a^2*1i)/112 - (a* b)/14 + a^2/112 + b^2/56)^(1/2)*1i)/((7^(1/2)*a^3*1i)/2 + a*b^2 + 2*a^2*b - a^3/2 - 4*b^3 - 7^(1/2)*a*b^2*1i) - (7^(1/2)*b^2*x*((7^(1/2)*b^2*1i)/56 - (7^(1/2)*a^2*1i)/112 - (a*b)/14 + a^2/112 + b^2/56)^(1/2)*2i)/((7^(1/2)* a^3*1i)/2 + a*b^2 + 2*a^2*b - a^3/2 - 4*b^3 - 7^(1/2)*a*b^2*1i))*((7^(1...