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

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

[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]])

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Rubi [A]  time = 0.312186, antiderivative size = 299, normalized size of antiderivative = 1., 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} \[ -\frac{\log \left (-\sqrt{2} x \sqrt{\sqrt{a^2+b}-a}+\sqrt{a^2+b}+x^2\right )}{4 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}-a}}+\frac{\log \left (\sqrt{2} x \sqrt{\sqrt{a^2+b}-a}+\sqrt{a^2+b}+x^2\right )}{4 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}-a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b}-a}-\sqrt{2} x}{\sqrt{\sqrt{a^2+b}+a}}\right )}{2 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}+a}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b}-a}+\sqrt{2} x}{\sqrt{\sqrt{a^2+b}+a}}\right )}{2 \sqrt{2} \sqrt{a^2+b} \sqrt{\sqrt{a^2+b}+a}} \]

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 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]

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 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 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 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]

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*}

Mathematica [C]  time = 0.0442712, size = 81, normalized size = 0.27 \[ -\frac{i \left (\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a-i \sqrt{b}}}\right )}{\sqrt{a-i \sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a+i \sqrt{b}}}\right )}{\sqrt{a+i \sqrt{b}}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*(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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Maple [B]  time = 0.199, size = 1099, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} + 2 \, a x^{2} + a^{2} + b}\,{d x} \end{align*}

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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Fricas [B]  time = 1.41114, size = 1206, normalized size = 4.03 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.736991, size = 63, normalized size = 0.21 \begin{align*} \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{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError