\(\int \frac {a+b x^2}{2+x^2+x^4} \, dx\) [100]

Optimal result

Integrand size = 18, antiderivative size = 234 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1183, 648, 632, 210, 642} \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (a+\sqrt {2} b\right ) \arctan \left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\left (a-\sqrt {2} b\right ) \log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}} \]

[In]

Int[(a + b*x^2)/(2 + x^2 + x^4),x]

[Out]

-1/2*(Sqrt[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]]) + (S
qrt[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 - ((a - S
qrt[2]*b)*Log[Sqrt[2] - Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])]) + ((a - Sqrt[2]*b)*Log[Sqr
t[2] + Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])])

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1183

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]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a-\left (a-\sqrt {2} b\right ) x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} a+\left (a-\sqrt {2} b\right ) x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ & = \frac {1}{8} \left (\sqrt {2} a+2 b\right ) \int \frac {1}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx+\frac {1}{8} \left (\sqrt {2} a+2 b\right ) \int \frac {1}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx-\frac {\left (a-\sqrt {2} b\right ) \int \frac {-\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (a-\sqrt {2} b\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{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 )}}+\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 {1}{4} \left (\sqrt {2} a+2 b\right ) \text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,-\sqrt {-1+2 \sqrt {2}}+2 x\right )-\frac {1}{4} \left (\sqrt {2} a+2 b\right ) \text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,\sqrt {-1+2 \sqrt {2}}+2 x\right ) \\ & = -\frac {\left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 \sqrt {2}\right )}}+\frac {\left (a+\sqrt {2} b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \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 )}}+\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 )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.47 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\frac {\left (-2 i a+\left (i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{\sqrt {14-14 i \sqrt {7}}}+\frac {\left (2 i a+\left (-i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{\sqrt {14+14 i \sqrt {7}}} \]

[In]

Integrate[(a + b*x^2)/(2 + x^2 + x^4),x]

[Out]

(((-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]]

Maple [C] (verified)

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	\(\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}+x \sqrt {-1+2 \sqrt {2}}\right )}{56}+\frac {\left (7 \sqrt {2}\, a -\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}-\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}-x \sqrt {-1+2 \sqrt {2}}\right )}{56}-\frac {\left (-7 \sqrt {2}\, a +\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -4 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +4 \sqrt {-1+2 \sqrt {2}}\, a -2 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}\)	\(369\)

[In]

int((b*x^2+a)/(x^4+x^2+2),x,method=_RETURNVERBOSE)

[Out]

1/2*sum((_R^2*b+a)/(2*_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4+_Z^2+2))

Fricas [B] (verification not implemented)

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 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=-\frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x + \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} + \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) + \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x - \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} + \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} + \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) - \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x + \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} - \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) + \frac {1}{28} \, \sqrt {7} \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} \log \left (-4 \, {\left (a^{4} - a^{3} b + 2 \, a b^{3} - 4 \, b^{4}\right )} x - \sqrt {a^{2} - 8 \, a b + 2 \, b^{2} - \sqrt {7} \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}}} {\left (\sqrt {7} {\left (a^{3} - 2 \, a b^{2}\right )} - \sqrt {-a^{4} + 4 \, a^{2} b^{2} - 4 \, b^{4}} {\left (a - 4 \, b\right )}\right )}\right ) \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2),x, algorithm="fricas")

[Out]

-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) + sq
rt(-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)))

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.52 \[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\operatorname {RootSum} {\left (1568 t^{4} + t^{2} \left (- 28 a^{2} + 224 a b - 56 b^{2}\right ) + a^{4} - 2 a^{3} b + 5 a^{2} b^{2} - 4 a b^{3} + 4 b^{4}, \left ( t \mapsto t \log {\left (x + \frac {112 t^{3} a - 448 t^{3} b + 6 t a^{3} + 12 t a^{2} b - 48 t a b^{2} + 8 t b^{3}}{a^{4} - a^{3} b + 2 a b^{3} - 4 b^{4}} \right )} \right )\right )} \]

[In]

integrate((b*x**2+a)/(x**4+x**2+2),x)

[Out]

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))))

Maxima [F]

\[ \int \frac {a+b x^2}{2+x^2+x^4} \, dx=\int { \frac {b x^{2} + a}{x^{4} + x^{2} + 2} \,d x } \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/(x^4 + x^2 + 2), x)

Giac [B] (verification not implemented)

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 ) \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2),x, algorithm="giac")

[Out]

-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*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(sqrt(2) + 4)) - 1/896*sqrt(7)*(sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(s
qrt(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*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(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*s
qrt(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*sqrt(2*sqrt(2) + 8))*log(x
^2 - 2*2^(1/4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqrt(2))

Mupad [B] (verification not implemented)

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}} \]

[In]

int((a + b*x^2)/(x^2 + x^4 + 2),x)

[Out]

- 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*1i)/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^(1/2)*b^2*1i)/56 - (7^(1/2)*a^2*1i)/112 - (a*b)/14 + a^2/112 + b^
2/56)^(1/2)