3.18 \(\int \frac{1+x^4}{1-5 x^4+x^8} \, dx\)

Optimal. Leaf size=171 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTan[Sqr
t[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqrt[
3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTanh[Sqrt[2/(Sqrt[3] + Sqrt[
7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])]

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Rubi [A]  time = 0.272449, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)/(1 - 5*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTan[Sqr
t[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqrt[
3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTanh[Sqrt[2/(Sqrt[3] + Sqrt[
7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])]

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Rubi in Sympy [A]  time = 17.7604, size = 168, normalized size = 0.98 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{6 \sqrt{- \sqrt{3} + \sqrt{7}}} - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{6 \sqrt{\sqrt{3} + \sqrt{7}}} + \frac{\sqrt{6} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{6 \sqrt{- \sqrt{3} + \sqrt{7}}} - \frac{\sqrt{6} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{6 \sqrt{\sqrt{3} + \sqrt{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+1)/(x**8-5*x**4+1),x)

[Out]

sqrt(6)*atan(sqrt(2)*x/sqrt(-sqrt(3) + sqrt(7)))/(6*sqrt(-sqrt(3) + sqrt(7))) -
sqrt(6)*atan(sqrt(2)*x/sqrt(sqrt(3) + sqrt(7)))/(6*sqrt(sqrt(3) + sqrt(7))) + sq
rt(6)*atanh(sqrt(2)*x/sqrt(-sqrt(3) + sqrt(7)))/(6*sqrt(-sqrt(3) + sqrt(7))) - s
qrt(6)*atanh(sqrt(2)*x/sqrt(sqrt(3) + sqrt(7)))/(6*sqrt(sqrt(3) + sqrt(7)))

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Mathematica [C]  time = 0.0209534, size = 55, normalized size = 0.32 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-5 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)/(1 - 5*x^4 + x^8),x]

[Out]

RootSum[1 - 5*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-5*#1^3 + 2*#1^7
) & ]/4

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Maple [C]  time = 0.012, size = 42, normalized size = 0.3 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-5\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-5\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8-5*x^4+1),x)

[Out]

1/4*sum((_R^4+1)/(2*_R^7-5*_R^3)*ln(x-_R),_R=RootOf(_Z^8-5*_Z^4+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} + 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^8 - 5*x^4 + 1),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/(x^8 - 5*x^4 + 1), x)

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Fricas [A]  time = 0.289364, size = 701, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^8 - 5*x^4 + 1),x, algorithm="fricas")

[Out]

sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*arctan(3/2*sqrt
(1/3)*sqrt(sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*(sqrt(7) + sqrt(3))
/(sqrt(3)*x + sqrt(3)*sqrt(1/2*sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3)))*
(sqrt(7)*sqrt(3) + 5) + x^2))) - sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*sqrt(7
) + 5*sqrt(3))))*arctan(3/2*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*sqrt(7) + 5
*sqrt(3))))*(sqrt(7) - sqrt(3))/(sqrt(3)*x + sqrt(3)*sqrt(-1/2*sqrt(1/6)*sqrt(sq
rt(3)*(3*sqrt(7) + 5*sqrt(3)))*(sqrt(7)*sqrt(3) - 5) + x^2))) - 1/4*sqrt(1/3)*sq
rt(sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*log(3/2*sqrt(1/3)*sqrt(sqrt
(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*(sqrt(7) + sqrt(3)) + sqrt(3)*x) +
1/4*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*log(-3/2*s
qrt(1/3)*sqrt(sqrt(1/6)*sqrt(-sqrt(3)*(3*sqrt(7) - 5*sqrt(3))))*(sqrt(7) + sqrt(
3)) + sqrt(3)*x) + 1/4*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*sqrt(7) + 5*sqrt
(3))))*log(3/2*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*sqrt(7) + 5*sqrt(3))))*(
sqrt(7) - sqrt(3)) + sqrt(3)*x) - 1/4*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*s
qrt(7) + 5*sqrt(3))))*log(-3/2*sqrt(1/3)*sqrt(sqrt(1/6)*sqrt(sqrt(3)*(3*sqrt(7)
+ 5*sqrt(3))))*(sqrt(7) - sqrt(3)) + sqrt(3)*x)

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Sympy [A]  time = 0.562695, size = 24, normalized size = 0.14 \[ \operatorname{RootSum}{\left (5308416 t^{8} - 11520 t^{4} + 1, \left ( t \mapsto t \log{\left (9216 t^{5} - 16 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)/(x**8-5*x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 11520*_t**4 + 1, Lambda(_t, _t*log(9216*_t**5 - 16*_t +
x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} + 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)/(x^8 - 5*x^4 + 1),x, algorithm="giac")

[Out]

integrate((x^4 + 1)/(x^8 - 5*x^4 + 1), x)