3.1506 \(\int \frac{1}{x^4 (1+x^8)} \, dx\)

Optimal. Leaf size=354 \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]

[Out]

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

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Rubi [A]  time = 0.186717, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {325, 299, 1127, 1161, 618, 204, 1164, 628} \[ -\frac{1}{3 x^3}+\frac{\log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )}{8 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 + x^8)),x]

[Out]

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 299

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b,
4]]}, Dist[s^3/(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Dist[s^3/
(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGt
Q[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]

Rule 1127

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[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, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 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 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 1164

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

Rubi steps

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

Mathematica [A]  time = 0.0066367, size = 216, normalized size = 0.61 \[ -\frac{1}{3 x^3}+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )+\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 + x^8)),x]

[Out]

-1/(3*x^3) - (ArcTan[(x - Cos[Pi/8])*Csc[Pi/8]]*Cos[Pi/8])/4 - (ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*Cos[Pi/8])/4
+ (Cos[Pi/8]*Log[1 + x^2 - 2*x*Sin[Pi/8]])/8 - (Cos[Pi/8]*Log[1 + x^2 + 2*x*Sin[Pi/8]])/8 + (ArcTan[Sec[Pi/8]
*(x - Sin[Pi/8])]*Sin[Pi/8])/4 + (ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*Sin[Pi/8])/4 - (Log[1 + x^2 - 2*x*Cos[Pi/8
]]*Sin[Pi/8])/8 + (Log[1 + x^2 + 2*x*Cos[Pi/8]]*Sin[Pi/8])/8

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Maple [C]  time = 0.007, size = 28, normalized size = 0.1 \begin{align*} -{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}}-{\frac{1}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^8+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3 \, x^{3}} - \int \frac{x^{4}}{x^{8} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.41375, size = 3287, normalized size = 9.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(24*x^3*sqrt(-sqrt(2) + 2)*arctan(-(2*x - 2*sqrt(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + sqrt(-sqrt(2) + 2))/
sqrt(sqrt(2) + 2)) + 24*x^3*sqrt(-sqrt(2) + 2)*arctan(-(2*x - 2*sqrt(x^2 - x*sqrt(-sqrt(2) + 2) + 1) - sqrt(-s
qrt(2) + 2))/sqrt(sqrt(2) + 2)) - 24*x^3*sqrt(sqrt(2) + 2)*arctan(-(2*x - 2*sqrt(x^2 + x*sqrt(sqrt(2) + 2) + 1
) + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 24*x^3*sqrt(sqrt(2) + 2)*arctan(-(2*x - 2*sqrt(x^2 - x*sqrt(sqrt(
2) + 2) + 1) - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) - 6*x^3*sqrt(-sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2)
+ 1) + 6*x^3*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) + 6*x^3*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt
(-sqrt(2) + 2) + 1) - 6*x^3*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2) + 2) + 1) + 12*(sqrt(2)*x^3*sqrt(sqrt(
2) + 2) - sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*arctan(-(2*sqrt(2)*x - 2*sqrt(2)*sqrt(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(
2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) +
sqrt(-sqrt(2) + 2))) + 12*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) - sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*arctan(-(2*sqrt(2)
*x - 2*sqrt(2)*sqrt(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(
2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) + 12*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) +
sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*arctan((2*sqrt(2)*x - 2*sqrt(2)*sqrt(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1
/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt
(2) + 2))) + 12*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) + sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*arctan((2*sqrt(2)*x - 2*sqrt(
2)*sqrt(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2) - sq
rt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 3*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) - sqrt(2)*x^3*s
qrt(-sqrt(2) + 2))*log(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + 3*(sqrt
(2)*x^3*sqrt(sqrt(2) + 2) + sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*log(x^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sq
rt(2)*x*sqrt(-sqrt(2) + 2) + 1) - 3*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) + sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*log(x^2 -
1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + 3*(sqrt(2)*x^3*sqrt(sqrt(2) + 2) -
sqrt(2)*x^3*sqrt(-sqrt(2) + 2))*log(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) +
1) + 64)/x^3

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Sympy [A]  time = 1.10075, size = 22, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (32768 t^{5} + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(32768*_t**5 + x))) - 1/(3*x**3)

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Giac [A]  time = 1.30793, size = 329, normalized size = 0.93 \begin{align*} \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2} + 2} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2} + 2} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{\sqrt{2} + 2} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{\sqrt{2} + 2} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) - \frac{1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(-sqrt(2) + 2)*arctan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/8*sqrt(-sqrt(2) + 2)*arctan((2
*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/8*sqrt(sqrt(2) + 2)*arctan((2*x + sqrt(sqrt(2) + 2))/sqrt(-sqr
t(2) + 2)) - 1/8*sqrt(sqrt(2) + 2)*arctan((2*x - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 1/16*sqrt(-sqrt(2) +
2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/16*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) - 1/16*sqr
t(sqrt(2) + 2)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + 1/16*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2) + 2) + 1
) - 1/3/x^3