3.1.8 \(\int \frac {1}{a^2+b+2 a x^2+x^4} \, dx\)

Optimal. Leaf size=299 \[ -\frac {\log \left (-\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}}+\frac {\log \left (\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b}-a}-\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b}-a}+\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}} \]

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Rubi [A]  time = 0.31, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1094, 634, 618, 204, 628} \begin {gather*} -\frac {\log \left (-\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}}+\frac {\log \left (\sqrt {2} x \sqrt {\sqrt {a^2+b}-a}+\sqrt {a^2+b}+x^2\right )}{4 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}-a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b}-a}-\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b}-a}+\sqrt {2} x}{\sqrt {\sqrt {a^2+b}+a}}\right )}{2 \sqrt {2} \sqrt {a^2+b} \sqrt {\sqrt {a^2+b}+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + b + 2*a*x^2 + x^4)^(-1),x]

[Out]

-ArcTan[(Sqrt[-a + Sqrt[a^2 + b]] - Sqrt[2]*x)/Sqrt[a + Sqrt[a^2 + b]]]/(2*Sqrt[2]*Sqrt[a^2 + b]*Sqrt[a + Sqrt
[a^2 + b]]) + ArcTan[(Sqrt[-a + Sqrt[a^2 + b]] + Sqrt[2]*x)/Sqrt[a + Sqrt[a^2 + b]]]/(2*Sqrt[2]*Sqrt[a^2 + b]*
Sqrt[a + Sqrt[a^2 + b]]) - Log[Sqrt[a^2 + b] - Sqrt[2]*Sqrt[-a + Sqrt[a^2 + b]]*x + x^2]/(4*Sqrt[2]*Sqrt[a^2 +
b]*Sqrt[-a + Sqrt[a^2 + b]]) + Log[Sqrt[a^2 + b] + Sqrt[2]*Sqrt[-a + Sqrt[a^2 + b]]*x + x^2]/(4*Sqrt[2]*Sqrt[
a^2 + b]*Sqrt[-a + Sqrt[a^2 + b]])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

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 628

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 634

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 1094

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

Rubi steps

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

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Mathematica [C]  time = 0.04, size = 81, normalized size = 0.27 \begin {gather*} -\frac {i \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a-i \sqrt {b}}}\right )}{\sqrt {a-i \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {a+i \sqrt {b}}}\right )}{\sqrt {a+i \sqrt {b}}}\right )}{2 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + b + 2*a*x^2 + x^4)^(-1),x]

[Out]

((-1/2*I)*(ArcTan[x/Sqrt[a - I*Sqrt[b]]]/Sqrt[a - I*Sqrt[b]] - ArcTan[x/Sqrt[a + I*Sqrt[b]]]/Sqrt[a + I*Sqrt[b
]]))/Sqrt[b]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a^2+b+2 a x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + b + 2*a*x^2 + x^4)^(-1),x]

[Out]

IntegrateAlgebraic[(a^2 + b + 2*a*x^2 + x^4)^(-1), x]

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fricas [B]  time = 0.67, size = 583, normalized size = 1.95 \begin {gather*} \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} \log \left ({\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + b\right )} \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} \log \left (-{\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + b\right )} \sqrt {\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} + a}{a^{2} b + b^{2}}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} \log \left ({\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - b\right )} \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} \log \left (-{\left ({\left (a^{3} b + a b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - b\right )} \sqrt {-\frac {{\left (a^{2} b + b^{2}\right )} \sqrt {-\frac {1}{a^{4} b + 2 \, a^{2} b^{2} + b^{3}}} - a}{a^{2} b + b^{2}}} + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+2*a*x^2+a^2+b),x, algorithm="fricas")

[Out]

