3.55 \(\int \frac {1}{1+4 x+4 x^2+4 x^4} \, dx\)

Optimal. Leaf size=234 \[ -\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )+\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )+\frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \left (\left (\frac {1}{x}+1\right )^2-1\right )\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\frac {2}{x}-\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\frac {2}{x}+\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right ) \]

[Out]

1/2*arctan(-1/2+1/2*(1+1/x)^2)-1/20*ln((1+1/x)^2+5^(1/2)-(1+1/x)*(2+2*5^(1/2))^(1/2))*(-10+5*5^(1/2))^(1/2)+1/
20*ln((1+1/x)^2+5^(1/2)+(1+1/x)*(2+2*5^(1/2))^(1/2))*(-10+5*5^(1/2))^(1/2)-1/10*arctan((2+2/x-(2+2*5^(1/2))^(1
/2))/(-2+2*5^(1/2))^(1/2))*(10+5*5^(1/2))^(1/2)-1/10*arctan((2+2/x+(2+2*5^(1/2))^(1/2))/(-2+2*5^(1/2))^(1/2))*
(10+5*5^(1/2))^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {2069, 1673, 1169, 634, 618, 204, 628, 12, 1107} \[ -\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )+\frac {1}{4} \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \log \left (\left (\frac {1}{x}+1\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )+\frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \left (\left (\frac {1}{x}+1\right )^2-1\right )\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\frac {2}{x}-\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\frac {2}{x}+\sqrt {2 \left (1+\sqrt {5}\right )}+2}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-1),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1169

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

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 2069

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1*((a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e -
32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4))/(b - 4*a*x)^4)^p)/(b - 4*a*x)^2, x], x, b/(4*a) + 1/x], x] /; NeQ[a
, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] &&  !
IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {1}{1+4 x+4 x^2+4 x^4} \, dx &=-\left (16 \operatorname {Subst}\left (\int \frac {(4-4 x)^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\right )\\ &=-\left (16 \operatorname {Subst}\left (\int -\frac {32 x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\right )-16 \operatorname {Subst}\left (\int \frac {16+16 x^2}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )\\ &=512 \operatorname {Subst}\left (\int \frac {x}{1280-512 x^2+256 x^4} \, dx,x,1+\frac {1}{x}\right )-\frac {\operatorname {Subst}\left (\int \frac {16 \sqrt {2 \left (1+\sqrt {5}\right )}-\left (16-16 \sqrt {5}\right ) x}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{32 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {16 \sqrt {2 \left (1+\sqrt {5}\right )}+\left (16-16 \sqrt {5}\right ) x}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{32 \sqrt {10 \left (1+\sqrt {5}\right )}}\\ &=256 \operatorname {Subst}\left (\int \frac {1}{1280-512 x+256 x^2} \, dx,x,\left (1+\frac {1}{x}\right )^2\right )+\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {5}\right )}+2 x}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{4 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {5}\right )}+2 x}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )}{4 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {1}{20} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )-\frac {1}{20} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} x+x^2} \, dx,x,1+\frac {1}{x}\right )\\ &=-\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )+\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )-512 \operatorname {Subst}\left (\int \frac {1}{-1048576-x^2} \, dx,x,-512+512 \left (1+\frac {1}{x}\right )^2\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {5}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {5}\right )}+2 \left (1+\frac {1}{x}\right )\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {5}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {5}\right )}+2 \left (1+\frac {1}{x}\right )\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {\left (1+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{4 \sqrt {10}}-\frac {\left (1+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{4 \sqrt {10}}-\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )+\frac {1}{4} \sqrt {-\frac {2}{5}+\frac {1}{\sqrt {5}}} \log \left (\sqrt {5}+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\left (1+\frac {1}{x}\right )^2\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 47, normalized size = 0.20 \[ \frac {1}{4} \text {RootSum}\left [4 \text {$\#$1}^4+4 \text {$\#$1}^2+4 \text {$\#$1}+1\& ,\frac {\log (x-\text {$\#$1})}{4 \text {$\#$1}^3+2 \text {$\#$1}+1}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-1),x]

[Out]

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

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fricas [C]  time = 1.33, size = 499, normalized size = 2.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(sqrt(10)*sqrt(-15/8*(2*sqrt(1/10*I - 1/5) - I)^2 - 5/4*(2*sqrt(1/10*I - 1/5) - I)*(2*sqrt(-1/10*I - 1/5
) + I) - 15/8*(2*sqrt(-1/10*I - 1/5) + I)^2 - 9) - 5*sqrt(1/10*I - 1/5) - 5*sqrt(-1/10*I - 1/5))*log(5/2*(2*sq
rt(1/10*I - 1/5) - I)^2*(12*sqrt(-1/10*I - 1/5) + 6*I - 1) + 15*(2*sqrt(1/10*I - 1/5) - I)*(2*sqrt(-1/10*I - 1
/5) + I)^2 - 5/2*(2*sqrt(-1/10*I - 1/5) + I)^2 + ((6*sqrt(10)*(2*sqrt(-1/10*I - 1/5) + I) - sqrt(10))*(2*sqrt(
1/10*I - 1/5) - I) - sqrt(10)*(2*sqrt(-1/10*I - 1/5) + I))*sqrt(-15/8*(2*sqrt(1/10*I - 1/5) - I)^2 - 5/4*(2*sq
rt(1/10*I - 1/5) - I)*(2*sqrt(-1/10*I - 1/5) + I) - 15/8*(2*sqrt(-1/10*I - 1/5) + I)^2 - 9) + 8*x + 3) + 1/20*
(sqrt(10)*sqrt(-15/8*(2*sqrt(1/10*I - 1/5) - I)^2 - 5/4*(2*sqrt(1/10*I - 1/5) - I)*(2*sqrt(-1/10*I - 1/5) + I)
- 15/8*(2*sqrt(-1/10*I - 1/5) + I)^2 - 9) + 5*sqrt(1/10*I - 1/5) + 5*sqrt(-1/10*I - 1/5))*log(5/2*(2*sqrt(1/1
0*I - 1/5) - I)^2*(12*sqrt(-1/10*I - 1/5) + 6*I - 1) + 15*(2*sqrt(1/10*I - 1/5) - I)*(2*sqrt(-1/10*I - 1/5) +
I)^2 - 5/2*(2*sqrt(-1/10*I - 1/5) + I)^2 - ((6*sqrt(10)*(2*sqrt(-1/10*I - 1/5) + I) - sqrt(10))*(2*sqrt(1/10*I
- 1/5) - I) - sqrt(10)*(2*sqrt(-1/10*I - 1/5) + I))*sqrt(-15/8*(2*sqrt(1/10*I - 1/5) - I)^2 - 5/4*(2*sqrt(1/1
0*I - 1/5) - I)*(2*sqrt(-1/10*I - 1/5) + I) - 15/8*(2*sqrt(-1/10*I - 1/5) + I)^2 - 9) + 8*x + 3) - 1/4*(2*sqrt
(1/10*I - 1/5) - I)*log(-5*(2*sqrt(1/10*I - 1/5) - I)^2*(12*sqrt(-1/10*I - 1/5) + 6*I - 1) - 30*(2*sqrt(1/10*I
- 1/5) - I)*(2*sqrt(-1/10*I - 1/5) + I)^2 - 30*(2*sqrt(-1/10*I - 1/5) + I)^3 + 8*x - 216*sqrt(-1/10*I - 1/5)
- 108*I + 21) - 1/4*(2*sqrt(-1/10*I - 1/5) + I)*log(30*(2*sqrt(-1/10*I - 1/5) + I)^3 + 5*(2*sqrt(-1/10*I - 1/5
) + I)^2 + 8*x + 216*sqrt(-1/10*I - 1/5) + 108*I - 27)

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giac [C]  time = 0.52, size = 265, normalized size = 1.13 \[ -\frac {1}{20} \, {\left (\left (i + 2\right ) \, \sqrt {\sqrt {5} - 2} {\left (\frac {i}{\sqrt {5} - 2} + 1\right )} + 5 i\right )} \log \left (\left (406 i + 174\right ) \, \sqrt {5} x + \left (868 i + 372\right ) \, x + 29 \, \sqrt {5} \sqrt {29 \, \sqrt {5} + 62} + \left (87 i - 203\right ) \, \sqrt {5} + \left (19 i + 62\right ) \, \sqrt {29 \, \sqrt {5} + 62} + 186 i - 434\right ) - \frac {1}{20} \, {\left (\left (i + 2\right ) \, \sqrt {\sqrt {5} - 2} {\left (-\frac {i}{\sqrt {5} - 2} - 1\right )} + 5 i\right )} \log \left (\left (406 i + 174\right ) \, \sqrt {5} x + \left (868 i + 372\right ) \, x - 29 \, \sqrt {5} \sqrt {29 \, \sqrt {5} + 62} + \left (87 i - 203\right ) \, \sqrt {5} - \left (19 i + 62\right ) \, \sqrt {29 \, \sqrt {5} + 62} + 186 i - 434\right ) - \frac {1}{20} \, {\left (\left (2 i + 1\right ) \, \sqrt {\sqrt {5} + 2} {\left (-\frac {i}{\sqrt {5} + 2} - 1\right )} - 5 i\right )} \log \left (\left (26 i + 130\right ) \, \sqrt {5} x - \left (44 i + 220\right ) \, x + 13 \, \sqrt {5} \sqrt {13 \, \sqrt {5} - 22} - \left (65 i - 13\right ) \, \sqrt {5} + \left (19 i - 22\right ) \, \sqrt {13 \, \sqrt {5} - 22} + 110 i - 22\right ) - \frac {1}{20} \, {\left (\left (2 i + 1\right ) \, \sqrt {\sqrt {5} + 2} {\left (\frac {i}{\sqrt {5} + 2} + 1\right )} - 5 i\right )} \log \left (\left (26 i + 130\right ) \, \sqrt {5} x - \left (44 i + 220\right ) \, x - 13 \, \sqrt {5} \sqrt {13 \, \sqrt {5} - 22} - \left (65 i - 13\right ) \, \sqrt {5} - \left (19 i - 22\right ) \, \sqrt {13 \, \sqrt {5} - 22} + 110 i - 22\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*((I + 2)*sqrt(sqrt(5) - 2)*(I/(sqrt(5) - 2) + 1) + 5*I)*log((406*I + 174)*sqrt(5)*x + (868*I + 372)*x +
29*sqrt(5)*sqrt(29*sqrt(5) + 62) + (87*I - 203)*sqrt(5) + (19*I + 62)*sqrt(29*sqrt(5) + 62) + 186*I - 434) - 1
/20*((I + 2)*sqrt(sqrt(5) - 2)*(-I/(sqrt(5) - 2) - 1) + 5*I)*log((406*I + 174)*sqrt(5)*x + (868*I + 372)*x - 2
9*sqrt(5)*sqrt(29*sqrt(5) + 62) + (87*I - 203)*sqrt(5) - (19*I + 62)*sqrt(29*sqrt(5) + 62) + 186*I - 434) - 1/
20*((2*I + 1)*sqrt(sqrt(5) + 2)*(-I/(sqrt(5) + 2) - 1) - 5*I)*log((26*I + 130)*sqrt(5)*x - (44*I + 220)*x + 13
*sqrt(5)*sqrt(13*sqrt(5) - 22) - (65*I - 13)*sqrt(5) + (19*I - 22)*sqrt(13*sqrt(5) - 22) + 110*I - 22) - 1/20*
((2*I + 1)*sqrt(sqrt(5) + 2)*(I/(sqrt(5) + 2) + 1) - 5*I)*log((26*I + 130)*sqrt(5)*x - (44*I + 220)*x - 13*sqr
t(5)*sqrt(13*sqrt(5) - 22) - (65*I - 13)*sqrt(5) - (19*I - 22)*sqrt(13*sqrt(5) - 22) + 110*I - 22)

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maple [C]  time = 0.01, size = 41, normalized size = 0.18 \[ \frac {\ln \left (-\RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x \right )}{16 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{3}+8 \RootOf \left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(4*x^4 + 4*x^2 + 4*x + 1), x)

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mupad [B]  time = 2.36, size = 87, normalized size = 0.37 \[ \sum _{k=1}^4\ln \left (-\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (\frac {x}{4}+\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (6\,x+\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right )\,\left (36\,x+16\right )\right )\right )\right )\,\mathrm {root}\left (z^4+\frac {9\,z^2}{40}+\frac {z}{40}+\frac {1}{1280},z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x + 4*x^2 + 4*x^4 + 1),x)

[Out]

symsum(log(-root(z^4 + (9*z^2)/40 + z/40 + 1/1280, z, k)*(x/4 + root(z^4 + (9*z^2)/40 + z/40 + 1/1280, z, k)*(
6*x + root(z^4 + (9*z^2)/40 + z/40 + 1/1280, z, k)*(36*x + 16))))*root(z^4 + (9*z^2)/40 + z/40 + 1/1280, z, k)
, k, 1, 4)

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sympy [B]  time = 2.57, size = 3432, normalized size = 14.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x**4+4*x**2+4*x+1),x)

[Out]

sqrt(-1/40 + sqrt(5)/80)*log(x**2 + x*(-8 - 21*sqrt(5)*sqrt(-2 + sqrt(5))/10 - sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(
5)) + sqrt(5) + 19)/2 - sqrt(5)/2 + 12*sqrt(-2 + sqrt(5)) + 9*sqrt(5)*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(5)*sqrt(
-2 + sqrt(5)) + sqrt(5) + 19)/5) - 841*sqrt(5)*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/20 - 14351/4
0 - 441*sqrt(-2 + sqrt(5))/4 - 75*sqrt(5)*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19
)/8 - 3*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 301*sqrt(5)*sqrt(-2 + sqrt(5))
/10 + 7407*sqrt(5)/40 + 3913*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/40) - sqrt(-1/40 + sqrt(5)/80)
*log(x**2 + x*(-8 - 12*sqrt(-2 + sqrt(5)) - sqrt(5)/2 + 21*sqrt(5)*sqrt(-2 + sqrt(5))/10 + sqrt(2*sqrt(5)*sqrt
(-2 + sqrt(5)) + sqrt(5) + 19)/2 + 9*sqrt(5)*sqrt(-2 + sqrt(5))*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) +
19)/5) - 3913*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/40 - 14351/40 - 75*sqrt(5)*sqrt(-2 + sqrt(5))*
sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/8 - 301*sqrt(5)*sqrt(-2 + sqrt(5))/10 - 3*sqrt(-2 + sqrt(5))
*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 441*sqrt(-2 + sqrt(5))/4 + 7407*sqrt(5)/40 + 841*sqrt(5)*
sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/20) - 2*sqrt(3/80 + 3*sqrt(5)/80 + sqrt(-2*sqrt(5)*sqrt(-2 +
sqrt(5)) + sqrt(5) + 19)/40)*atan(-20*x/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)
) + sqrt(5) + 19)) + 5*sqrt(-2 + sqrt(5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5)
+ 19)) + 6*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*
sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) - 18*sqrt(5)*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(
5) + 19)/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 5*sqrt(-2 +
sqrt(5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 6*sqrt(5)*sqrt(3 + 3*sq
rt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 1
9)) - 120*sqrt(-2 + sqrt(5))/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5)
+ 19)) + 5*sqrt(-2 + sqrt(5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 6*s
qrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(-2 + sq
rt(5)) + sqrt(5) + 19)) + 5*sqrt(5)/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + s
qrt(5) + 19)) + 5*sqrt(-2 + sqrt(5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)
) + 6*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(
-2 + sqrt(5)) + sqrt(5) + 19)) + 5*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/(-27*sqrt(5)*sqrt(3 + 3*
sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 5*sqrt(-2 + sqrt(5))*sqrt(3 + 3*sqrt(5) + 2*
sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 6*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2
+ sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 21*sqrt(5)*sqrt(-2 + sqrt(5
))/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 5*sqrt(-2 + sqrt(
5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 6*sqrt(5)*sqrt(3 + 3*sqrt(5)
+ 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) +
80/(-27*sqrt(5)*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 5*sqrt(-2 + sqrt(
5))*sqrt(3 + 3*sqrt(5) + 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)) + 6*sqrt(5)*sqrt(3 + 3*sqrt(5)
+ 2*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))*sqrt(-2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19))) -
2*sqrt(-sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/40 + 3/80 + 3*sqrt(5)/80)*atan(20*x/(5*sqrt(-2 + sq
rt(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sqrt(5)*sqrt(-2*sqrt(2*
sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sq
rt(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5))) - 80/(5*sqrt(-2 + sqrt
(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sqrt(5)*sqrt(-2*sqrt(2*sq
rt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt
(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5))) - 120*sqrt(-2 + sqrt(5))
/(5*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sqrt(5)
*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2
+ sqrt(5)) + sqrt(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5))) - 5*sq
rt(5)/(5*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sq
rt(5)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sq
rt(-2 + sqrt(5)) + sqrt(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5))) +
21*sqrt(5)*sqrt(-2 + sqrt(5))/(5*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)
+ 3 + 3*sqrt(5)) + 27*sqrt(5)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*
sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5)
+ 19) + 3 + 3*sqrt(5))) + 5*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/(5*sqrt(-2 + sqrt(5))*sqrt(-2*sq
rt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sqrt(5)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 +
sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)*sqrt(
-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*sqrt(5))) + 18*sqrt(5)*sqrt(-2 + sqrt(5))*sqrt(2*
sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)/(5*sqrt(-2 + sqrt(5))*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + s
qrt(5) + 19) + 3 + 3*sqrt(5)) + 27*sqrt(5)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19) + 3 + 3*s
qrt(5)) + 6*sqrt(5)*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)) + sqrt(5) + 19)*sqrt(-2*sqrt(2*sqrt(5)*sqrt(-2 + sqrt(5)
) + sqrt(5) + 19) + 3 + 3*sqrt(5))))

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