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

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

[Out]

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

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Rubi [A]  time = 0.574841, 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 \[ -\frac{\log \left (-\sqrt{2} \sqrt{\sqrt{a^2+1}-a} x+\sqrt{a^2+1}+x^2\right )}{4 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}-a}}+\frac{\log \left (\sqrt{2} \sqrt{\sqrt{a^2+1}-a} x+\sqrt{a^2+1}+x^2\right )}{4 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}-a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+1}-a}-\sqrt{2} x}{\sqrt{\sqrt{a^2+1}+a}}\right )}{2 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}+a}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a^2+1}-a}+\sqrt{2} x}{\sqrt{\sqrt{a^2+1}+a}}\right )}{2 \sqrt{2} \sqrt{a^2+1} \sqrt{\sqrt{a^2+1}+a}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a^2 + 2*a*x^2 + x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.4471, size = 264, normalized size = 0.88 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{- 2 a + 2 \sqrt{a^{2} + 1}}}{2}\right )}{\sqrt{a + \sqrt{a^{2} + 1}}} \right )}}{4 \sqrt{a + \sqrt{a^{2} + 1}} \sqrt{a^{2} + 1}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{- 2 a + 2 \sqrt{a^{2} + 1}}}{2}\right )}{\sqrt{a + \sqrt{a^{2} + 1}}} \right )}}{4 \sqrt{a + \sqrt{a^{2} + 1}} \sqrt{a^{2} + 1}} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x \sqrt{- a + \sqrt{a^{2} + 1}} + \sqrt{a^{2} + 1} \right )}}{8 \sqrt{- a + \sqrt{a^{2} + 1}} \sqrt{a^{2} + 1}} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x \sqrt{- a + \sqrt{a^{2} + 1}} + \sqrt{a^{2} + 1} \right )}}{8 \sqrt{- a + \sqrt{a^{2} + 1}} \sqrt{a^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4+2*a*x**2+a**2+1),x)

[Out]

sqrt(2)*atan(sqrt(2)*(x - sqrt(-2*a + 2*sqrt(a**2 + 1))/2)/sqrt(a + sqrt(a**2 +
1)))/(4*sqrt(a + sqrt(a**2 + 1))*sqrt(a**2 + 1)) + sqrt(2)*atan(sqrt(2)*(x + sqr
t(-2*a + 2*sqrt(a**2 + 1))/2)/sqrt(a + sqrt(a**2 + 1)))/(4*sqrt(a + sqrt(a**2 +
1))*sqrt(a**2 + 1)) - sqrt(2)*log(x**2 - sqrt(2)*x*sqrt(-a + sqrt(a**2 + 1)) + s
qrt(a**2 + 1))/(8*sqrt(-a + sqrt(a**2 + 1))*sqrt(a**2 + 1)) + sqrt(2)*log(x**2 +
sqrt(2)*x*sqrt(-a + sqrt(a**2 + 1)) + sqrt(a**2 + 1))/(8*sqrt(-a + sqrt(a**2 +
1))*sqrt(a**2 + 1))

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Mathematica [C]  time = 0.0471354, size = 52, normalized size = 0.17 \[ -\frac{1}{2} i \left (\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a-i}}\right )}{\sqrt{a-i}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{a+i}}\right )}{\sqrt{a+i}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.095, size = 1073, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+2*a*x^2+a^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} + 2 \, a x^{2} + a^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28996, size = 1048, normalized size = 3.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*((a + sqrt(a^2 + 1))*log(x^2 + sqrt(2)*x*sqrt((a^2 + (a^3 + a)/sqrt(
a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1))/(a^2 + 1)^(1/4) + sqrt(a^
2 + 1))/(a^2 + 1)^(1/4) - (a + sqrt(a^2 + 1))*log(x^2 - sqrt(2)*x*sqrt((a^2 + (a
^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1))/(a^2 + 1)^(1
/4) + sqrt(a^2 + 1))/(a^2 + 1)^(1/4) - 4*(a^2 + 1)^(3/4)*arctan((sqrt(a^4 + 2*a^
2 + 1) + (a^5 + 2*a^3 + a)/(sqrt(a^4 + 2*a^2 + 1)*sqrt(a^2 + 1)))/((sqrt(2)*sqrt
(x^2 + sqrt(2)*x*sqrt((a^2 + (a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/s
qrt(a^2 + 1) + 1))/(a^2 + 1)^(1/4) + sqrt(a^2 + 1))*(a + sqrt(a^2 + 1))*sqrt((a^
2 + (a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1)) + sqrt
(2)*(a*x + sqrt(a^2 + 1)*x)*sqrt((a^2 + (a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*
(a^3 + a)/sqrt(a^2 + 1) + 1)) + (a^2 + 1)^(1/4))*(a^2 + 1)^(1/4)))/sqrt(a^4 + 2*
a^2 + 1) - 4*(a^2 + 1)^(3/4)*arctan((sqrt(a^4 + 2*a^2 + 1) + (a^5 + 2*a^3 + a)/(
sqrt(a^4 + 2*a^2 + 1)*sqrt(a^2 + 1)))/((sqrt(2)*sqrt(x^2 - sqrt(2)*x*sqrt((a^2 +
(a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1))/(a^2 + 1)
^(1/4) + sqrt(a^2 + 1))*(a + sqrt(a^2 + 1))*sqrt((a^2 + (a^3 + a)/sqrt(a^2 + 1)
+ 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1)) + sqrt(2)*(a*x + sqrt(a^2 + 1)*x)*
sqrt((a^2 + (a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) + 1)
) - (a^2 + 1)^(1/4))*(a^2 + 1)^(1/4)))/sqrt(a^4 + 2*a^2 + 1))/((a + sqrt(a^2 + 1
))*sqrt((a^2 + (a^3 + a)/sqrt(a^2 + 1) + 1)/(2*a^2 + 2*(a^3 + a)/sqrt(a^2 + 1) +
1)))

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Sympy [A]  time = 1.37955, size = 48, normalized size = 0.16 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} + 256\right ) - 32 t^{2} a + 1, \left ( t \mapsto t \log{\left (64 t^{3} a^{3} + 64 t^{3} a - 4 t a^{2} + 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+2*a*x**2+a**2+1),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} + 2 \, a x^{2} + a^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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