1/4*sqrt(((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) + a)/(a^2*b + b^2))*log(((a^3*b + a*b^2)*sqrt(-1/(a
^4*b + 2*a^2*b^2 + b^3)) + b)*sqrt(((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) + a)/(a^2*b + b^2)) + x)
- 1/4*sqrt(((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) + a)/(a^2*b + b^2))*log(-((a^3*b + a*b^2)*sqrt(-1
/(a^4*b + 2*a^2*b^2 + b^3)) + b)*sqrt(((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) + a)/(a^2*b + b^2)) +
x) - 1/4*sqrt(-((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) - a)/(a^2*b + b^2))*log(((a^3*b + a*b^2)*sqrt
(-1/(a^4*b + 2*a^2*b^2 + b^3)) - b)*sqrt(-((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) - a)/(a^2*b + b^2)
) + x) + 1/4*sqrt(-((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) - a)/(a^2*b + b^2))*log(-((a^3*b + a*b^2)
*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) - b)*sqrt(-((a^2*b + b^2)*sqrt(-1/(a^4*b + 2*a^2*b^2 + b^3)) - a)/(a^2*b +
b^2)) + x)

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giac [A]  time = 0.16, size = 75, normalized size = 0.25 \begin {gather*} -\frac {\sqrt {a + \sqrt {-b}} \arctan \left (\frac {x}{\sqrt {a + \sqrt {-b}}}\right )}{2 \, {\left (a \sqrt {-b} - b\right )}} + \frac {\sqrt {a - \sqrt {-b}} \arctan \left (\frac {x}{\sqrt {a - \sqrt {-b}}}\right )}{2 \, {\left (a \sqrt {-b} + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+2*a*x^2+a^2+b),x, algorithm="giac")

[Out]

-1/2*sqrt(a + sqrt(-b))*arctan(x/sqrt(a + sqrt(-b)))/(a*sqrt(-b) - b) + 1/2*sqrt(a - sqrt(-b))*arctan(x/sqrt(a
- sqrt(-b)))/(a*sqrt(-b) + b)

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maple [B]  time = 0.13, size = 1099, normalized size = 3.68 \begin {gather*} \frac {a^{4} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}\, b}-\frac {a^{4} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}\, b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a^{3} \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x +\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right )^{\frac {3}{2}} b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a^{3} \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x -\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right )^{\frac {3}{2}} b}+\frac {3 a^{2} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}}-\frac {3 a^{2} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}}-\frac {a^{2} \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \sqrt {a^{2}+b}\, \sqrt {2 a +2 \sqrt {a^{2}+b}}\, b}+\frac {a^{2} \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \sqrt {a^{2}+b}\, \sqrt {2 a +2 \sqrt {a^{2}+b}}\, b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a^{2} \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x +\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right ) b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a^{2} \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x -\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right ) b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x +\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right )^{\frac {3}{2}}}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, a \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x -\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right )^{\frac {3}{2}}}+\frac {b \arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{\left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}}-\frac {b \arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{\left (a^{2}+b \right )^{\frac {3}{2}} \sqrt {2 a +2 \sqrt {a^{2}+b}}}-\frac {\arctan \left (\frac {2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \sqrt {a^{2}+b}\, \sqrt {2 a +2 \sqrt {a^{2}+b}}}+\frac {\arctan \left (\frac {-2 x +\sqrt {-2 a +2 \sqrt {a^{2}+b}}}{\sqrt {2 a +2 \sqrt {a^{2}+b}}}\right )}{2 \sqrt {a^{2}+b}\, \sqrt {2 a +2 \sqrt {a^{2}+b}}}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, \ln \left (x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x +\sqrt {a^{2}+b}\right )}{8 a^{2}+8 b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, \ln \left (-x^{2}+\sqrt {-2 a +2 \sqrt {a^{2}+b}}\, x -\sqrt {a^{2}+b}\right )}{8 \left (a^{2}+b \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4+2*a*x^2+a^2+b),x)

[Out]

