\(\int \frac {1}{1+2 x^4+x^8} \, dx\) [285]

Optimal result

Integrand size = 12, antiderivative size = 97 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {x}{4 \left (1+x^4\right )}-\frac {3 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \]

[Out]

1/4*x/(x^4+1)+3/16*arctan(-1+x*2^(1/2))*2^(1/2)+3/16*arctan(1+x*2^(1/2))*2^(1/2)-3/32*ln(1+x^2-x*2^(1/2))*2^(1
/2)+3/32*ln(1+x^2+x*2^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {28, 205, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{1+2 x^4+x^8} \, dx=-\frac {3 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {x}{4 \left (x^4+1\right )}-\frac {3 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {3 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}} \]

[In]

Int[(1 + 2*x^4 + x^8)^(-1),x]

[Out]

x/(4*(1 + x^4)) - (3*ArcTan[1 - Sqrt[2]*x])/(8*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*x])/(8*Sqrt[2]) - (3*Log[1 - S
qrt[2]*x + x^2])/(16*Sqrt[2]) + (3*Log[1 + Sqrt[2]*x + x^2])/(16*Sqrt[2])

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 210

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

Rule 217

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 631

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 642

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 1176

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 1179

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (1+x^4\right )^2} \, dx \\ & = \frac {x}{4 \left (1+x^4\right )}+\frac {3}{4} \int \frac {1}{1+x^4} \, dx \\ & = \frac {x}{4 \left (1+x^4\right )}+\frac {3}{8} \int \frac {1-x^2}{1+x^4} \, dx+\frac {3}{8} \int \frac {1+x^2}{1+x^4} \, dx \\ & = \frac {x}{4 \left (1+x^4\right )}+\frac {3}{16} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {3}{16} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx-\frac {3 \int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}-\frac {3 \int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}} \\ & = \frac {x}{4 \left (1+x^4\right )}-\frac {3 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{8 \sqrt {2}} \\ & = \frac {x}{4 \left (1+x^4\right )}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {1}{32} \left (\frac {8 x}{1+x^4}-6 \sqrt {2} \arctan \left (1-\sqrt {2} x\right )+6 \sqrt {2} \arctan \left (1+\sqrt {2} x\right )-3 \sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )+3 \sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \]

[In]

Integrate[(1 + 2*x^4 + x^8)^(-1),x]

[Out]

((8*x)/(1 + x^4) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*x] - 3*Sqrt[2]*Log[1 - Sqrt[
2]*x + x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*x + x^2])/32

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.34

method	result	size
risch	\(\frac {x}{4 x^{4}+4}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16}\)	\(33\)
default	\(\frac {x}{4 x^{4}+4}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {1+x^{2}+x \sqrt {2}}{1+x^{2}-x \sqrt {2}}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{32}\)	\(63\)

[In]

int(1/(x^8+2*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/4*x/(x^4+1)+3/16*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^4+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=-\frac {3 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} - i - 1\right )} \log \left (2 \, x + \left (i + 1\right ) \, \sqrt {2}\right ) + 3 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} + i - 1\right )} \log \left (2 \, x - \left (i - 1\right ) \, \sqrt {2}\right ) + 3 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} - i + 1\right )} \log \left (2 \, x + \left (i - 1\right ) \, \sqrt {2}\right ) + 3 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} + i + 1\right )} \log \left (2 \, x - \left (i + 1\right ) \, \sqrt {2}\right ) - 8 \, x}{32 \, {\left (x^{4} + 1\right )}} \]

[In]

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

[Out]

-1/32*(3*sqrt(2)*(-(I + 1)*x^4 - I - 1)*log(2*x + (I + 1)*sqrt(2)) + 3*sqrt(2)*((I - 1)*x^4 + I - 1)*log(2*x -
(I - 1)*sqrt(2)) + 3*sqrt(2)*(-(I - 1)*x^4 - I + 1)*log(2*x + (I - 1)*sqrt(2)) + 3*sqrt(2)*((I + 1)*x^4 + I +
1)*log(2*x - (I + 1)*sqrt(2)) - 8*x)/(x^4 + 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {x}{4 x^{4} + 4} - \frac {3 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} + \frac {3 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} + \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} + \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} \]

[In]

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

[Out]

x/(4*x**4 + 4) - 3*sqrt(2)*log(x**2 - sqrt(2)*x + 1)/32 + 3*sqrt(2)*log(x**2 + sqrt(2)*x + 1)/32 + 3*sqrt(2)*a
tan(sqrt(2)*x - 1)/16 + 3*sqrt(2)*atan(sqrt(2)*x + 1)/16

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {3}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {3}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]

[In]

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

[Out]

3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) + 3/32*sqr
t(2)*log(x^2 + sqrt(2)*x + 1) - 3/32*sqrt(2)*log(x^2 - sqrt(2)*x + 1) + 1/4*x/(x^4 + 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {3}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {3}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]

[In]

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

[Out]

3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 3/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) + 3/32*sqr
t(2)*log(x^2 + sqrt(2)*x + 1) - 3/32*sqrt(2)*log(x^2 - sqrt(2)*x + 1) + 1/4*x/(x^4 + 1)

Mupad [B] (verification not implemented)

Time = 8.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.45 \[ \int \frac {1}{1+2 x^4+x^8} \, dx=\frac {x}{4\,\left (x^4+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{16}+\frac {3}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{16}-\frac {3}{16}{}\mathrm {i}\right ) \]

[In]

int(1/(2*x^4 + x^8 + 1),x)

[Out]

2^(1/2)*atan(2^(1/2)*x*(1/2 - 1i/2))*(3/16 + 3i/16) + 2^(1/2)*atan(2^(1/2)*x*(1/2 + 1i/2))*(3/16 - 3i/16) + x/
(4*(x^4 + 1))