3.384 \(\int \frac{1}{x^4 \left (1+3 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=484 \[ -\frac{1}{3 x^3}+\frac{\sqrt [4]{843+377 \sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843+377 \sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843-377 \sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843-377 \sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843+377 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843+377 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843-377 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843-377 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]

[Out]

-1/(3*x^3) + ((843 + 377*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/
4)])/(2*2^(3/4)*Sqrt[5]) - ((843 + 377*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3
- Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((843 - 377*Sqrt[5])^(1/4)*ArcTan[1 - (
2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4
)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 + 377
*Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] - 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sqrt[2]*
x^2])/(4*2^(3/4)*Sqrt[5]) - ((843 + 377*Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] + 2
^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)*Sqrt[5]) - ((843 - 377*S
qrt[5])^(1/4)*Log[Sqrt[3 + Sqrt[5]] - 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^
2])/(4*2^(3/4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4)*Log[Sqrt[3 + Sqrt[5]] + 2^(
3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)*Sqrt[5])

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Rubi [A]  time = 0.808392, antiderivative size = 466, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{3 x^3}+\frac{\sqrt [4]{843+377 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843+377 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843-377 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843-377 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843+377 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843+377 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{843-377 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4} \sqrt{5}}+\frac{\sqrt [4]{843-377 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4} \sqrt{5}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/(x^4*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(3*x^3) + ((843 + 377*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/
4)])/(2*2^(3/4)*Sqrt[5]) - ((843 + 377*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3
- Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((843 - 377*Sqrt[5])^(1/4)*ArcTan[1 - (
2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4
)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((843 + 377
*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2]
)/(4*2^(3/4)*Sqrt[5]) - ((843 + 377*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2
*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5]) - ((843 - 377*Sqrt[5])^
(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/
4)*Sqrt[5]) + ((843 - 377*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + S
qrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)*Sqrt[5])

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Rubi in Sympy [A]  time = 93.5898, size = 610, normalized size = 1.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**8+3*x**4+1),x)

[Out]

2**(3/4)*(3/2 + 9*sqrt(5)/10)*sqrt(-2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(-s
qrt(5) + 3)**(1/4) + sqrt(-2*sqrt(5) + 6))/(24*(-sqrt(5) + 3)**(5/4)) - 2**(3/4)
*(3/2 + 9*sqrt(5)/10)*sqrt(-2*sqrt(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(-sqrt(5) +
3)**(1/4) + sqrt(-2*sqrt(5) + 6))/(24*(-sqrt(5) + 3)**(5/4)) + 2**(3/4)*(-9*sqr
t(5)/10 + 3/2)*sqrt(2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4
) + sqrt(2*sqrt(5) + 6))/(24*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(-9*sqrt(5)/10 + 3
/2)*sqrt(2*sqrt(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*
sqrt(5) + 6))/(24*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(3/2 + 9*sqrt(5)/10)*atan(2**
(3/4)*(x - (-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(6*sqrt(-2*sqrt(5)
+ 6)*(-sqrt(5) + 3)**(1/4)) - 2**(3/4)*(3/2 + 9*sqrt(5)/10)*atan(2**(3/4)*(x + (
-2*sqrt(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(6*sqrt(-2*sqrt(5) + 6)*(-sqrt(
5) + 3)**(1/4)) - 2**(3/4)*(-9*sqrt(5)/10 + 3/2)*atan(2**(3/4)*(x - (2*sqrt(5) +
6)**(1/4)/2)/(sqrt(5) + 3)**(1/4))/(6*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6))
- 2**(3/4)*(-9*sqrt(5)/10 + 3/2)*atan(2**(3/4)*(x + (2*sqrt(5) + 6)**(1/4)/2)/(
sqrt(5) + 3)**(1/4))/(6*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6)) - 1/(3*x**3)

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Mathematica [C]  time = 0.0235776, size = 65, normalized size = 0.13 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right ]-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(3*x^3) - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3
*#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.014, size = 50, normalized size = 0.1 \[ -{\frac{1}{3\,{x}^{3}}}+{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}-3 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^8+3*x^4+1),x)

[Out]

-1/3/x^3+1/4*sum((-_R^4-3)/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{3 \, x^{3}} - \int \frac{x^{4} + 3}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^4),x, algorithm="maxima")

[Out]

-1/3/x^3 - integrate((x^4 + 3)/(x^8 + 3*x^4 + 1), x)

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Fricas [A]  time = 0.328997, size = 1897, normalized size = 3.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^4),x, algorithm="fricas")

[Out]

