3.399 \(\int \frac{x^2}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=145 \[ \frac{1}{20} \sqrt{10 \sqrt{5}-10} \tan ^{-1}\left (\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right )-\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tan ^{-1}\left (\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right )-\frac{1}{20} \sqrt{10 \sqrt{5}-10} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2 \sqrt{5}-2} x\right )+\frac{1}{20} \sqrt{10+10 \sqrt{5}} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2+2 \sqrt{5}} x\right ) \]

[Out]

(Sqrt[-10 + 10*Sqrt[5]]*ArcTan[(Sqrt[-2 + 2*Sqrt[5]]*x)/2])/20 - (Sqrt[10 + 10*Sqrt[5]]*ArcTan[(Sqrt[2 + 2*Sqr
t[5]]*x)/2])/20 - (Sqrt[-10 + 10*Sqrt[5]]*ArcTanh[(Sqrt[-2 + 2*Sqrt[5]]*x)/2])/20 + (Sqrt[10 + 10*Sqrt[5]]*Arc
Tanh[(Sqrt[2 + 2*Sqrt[5]]*x)/2])/20

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Rubi [A]  time = 0.0599003, antiderivative size = 166, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1375, 298, 203, 206} \[ \frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}+\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 - 3*x^4 + x^8),x]

[Out]

ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(3/4)*Sqrt[5]*(3 + Sqrt[5])^(1/4)) - (((3 + Sqrt[5])/2)^(1/4)*ArcTan[((3
+ Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(3/4)*Sqrt[5]*(3 + Sqrt[5])^(1/4))
+ (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5])

Rule 1375

Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]
}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

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 206

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

Rubi steps

\begin{align*} \int \frac{x^2}{1-3 x^4+x^8} \, dx &=\frac{\int \frac{x^2}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{\sqrt{5}}-\frac{\int \frac{x^2}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{\sqrt{5}}\\ &=\frac{\int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx}{\sqrt{10}}-\frac{\int \frac{1}{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2} \, dx}{\sqrt{10}}-\frac{\int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx}{\sqrt{10}}+\frac{\int \frac{1}{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2} \, dx}{\sqrt{10}}\\ &=\frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2^{3/4} \sqrt{5} \sqrt [4]{3+\sqrt{5}}}+\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0451395, size = 131, normalized size = 0.9 \[ -\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(1 - 3*x^4 + x^8),x]

[Out]

-(ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])]) + ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqr
t[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*
(1 + Sqrt[5])]

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Maple [A]  time = 0.02, size = 110, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^8-3*x^4+1),x)

[Out]

-1/5*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))-1/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/
(-2+2*5^(1/2))^(1/2))+1/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+1/5*5^(1/2)/(2+2*5^(1
/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

integrate(x^2/(x^8 - 3*x^4 + 1), x)

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Fricas [B]  time = 1.92728, size = 903, normalized size = 6.23 \begin{align*} \frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} + 1} \arctan \left (\frac{1}{20} \, \sqrt{10} \sqrt{5} \sqrt{2} \sqrt{2 \, x^{2} + \sqrt{5} - 1} \sqrt{\sqrt{5} + 1} - \frac{1}{10} \, \sqrt{10} \sqrt{5} x \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} - 1} \arctan \left (\frac{1}{20} \, \sqrt{10} \sqrt{5} \sqrt{2} \sqrt{2 \, x^{2} + \sqrt{5} + 1} \sqrt{\sqrt{5} - 1} - \frac{1}{10} \, \sqrt{10} \sqrt{5} x \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \, \sqrt{10} \sqrt{\sqrt{5} - 1} \log \left (\sqrt{10}{\left (\sqrt{5} + 5\right )} \sqrt{\sqrt{5} - 1} + 20 \, x\right ) + \frac{1}{40} \, \sqrt{10} \sqrt{\sqrt{5} - 1} \log \left (-\sqrt{10}{\left (\sqrt{5} + 5\right )} \sqrt{\sqrt{5} - 1} + 20 \, x\right ) - \frac{1}{40} \, \sqrt{10} \sqrt{\sqrt{5} + 1} \log \left (\sqrt{10} \sqrt{\sqrt{5} + 1}{\left (\sqrt{5} - 5\right )} + 20 \, x\right ) + \frac{1}{40} \, \sqrt{10} \sqrt{\sqrt{5} + 1} \log \left (-\sqrt{10} \sqrt{\sqrt{5} + 1}{\left (\sqrt{5} - 5\right )} + 20 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

1/10*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(1/20*sqrt(10)*sqrt(5)*sqrt(2)*sqrt(2*x^2 + sqrt(5) - 1)*sqrt(sqrt(5) +
1) - 1/10*sqrt(10)*sqrt(5)*x*sqrt(sqrt(5) + 1)) - 1/10*sqrt(10)*sqrt(sqrt(5) - 1)*arctan(1/20*sqrt(10)*sqrt(5)
*sqrt(2)*sqrt(2*x^2 + sqrt(5) + 1)*sqrt(sqrt(5) - 1) - 1/10*sqrt(10)*sqrt(5)*x*sqrt(sqrt(5) - 1)) - 1/40*sqrt(
10)*sqrt(sqrt(5) - 1)*log(sqrt(10)*(sqrt(5) + 5)*sqrt(sqrt(5) - 1) + 20*x) + 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*l
og(-sqrt(10)*(sqrt(5) + 5)*sqrt(sqrt(5) - 1) + 20*x) - 1/40*sqrt(10)*sqrt(sqrt(5) + 1)*log(sqrt(10)*sqrt(sqrt(
5) + 1)*(sqrt(5) - 5) + 20*x) + 1/40*sqrt(10)*sqrt(sqrt(5) + 1)*log(-sqrt(10)*sqrt(sqrt(5) + 1)*(sqrt(5) - 5)
+ 20*x)

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Sympy [A]  time = 0.891353, size = 53, normalized size = 0.37 \begin{align*} \operatorname{RootSum}{\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log{\left (6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log{\left (6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(6144000*_t**7 - 2240*_t**3 + x))) + RootSum(6400*_t**4 +
80*_t**2 - 1, Lambda(_t, _t*log(6144000*_t**7 - 2240*_t**3 + x)))

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Giac [A]  time = 1.21715, size = 198, normalized size = 1.37 \begin{align*} \frac{1}{20} \, \sqrt{10 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{10 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

1/20*sqrt(10*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/20*sqrt(10*sqrt(5) + 10)*arctan(x/sqrt(1/2*sq
rt(5) - 1/2)) - 1/40*sqrt(10*sqrt(5) - 10)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) - 10)*
log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/4
0*sqrt(10*sqrt(5) + 10)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))