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

Optimal. Leaf size=411 \[ -\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4}} \]

[Out]

-1/4*arctan(-1+2^(3/4)*x/(3+5^(1/2))^(1/4))*(3-5^(1/2))^(1/4)*2^(1/4)-1/4*arctan(1+2^(3/4)*x/(3+5^(1/2))^(1/4)
)*(3-5^(1/2))^(1/4)*2^(1/4)+1/8*ln(2*x^2-2*2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+1)*(3-5^(1/2))^(1/4)*2^(1/4)-1/
8*ln(2*x^2+2*2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+1)*(3-5^(1/2))^(1/4)*2^(1/4)+1/4*arctan(-1+2^(3/4)*x/(3-5^(1/
2))^(1/4))*(3+5^(1/2))^(1/4)*2^(1/4)+1/4*arctan(1+2^(3/4)*x/(3-5^(1/2))^(1/4))*(3+5^(1/2))^(1/4)*2^(1/4)-1/8*l
n(2*x^2-2*2^(1/4)*x*(3-5^(1/2))^(1/4)+5^(1/2)-1)*(3+5^(1/2))^(1/4)*2^(1/4)+1/8*ln(2*x^2+2*2^(1/4)*x*(3-5^(1/2)
)^(1/4)+5^(1/2)-1)*(3+5^(1/2))^(1/4)*2^(1/4)

________________________________________________________________________________________

Rubi [A]  time = 0.32, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1420, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 + 3*x^4 + x^8),x]

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) + ((3 + Sqrt[5])^(1/4)*ArcTan[1
+ (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(
1/4)])/(2*2^(3/4)) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) - ((3 + Sqr
t[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) + ((3 + Sqrt[5])^(1/
4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*Log[Sq
rt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) - ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 +
Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4))

Rule 204

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

Rule 211

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

Rule 617

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1420

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, 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[b^2 - 4*a*c, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 57, normalized size = 0.14 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+3 \text {$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)/(1 + 3*x^4 + x^8),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.97, size = 894, normalized size = 2.18 \[ \text {result too large to display} \]

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]

1/16*(sqrt(5)*sqrt(2) - 3*sqrt(2))*(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3)*arctan(1/16*sqrt(4*x^2 - sqrt(2*sqr
t(5) + 6)*(sqrt(5) - 3) + 2*(sqrt(5)*x - x)*(2*sqrt(5) + 6)^(1/4))*(sqrt(5)*sqrt(2) - 2*sqrt(2))*(2*sqrt(5) +
6)^(5/4)*sqrt(sqrt(5) + 3) - 1/8*(sqrt(5)*sqrt(2)*x - 2*sqrt(2)*x)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) + 1
/8*(sqrt(5)*sqrt(2) - 3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) + 1/16*(sqrt(5)*sqrt(2) - 3*sqrt(2))*(
2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3)*arctan(1/16*sqrt(4*x^2 - sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3) - 2*(sqrt(5)
*x - x)*(2*sqrt(5) + 6)^(1/4))*(sqrt(5)*sqrt(2) - 2*sqrt(2))*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) - 1/8*(sq
rt(5)*sqrt(2)*x - 2*sqrt(2)*x)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) - 1/8*(sqrt(5)*sqrt(2) - 3*sqrt(2))*sqr
t(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) + 1/16*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(
3/4)*arctan(1/16*sqrt(4*x^2 + (sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6) + 2*(sqrt(5)*x + x)*(-2*sqrt(5) + 6)^(1/4))*(
sqrt(5)*sqrt(2) + 2*sqrt(2))*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(5/4) - 1/8*((sqrt(5)*sqrt(2)*x + 2*sqrt(2)*x
)*(-2*sqrt(5) + 6)^(5/4) + (sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sqrt(5) + 3)) + 1/16*(sqr
t(5)*sqrt(2) + 3*sqrt(2))*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4)*arctan(1/16*sqrt(4*x^2 + (sqrt(5) + 3)*sqr
t(-2*sqrt(5) + 6) - 2*(sqrt(5)*x + x)*(-2*sqrt(5) + 6)^(1/4))*(sqrt(5)*sqrt(2) + 2*sqrt(2))*sqrt(-sqrt(5) + 3)
*(-2*sqrt(5) + 6)^(5/4) - 1/8*((sqrt(5)*sqrt(2)*x + 2*sqrt(2)*x)*(-2*sqrt(5) + 6)^(5/4) - (sqrt(5)*sqrt(2) + 3
*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sqrt(5) + 3)) + 1/8*(2*sqrt(5) + 6)^(1/4)*log(4*x^2 - sqrt(2*sqrt(5) + 6
)*(sqrt(5) - 3) + 2*(sqrt(5)*x - x)*(2*sqrt(5) + 6)^(1/4)) - 1/8*(2*sqrt(5) + 6)^(1/4)*log(4*x^2 - sqrt(2*sqrt
(5) + 6)*(sqrt(5) - 3) - 2*(sqrt(5)*x - x)*(2*sqrt(5) + 6)^(1/4)) - 1/8*(-2*sqrt(5) + 6)^(1/4)*log(4*x^2 + (sq
rt(5) + 3)*sqrt(-2*sqrt(5) + 6) + 2*(sqrt(5)*x + x)*(-2*sqrt(5) + 6)^(1/4)) + 1/8*(-2*sqrt(5) + 6)^(1/4)*log(4
*x^2 + (sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6) - 2*(sqrt(5)*x + x)*(-2*sqrt(5) + 6)^(1/4))