-1/120*sqrt(5)*sqrt(2)*(12*(1/250)^(1/4)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(sqr
t(5)*(843*sqrt(5) - 1885))^(3/4)*x^3*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*
(843*sqrt(5) - 1885))^(3/4)*(3*sqrt(5) + 7)/(2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(s
qrt(5)*(843*sqrt(5) - 1885))*x + 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5
) - 1885))^(3/4)*(3*sqrt(5) + 7) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(84
3*sqrt(5) - 1885))*sqrt((843*sqrt(5)*x^2 - 1885*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*
(843*sqrt(5) - 1885))*(9*sqrt(5) - 20) - 5*(1/250)^(1/4)*(55*sqrt(5)*sqrt(2)*x -
123*sqrt(2)*x)*(sqrt(5)*(843*sqrt(5) - 1885))^(1/4))/(843*sqrt(5) - 1885)))) +
12*(1/250)^(1/4)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(sqrt(5)*(843*sqrt(5) - 1885
))^(3/4)*x^3*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5) - 1885))^(3
/4)*(3*sqrt(5) + 7)/(2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) - 18
85))*x - 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5) - 1885))^(3/4)*(3*sqrt
(5) + 7) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) - 1885))*sqrt(
(843*sqrt(5)*x^2 - 1885*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) - 1885))*(9
*sqrt(5) - 20) + 5*(1/250)^(1/4)*(55*sqrt(5)*sqrt(2)*x - 123*sqrt(2)*x)*(sqrt(5)
*(843*sqrt(5) - 1885))^(1/4))/(843*sqrt(5) - 1885)))) + 12*(1/250)^(1/4)*(sqrt(5
)*(843*sqrt(5) + 1885))^(3/4)*sqrt(sqrt(5)*(843*sqrt(5) - 1885))*x^3*arctan(5*sq
rt(1/10)*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5) + 1885))^(3/4)*(3*sqrt(5) - 7)/(2*s
qrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*x + 5*sqrt(1/10)*(1
/250)^(1/4)*(sqrt(5)*(843*sqrt(5) + 1885))^(3/4)*(3*sqrt(5) - 7) + 2*sqrt(5)*sqr
t(2)*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*sqrt((843*sqrt(5)*x^2 + 1885*
x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(9*sqrt(5) + 20) - 5*(1/25
0)^(1/4)*(55*sqrt(5)*sqrt(2)*x + 123*sqrt(2)*x)*(sqrt(5)*(843*sqrt(5) + 1885))^(
1/4))/(843*sqrt(5) + 1885)))) + 12*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5) + 1885))^
(3/4)*sqrt(sqrt(5)*(843*sqrt(5) - 1885))*x^3*arctan(5*sqrt(1/10)*(1/250)^(1/4)*(
sqrt(5)*(843*sqrt(5) + 1885))^(3/4)*(3*sqrt(5) - 7)/(2*sqrt(5)*sqrt(2)*sqrt(1/10
)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*x - 5*sqrt(1/10)*(1/250)^(1/4)*(sqrt(5)*(84
3*sqrt(5) + 1885))^(3/4)*(3*sqrt(5) - 7) + 2*sqrt(5)*sqrt(2)*sqrt(1/10)*sqrt(sqr
t(5)*(843*sqrt(5) + 1885))*sqrt((843*sqrt(5)*x^2 + 1885*x^2 + 2*sqrt(1/10)*sqrt(
sqrt(5)*(843*sqrt(5) + 1885))*(9*sqrt(5) + 20) + 5*(1/250)^(1/4)*(55*sqrt(5)*sqr
t(2)*x + 123*sqrt(2)*x)*(sqrt(5)*(843*sqrt(5) + 1885))^(1/4))/(843*sqrt(5) + 188
5)))) + 3*(1/250)^(1/4)*(sqrt(5)*(843*sqrt(5) + 1885))^(3/4)*sqrt(sqrt(5)*(843*s
qrt(5) - 1885))*x^3*log(843*sqrt(5)*x^2 + 1885*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(
843*sqrt(5) + 1885))*(9*sqrt(5) + 20) + 5*(1/250)^(1/4)*(55*sqrt(5)*sqrt(2)*x +
123*sqrt(2)*x)*(sqrt(5)*(843*sqrt(5) + 1885))^(1/4)) - 3*(1/250)^(1/4)*(sqrt(5)*
(843*sqrt(5) + 1885))^(3/4)*sqrt(sqrt(5)*(843*sqrt(5) - 1885))*x^3*log(843*sqrt(
5)*x^2 + 1885*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(9*sqrt(5) +
20) - 5*(1/250)^(1/4)*(55*sqrt(5)*sqrt(2)*x + 123*sqrt(2)*x)*(sqrt(5)*(843*sqrt
(5) + 1885))^(1/4)) + 3*(1/250)^(1/4)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(sqrt(5
)*(843*sqrt(5) - 1885))^(3/4)*x^3*log(843*sqrt(5)*x^2 - 1885*x^2 + 2*sqrt(1/10)*
sqrt(sqrt(5)*(843*sqrt(5) - 1885))*(9*sqrt(5) - 20) + 5*(1/250)^(1/4)*(55*sqrt(5
)*sqrt(2)*x - 123*sqrt(2)*x)*(sqrt(5)*(843*sqrt(5) - 1885))^(1/4)) - 3*(1/250)^(
1/4)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))*(sqrt(5)*(843*sqrt(5) - 1885))^(3/4)*x^3
*log(843*sqrt(5)*x^2 - 1885*x^2 + 2*sqrt(1/10)*sqrt(sqrt(5)*(843*sqrt(5) - 1885)
)*(9*sqrt(5) - 20) - 5*(1/250)^(1/4)*(55*sqrt(5)*sqrt(2)*x - 123*sqrt(2)*x)*(sqr
t(5)*(843*sqrt(5) - 1885))^(1/4)) + 4*sqrt(2)*sqrt(sqrt(5)*(843*sqrt(5) + 1885))
*sqrt(sqrt(5)*(843*sqrt(5) - 1885)))/(x^3*sqrt(843*sqrt(5) + 1885)*sqrt(843*sqrt
(5) - 1885))

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Sympy [A]  time = 4.0804, size = 34, normalized size = 0.07 \[ \operatorname{RootSum}{\left (40960000 t^{8} + 5395200 t^{4} + 1, \left ( t \mapsto t \log{\left (\frac{179200 t^{5}}{377} + \frac{23112 t}{377} + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 5395200*_t**4 + 1, Lambda(_t, _t*log(179200*_t**5/377 +
23112*_t/377 + x))) - 1/(3*x**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{8} + 3 \, x^{4} + 1\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 3*x^4 + 1)*x^4),x, algorithm="giac")

[Out]

integrate(1/((x^8 + 3*x^4 + 1)*x^4), x)