1/8/b/(a^2+b)*ln(x^2+x*(2*(a^2+b)^(1/2)-2*a)^(1/2)+(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a^2+1/8/b/(a^2+b
)^(3/2)*ln(x^2+x*(2*(a^2+b)^(1/2)-2*a)^(1/2)+(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a^3+1/8/(a^2+b)*ln(x^2
+x*(2*(a^2+b)^(1/2)-2*a)^(1/2)+(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)+1/8/(a^2+b)^(3/2)*ln(x^2+x*(2*(a^2+b
)^(1/2)-2*a)^(1/2)+(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a-1/2/b/(a^2+b)^(1/2)/(2*(a^2+b)^(1/2)+2*a)^(1/2
)*arctan((2*x+(2*(a^2+b)^(1/2)-2*a)^(1/2))/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^2+1/2/b/(a^2+b)^(3/2)/(2*(a^2+b)^(1/
2)+2*a)^(1/2)*arctan((2*x+(2*(a^2+b)^(1/2)-2*a)^(1/2))/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^4-1/2/(a^2+b)^(1/2)/(2*(
a^2+b)^(1/2)+2*a)^(1/2)*arctan((2*x+(2*(a^2+b)^(1/2)-2*a)^(1/2))/(2*(a^2+b)^(1/2)+2*a)^(1/2))+3/2/(a^2+b)^(3/2
)/(2*(a^2+b)^(1/2)+2*a)^(1/2)*arctan((2*x+(2*(a^2+b)^(1/2)-2*a)^(1/2))/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^2+b/(a^2
+b)^(3/2)/(2*(a^2+b)^(1/2)+2*a)^(1/2)*arctan((2*x+(2*(a^2+b)^(1/2)-2*a)^(1/2))/(2*(a^2+b)^(1/2)+2*a)^(1/2))-1/
8/b/(a^2+b)*ln(x*(2*(a^2+b)^(1/2)-2*a)^(1/2)-x^2-(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a^2-1/8/b/(a^2+b)^
(3/2)*ln(x*(2*(a^2+b)^(1/2)-2*a)^(1/2)-x^2-(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a^3-1/8/(a^2+b)*ln(x*(2*
(a^2+b)^(1/2)-2*a)^(1/2)-x^2-(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)-1/8/(a^2+b)^(3/2)*ln(x*(2*(a^2+b)^(1/2
)-2*a)^(1/2)-x^2-(a^2+b)^(1/2))*(2*(a^2+b)^(1/2)-2*a)^(1/2)*a+1/2/b/(a^2+b)^(1/2)/(2*(a^2+b)^(1/2)+2*a)^(1/2)*
arctan(((2*(a^2+b)^(1/2)-2*a)^(1/2)-2*x)/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^2-1/2/b/(a^2+b)^(3/2)/(2*(a^2+b)^(1/2)
+2*a)^(1/2)*arctan(((2*(a^2+b)^(1/2)-2*a)^(1/2)-2*x)/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^4+1/2/(a^2+b)^(1/2)/(2*(a^
2+b)^(1/2)+2*a)^(1/2)*arctan(((2*(a^2+b)^(1/2)-2*a)^(1/2)-2*x)/(2*(a^2+b)^(1/2)+2*a)^(1/2))-3/2/(a^2+b)^(3/2)/
(2*(a^2+b)^(1/2)+2*a)^(1/2)*arctan(((2*(a^2+b)^(1/2)-2*a)^(1/2)-2*x)/(2*(a^2+b)^(1/2)+2*a)^(1/2))*a^2-b/(a^2+b
)^(3/2)/(2*(a^2+b)^(1/2)+2*a)^(1/2)*arctan(((2*(a^2+b)^(1/2)-2*a)^(1/2)-2*x)/(2*(a^2+b)^(1/2)+2*a)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} + 2 \, a x^{2} + a^{2} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+2*a*x^2+a^2+b),x, algorithm="maxima")

[Out]

integrate(1/(x^4 + 2*a*x^2 + a^2 + b), x)

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mupad [B]  time = 4.38, size = 872, normalized size = 2.92 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {8\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a\,b^2}{a^2\,b^2+b^3}}-\frac {8\,a^2\,b^2\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}-\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}+\frac {8\,a\,b\,x\,\sqrt {\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}-\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}\,\sqrt {-b^3}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}-\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}-\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,b^2\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}+\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}-\frac {8\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}}{\frac {2\,b\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a\,b^2}{a^2\,b^2+b^3}}+\frac {8\,a\,b\,x\,\sqrt {\frac {\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}+\frac {a\,b}{16\,\left (a^2\,b^2+b^3\right )}}\,\sqrt {-b^3}}{\frac {2\,b^4\,\sqrt {-b^3}}{a^2\,b^2+b^3}+\frac {2\,a^3\,b^4}{a^2\,b^2+b^3}+\frac {2\,a\,b^5}{a^2\,b^2+b^3}+\frac {2\,a^2\,b^3\,\sqrt {-b^3}}{a^2\,b^2+b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {-b^3}}{16\,\left (a^2\,b^2+b^3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b + 2*a*x^2 + a^2 + x^4),x)

