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

Optimal. Leaf size=157 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}} \]

[Out]

ArcTan[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/
(2*2^(1/4)*Sqrt[1 + Sqrt[3]]) + ArcTanh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTa
nh[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[1 + Sqrt[3]])

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Rubi [A]  time = 0.0865303, antiderivative size = 157, 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, Rules used = {1419, 1093, 203, 207} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{\sqrt{3}-1}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 4*x^4 + x^8),x]

[Out]

ArcTan[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/
(2*2^(1/4)*Sqrt[1 + Sqrt[3]]) + ArcTanh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[-1 + Sqrt[3]]) - ArcTa
nh[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[1 + Sqrt[3]])

Rule 1419

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[(2*d)/e -
b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x^(n/2
) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2,
0] && IGtQ[n/2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !LtQ[(2*d)/e - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))

Rule 1093

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

Rule 203

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

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1+x^4}{1-4 x^4+x^8} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{6} x^2+x^4} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{6} x^2+x^4} \, dx\\ &=\frac{\int \frac{1}{-\sqrt{\frac{3}{2}}-\frac{1}{\sqrt{2}}+x^2} \, dx}{2 \sqrt{2}}+\frac{\int \frac{1}{\sqrt{\frac{3}{2}}-\frac{1}{\sqrt{2}}+x^2} \, dx}{2 \sqrt{2}}-\frac{\int \frac{1}{-\sqrt{\frac{3}{2}}+\frac{1}{\sqrt{2}}+x^2} \, dx}{2 \sqrt{2}}-\frac{\int \frac{1}{\sqrt{\frac{3}{2}}+\frac{1}{\sqrt{2}}+x^2} \, dx}{2 \sqrt{2}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{-1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{-1+\sqrt{3}}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{-1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{-1+\sqrt{3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{1+\sqrt{3}}}\\ \end{align*}

Mathematica [C]  time = 0.0129564, size = 53, normalized size = 0.34 \[ \frac{1}{8} \text{RootSum}\left [\text{$\#$1}^8-4 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{\text{$\#$1}^7-2 \text{$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.008, size = 40, normalized size = 0.3 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-4\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}-2\,{{\it \_R}}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.35917, size = 1062, normalized size = 6.76 \begin{align*} \frac{1}{2} \, \sqrt{2}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \arctan \left (\frac{1}{2} \, \sqrt{x^{2} +{\left (\sqrt{3} + 2\right )} \sqrt{-\sqrt{3} + 2}}{\left (\sqrt{3} \sqrt{2} + \sqrt{2}\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{3}{4}} - \frac{1}{2} \,{\left (\sqrt{3} \sqrt{2} x + \sqrt{2} x\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{3}{4}}\right ) - \frac{1}{2} \, \sqrt{2}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \arctan \left (\frac{1}{2} \,{\left (\sqrt{x^{2} - \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )}}{\left (\sqrt{3} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{3} + 2} -{\left (\sqrt{3} \sqrt{2} x - \sqrt{2} x\right )} \sqrt{\sqrt{3} + 2}\right )}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}}\right ) + \frac{1}{8} \, \sqrt{2}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left ({\left (\sqrt{3} \sqrt{2} - \sqrt{2}\right )}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} + 2 \, x\right ) - \frac{1}{8} \, \sqrt{2}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (-{\left (\sqrt{3} \sqrt{2} - \sqrt{2}\right )}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} + 2 \, x\right ) - \frac{1}{8} \, \sqrt{2}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left ({\left (\sqrt{3} \sqrt{2} + \sqrt{2}\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} + 2 \, x\right ) + \frac{1}{8} \, \sqrt{2}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (-{\left (\sqrt{3} \sqrt{2} + \sqrt{2}\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} + 2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.179358, size = 24, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (1048576 t^{8} - 4096 t^{4} + 1, \left ( t \mapsto t \log{\left (4096 t^{5} - 12 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(1048576*_t**8 - 4096*_t**4 + 1, Lambda(_t, _t*log(4096*_t**5 - 12*_t + x)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} + 1}{x^{8} - 4 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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