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

Optimal. Leaf size=165 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (\sqrt{3}-1\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (1+\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (\sqrt{3}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (1+\sqrt{3}\right )}} \]

[Out]

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

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Rubi [A]  time = 0.104054, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1419, 1093, 207, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (\sqrt{3}-1\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (1+\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{\sqrt{3}-1}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (\sqrt{3}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+\sqrt{3}}}\right )}{2 \sqrt [4]{2} \sqrt{3 \left (1+\sqrt{3}\right )}} \]

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

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])

Rubi steps

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

Mathematica [C]  time = 0.0136134, size = 55, normalized size = 0.33 \[ -\frac{1}{8} \text{RootSum}\left [\text{$\#$1}^8-4 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{\text{$\#$1}^7-2 \text{$\#$1}^3}\& \right ] \]

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 = 0.007, size = 42, 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 = 1.41732, size = 1004, normalized size = 6.08 \begin{align*} -\frac{1}{6} \, \sqrt{6}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \arctan \left (\frac{1}{6} \, \sqrt{6} \sqrt{x^{2} +{\left (\sqrt{3} + 2\right )} \sqrt{-\sqrt{3} + 2}}{\left (\sqrt{3} + 3\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{3}{4}} - \frac{1}{6} \, \sqrt{6}{\left (\sqrt{3} x + 3 \, x\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{3}{4}}\right ) + \frac{1}{6} \, \sqrt{6}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \arctan \left (\frac{1}{6} \,{\left (\sqrt{6} \sqrt{x^{2} - \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )}} \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 3\right )} - \sqrt{6}{\left (\sqrt{3} x - 3 \, x\right )} \sqrt{\sqrt{3} + 2}\right )}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}}\right ) - \frac{1}{24} \, \sqrt{6}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (\sqrt{6}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}}{\left (\sqrt{3} - 3\right )} + 6 \, x\right ) + \frac{1}{24} \, \sqrt{6}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (-\sqrt{6}{\left (\sqrt{3} + 2\right )}^{\frac{1}{4}}{\left (\sqrt{3} - 3\right )} + 6 \, x\right ) + \frac{1}{24} \, \sqrt{6}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (\sqrt{6}{\left (\sqrt{3} + 3\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} + 6 \, x\right ) - \frac{1}{24} \, \sqrt{6}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} \log \left (-\sqrt{6}{\left (\sqrt{3} + 3\right )}{\left (-\sqrt{3} + 2\right )}^{\frac{1}{4}} + 6 \, 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]

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

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Sympy [A]  time = 0.181679, size = 26, normalized size = 0.16 \begin{align*} - \operatorname{RootSum}{\left (84934656 t^{8} - 36864 t^{4} + 1, \left ( t \mapsto t \log{\left (36864 t^{5} - 20 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(84934656*_t**8 - 36864*_t**4 + 1, Lambda(_t, _t*log(36864*_t**5 - 20*_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)