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

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Rubi [A]  time = 0.151257, 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, Rules used = {1419, 1093, 203, 207} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}} \]

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

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}{1-5 x^4+x^8} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{7} x^2+x^4} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{7} x^2+x^4} \, dx\\ &=\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1}{\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}+\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1}{\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{2}{-\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (-\sqrt{3}+\sqrt{7}\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{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (-\sqrt{3}+\sqrt{7}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}\\ \end{align*}

Mathematica [C]  time = 0.0132405, 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 = 0.008, size = 42, normalized size = 0.3 \begin{align*}{\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}}}} \end{align*}

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. \begin{align*} \int \frac{x^{4} + 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \end{align*}

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 [B]  time = 1.45293, size = 1812, normalized size = 10.6 \begin{align*} \frac{1}{6} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{48} \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} + 3 \, \sqrt{6} \sqrt{2}\right )} \sqrt{4 \, x^{2} +{\left (\sqrt{7} \sqrt{3} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} - \frac{1}{24} \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} x + 3 \, \sqrt{6} \sqrt{2} x\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}}\right ) - \frac{1}{6} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{48} \,{\left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} - 3 \, \sqrt{6} \sqrt{2}\right )} \sqrt{4 \, x^{2} -{\left (\sqrt{7} \sqrt{3} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}} \sqrt{\sqrt{7} \sqrt{3} + 5} - 2 \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} x - 3 \, \sqrt{6} \sqrt{2} x\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}}\right ) + \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} - 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) - \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left (-{\left (\sqrt{7} \sqrt{6} \sqrt{3} - 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) - \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} + 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) + \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left (-{\left (\sqrt{7} \sqrt{6} \sqrt{3} + 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) \end{align*}

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]

1/6*sqrt(6)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))*arctan(1/48*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2) + 3*sqrt(6)*
sqrt(2))*sqrt(4*x^2 + (sqrt(7)*sqrt(3)*sqrt(2) + 5*sqrt(2))*sqrt(-sqrt(7)*sqrt(3) + 5))*sqrt(-sqrt(7)*sqrt(3)
+ 5)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)) - 1/24*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2)*x + 3*sqrt(6)*sqrt(2)*x)
*sqrt(-sqrt(7)*sqrt(3) + 5)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))) - 1/6*sqrt(6)*sqrt(sqrt(2)*sqrt(sqrt(7)*
sqrt(3) + 5))*arctan(1/48*((sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2) - 3*sqrt(6)*sqrt(2))*sqrt(4*x^2 - (sqrt(7)*sqrt(3)
*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(7)*sqrt(3) + 5))*sqrt(sqrt(7)*sqrt(3) + 5) - 2*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2
)*x - 3*sqrt(6)*sqrt(2)*x)*sqrt(sqrt(7)*sqrt(3) + 5))*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))) + 1/24*sqrt(6)*
sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))*log((sqrt(7)*sqrt(6)*sqrt(3) - 3*sqrt(6))*sqrt(sqrt(2)*sqrt(sqrt(7)*sq
rt(3) + 5)) + 12*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))*log(-(sqrt(7)*sqrt(6)*sqrt(3) - 3*s
qrt(6))*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5)) + 12*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)
)*log((sqrt(7)*sqrt(6)*sqrt(3) + 3*sqrt(6))*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)) + 12*x) + 1/24*sqrt(6)*sq
rt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))*log(-(sqrt(7)*sqrt(6)*sqrt(3) + 3*sqrt(6))*sqrt(sqrt(2)*sqrt(-sqrt(7)*s
qrt(3) + 5)) + 12*x)

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Sympy [A]  time = 0.190294, size = 24, normalized size = 0.14 \begin{align*} \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 )} \end{align*}

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

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)