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

Optimal. Leaf size=129 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[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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Rubi [A]  time = 0.117834, antiderivative size = 129, 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} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(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 + Sqrt[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])]

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

Mathematica [A]  time = 0.0772217, size = 129, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)/(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 + Sqrt[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 = 0.9 \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^4+1)/(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^{4} - 1}{x^{8} - 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.41056, size = 905, normalized size = 7.02 \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^4+1)/(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)*
log(-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 = 0.879065, size = 51, normalized size = 0.4 \begin{align*} - \operatorname{RootSum}{\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log{\left (25600 t^{5} - 16 t + x \right )} \right )\right )} - \operatorname{RootSum}{\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log{\left (25600 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**8-3*x**4+1),x)

[Out]

-RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(25600*_t**5 - 16*_t + x))) - RootSum(6400*_t**4 + 80*_t*
*2 - 1, Lambda(_t, _t*log(25600*_t**5 - 16*_t + x)))

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Giac [A]  time = 1.20993, size = 198, normalized size = 1.53 \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^4+1)/(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)))