[Out]

- 2*atanh((8*x*((a*b)/(16*(b^3 + a^2*b^2)) - (-b^3)^(1/2)/(16*(b^3 + a^2*b^2)))^(1/2))/((2*b*(-b^3)^(1/2))/(b^
3 + a^2*b^2) - (2*a*b^2)/(b^3 + a^2*b^2)) - (8*a^2*b^2*x*((a*b)/(16*(b^3 + a^2*b^2)) - (-b^3)^(1/2)/(16*(b^3 +
a^2*b^2)))^(1/2))/((2*b^4*(-b^3)^(1/2))/(b^3 + a^2*b^2) - (2*a^3*b^4)/(b^3 + a^2*b^2) - (2*a*b^5)/(b^3 + a^2*
b^2) + (2*a^2*b^3*(-b^3)^(1/2))/(b^3 + a^2*b^2)) + (8*a*b*x*((a*b)/(16*(b^3 + a^2*b^2)) - (-b^3)^(1/2)/(16*(b^
3 + a^2*b^2)))^(1/2)*(-b^3)^(1/2))/((2*b^4*(-b^3)^(1/2))/(b^3 + a^2*b^2) - (2*a^3*b^4)/(b^3 + a^2*b^2) - (2*a*
b^5)/(b^3 + a^2*b^2) + (2*a^2*b^3*(-b^3)^(1/2))/(b^3 + a^2*b^2)))*((a*b - (-b^3)^(1/2))/(16*(b^3 + a^2*b^2)))^
(1/2) - 2*atanh((8*a^2*b^2*x*((-b^3)^(1/2)/(16*(b^3 + a^2*b^2)) + (a*b)/(16*(b^3 + a^2*b^2)))^(1/2))/((2*b^4*(
-b^3)^(1/2))/(b^3 + a^2*b^2) + (2*a^3*b^4)/(b^3 + a^2*b^2) + (2*a*b^5)/(b^3 + a^2*b^2) + (2*a^2*b^3*(-b^3)^(1/
2))/(b^3 + a^2*b^2)) - (8*x*((-b^3)^(1/2)/(16*(b^3 + a^2*b^2)) + (a*b)/(16*(b^3 + a^2*b^2)))^(1/2))/((2*b*(-b^
3)^(1/2))/(b^3 + a^2*b^2) + (2*a*b^2)/(b^3 + a^2*b^2)) + (8*a*b*x*((-b^3)^(1/2)/(16*(b^3 + a^2*b^2)) + (a*b)/(
16*(b^3 + a^2*b^2)))^(1/2)*(-b^3)^(1/2))/((2*b^4*(-b^3)^(1/2))/(b^3 + a^2*b^2) + (2*a^3*b^4)/(b^3 + a^2*b^2) +
(2*a*b^5)/(b^3 + a^2*b^2) + (2*a^2*b^3*(-b^3)^(1/2))/(b^3 + a^2*b^2)))*((a*b + (-b^3)^(1/2))/(16*(b^3 + a^2*b
^2)))^(1/2)

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sympy [A]  time = 0.80, size = 63, normalized size = 0.21 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} b^{2} + 256 b^{3}\right ) - 32 t^{2} a b + 1, \left (t \mapsto t \log {\left (64 t^{3} a^{3} b + 64 t^{3} a b^{2} - 4 t a^{2} + 4 t b + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4+2*a*x**2+a**2+b),x)

[Out]

RootSum(_t**4*(256*a**2*b**2 + 256*b**3) - 32*_t**2*a*b + 1, Lambda(_t, _t*log(64*_t**3*a**3*b + 64*_t**3*a*b*
*2 - 4*_t*a**2 + 4*_t*b + x)))

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