________________________________________________________________________________________

giac [A]  time = 0.69, size = 223, normalized size = 0.54 \[ \frac {1}{16} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{16} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{16} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {\sqrt {5} - 1} + \frac {1}{16} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {\sqrt {5} - 1} - \frac {1}{8} \, \sqrt {\sqrt {5} - 1} \log \left (2500 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 2500 \, x^{2}\right ) + \frac {1}{8} \, \sqrt {\sqrt {5} - 1} \log \left (2500 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 2500 \, x^{2}\right ) + \frac {1}{8} \, \sqrt {\sqrt {5} + 1} \log \left (1156 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 1156 \, x^{2}\right ) - \frac {1}{8} \, \sqrt {\sqrt {5} + 1} \log \left (1156 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 1156 \, x^{2}\right ) \]

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]

1/16*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(sqrt(5) + 1) - 1/16*(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) + 1
))*sqrt(sqrt(5) + 1) - 1/16*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(sqrt(5) - 1) + 1/16*(pi + 4*arctan(-
x*sqrt(sqrt(5) - 1) - 1))*sqrt(sqrt(5) - 1) - 1/8*sqrt(sqrt(5) - 1)*log(2500*(x + sqrt(sqrt(5) + 1))^2 + 2500*
x^2) + 1/8*sqrt(sqrt(5) - 1)*log(2500*(x - sqrt(sqrt(5) + 1))^2 + 2500*x^2) + 1/8*sqrt(sqrt(5) + 1)*log(1156*(
x + sqrt(sqrt(5) - 1))^2 + 1156*x^2) - 1/8*sqrt(sqrt(5) + 1)*log(1156*(x - sqrt(sqrt(5) - 1))^2 + 1156*x^2)

________________________________________________________________________________________

maple [C]  time = 0.01, size = 44, normalized size = 0.11 \[ \frac {\left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{4} - 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.68, size = 447, normalized size = 1.09 \[ \frac {2^{3/4}\,\mathrm {atan}\left (\frac {1875\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (625\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-250\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {875\,2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (625\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-250\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{4}-\frac {2^{3/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,1875{}\mathrm {i}}{2\,\left (625\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-250\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,875{}\mathrm {i}}{2\,\left (625\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-250\,\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{4}+\frac {2^{3/4}\,\mathrm {atan}\left (\frac {1875\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (625\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+250\,\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {875\,2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (625\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+250\,\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{4}-\frac {2^{3/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1875{}\mathrm {i}}{2\,\left (625\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+250\,\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,875{}\mathrm {i}}{2\,\left (625\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+250\,\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(3*x^4 + x^8 + 1),x)

[Out]

(2^(3/4)*atan((1875*2^(3/4)*x*(5^(1/2) - 3)^(1/4))/(2*(625*2^(1/2)*(5^(1/2) - 3)^(1/2) - 250*2^(1/2)*5^(1/2)*(
5^(1/2) - 3)^(1/2))) - (875*2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4))/(2*(625*2^(1/2)*(5^(1/2) - 3)^(1/2) - 250*2
^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))))*(5^(1/2) - 3)^(1/4))/4 - (2^(3/4)*atan((2^(3/4)*x*(5^(1/2) - 3)^(1/4)*18
75i)/(2*(625*2^(1/2)*(5^(1/2) - 3)^(1/2) - 250*2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))) - (2^(3/4)*5^(1/2)*x*(5^(
1/2) - 3)^(1/4)*875i)/(2*(625*2^(1/2)*(5^(1/2) - 3)^(1/2) - 250*2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))))*(5^(1/2
) - 3)^(1/4)*1i)/4 + (2^(3/4)*atan((1875*2^(3/4)*x*(- 5^(1/2) - 3)^(1/4))/(2*(625*2^(1/2)*(- 5^(1/2) - 3)^(1/2
) + 250*2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))) + (875*2^(3/4)*5^(1/2)*x*(- 5^(1/2) - 3)^(1/4))/(2*(625*2^(1/2
)*(- 5^(1/2) - 3)^(1/2) + 250*2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))))*(- 5^(1/2) - 3)^(1/4))/4 - (2^(3/4)*ata
n((2^(3/4)*x*(- 5^(1/2) - 3)^(1/4)*1875i)/(2*(625*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 250*2^(1/2)*5^(1/2)*(- 5^(1/
2) - 3)^(1/2))) + (2^(3/4)*5^(1/2)*x*(- 5^(1/2) - 3)^(1/4)*875i)/(2*(625*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 250*2
^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))))*(- 5^(1/2) - 3)^(1/4)*1i)/4

________________________________________________________________________________________

sympy [A]  time = 1.45, size = 26, normalized size = 0.06 \[ - \operatorname {RootSum} {\left (65536 t^{8} + 768 t^{4} + 1, \left (t \mapsto t \log {\left (1024 t^{5} + 8 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(65536*_t**8 + 768*_t**4 + 1, Lambda(_t, _t*log(1024*_t**5 + 8*_t + x)))

________________________________________________________________________________________