3.1505 \(\int \frac{1}{x^2 (1+x^8)} \, dx\)

Optimal. Leaf size=344 \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqr
t[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])]) -
(Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/16 + (Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x
^2])/16 - (Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/16 + (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt
[2]]*x + x^2])/16

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Rubi [A]  time = 0.193116, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {325, 299, 1122, 1169, 634, 618, 204, 628} \[ -\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 + x^8)),x]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqr
t[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])]) -
(Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/16 + (Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x
^2])/16 - (Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/16 + (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt
[2]]*x + x^2])/16

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 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

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 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 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 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^2 \left (1+x^8\right )} \, dx &=-\frac{1}{x}-\int \frac{x^6}{1+x^8} \, dx\\ &=-\frac{1}{x}-\frac{\int \frac{x^4}{1-\sqrt{2} x^2+x^4} \, dx}{2 \sqrt{2}}+\frac{\int \frac{x^4}{1+\sqrt{2} x^2+x^4} \, dx}{2 \sqrt{2}}\\ &=-\frac{1}{x}+\frac{\int \frac{1-\sqrt{2} x^2}{1-\sqrt{2} x^2+x^4} \, dx}{2 \sqrt{2}}-\frac{\int \frac{1+\sqrt{2} x^2}{1+\sqrt{2} x^2+x^4} \, dx}{2 \sqrt{2}}\\ &=-\frac{1}{x}-\frac{\int \frac{\sqrt{2-\sqrt{2}}-\left (1-\sqrt{2}\right ) x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\int \frac{\sqrt{2-\sqrt{2}}+\left (1-\sqrt{2}\right ) x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{\int \frac{\sqrt{2+\sqrt{2}}-\left (1+\sqrt{2}\right ) x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}+\frac{\int \frac{\sqrt{2+\sqrt{2}}+\left (1+\sqrt{2}\right ) x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (2+\sqrt{2}\right )}}\\ &=-\frac{1}{x}-\frac{1}{8} \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} \int \frac{1}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx-\frac{1}{8} \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} \int \frac{1}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx-\frac{1}{16} \sqrt{2-\sqrt{2}} \int \frac{-\sqrt{2-\sqrt{2}}+2 x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx+\frac{1}{16} \sqrt{2-\sqrt{2}} \int \frac{\sqrt{2-\sqrt{2}}+2 x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx-\frac{1}{16} \sqrt{2+\sqrt{2}} \int \frac{-\sqrt{2+\sqrt{2}}+2 x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx+\frac{1}{16} \sqrt{2+\sqrt{2}} \int \frac{\sqrt{2+\sqrt{2}}+2 x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx-\frac{1}{8} \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} \int \frac{1}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx-\frac{1}{8} \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} \int \frac{1}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx\\ &=-\frac{1}{x}-\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (1-\sqrt{2-\sqrt{2}} x+x^2\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (1+\sqrt{2-\sqrt{2}} x+x^2\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (1-\sqrt{2+\sqrt{2}} x+x^2\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (1+\sqrt{2+\sqrt{2}} x+x^2\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,-\sqrt{2+\sqrt{2}}+2 x\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,\sqrt{2+\sqrt{2}}+2 x\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,-\sqrt{2-\sqrt{2}}+2 x\right )+\frac{1}{4} \sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,\sqrt{2-\sqrt{2}}+2 x\right )\\ &=-\frac{1}{x}+\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+2 x}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{8} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}+2 x}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (1-\sqrt{2-\sqrt{2}} x+x^2\right )+\frac{1}{16} \sqrt{2-\sqrt{2}} \log \left (1+\sqrt{2-\sqrt{2}} x+x^2\right )-\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (1-\sqrt{2+\sqrt{2}} x+x^2\right )+\frac{1}{16} \sqrt{2+\sqrt{2}} \log \left (1+\sqrt{2+\sqrt{2}} x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0066273, size = 214, normalized size = 0.62 \[ -\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \sin \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\frac{1}{8} \cos \left (\frac{\pi }{8}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-\frac{1}{x}-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x-\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x-\sin \left (\frac{\pi }{8}\right )\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 + x^8)),x]

[Out]

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

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Maple [C]  time = 0.005, 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}}}}-{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{x} - \int \frac{x^{6}}{x^{8} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x - integrate(x^6/(x^8 + 1), x)

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Fricas [B]  time = 1.50916, size = 8192, normalized size = 23.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(8*x*sqrt(-sqrt(2) + 2)*arctan((3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2) + 8*x - sqrt((
sqrt(2) + 2)^3 - 3*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) - 2)^3 + 48*x*(s
qrt(2) + 2)*sqrt(-sqrt(2) + 2) - 16*x*(-sqrt(2) + 2)^(3/2) + 64*x^2))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) +
2)*(sqrt(2) - 2))) + 8*x*sqrt(-sqrt(2) + 2)*arctan(-(3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2)
- 8*x + sqrt((sqrt(2) + 2)^3 - 3*(sqrt(2) + 2)^2*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) -
2)^3 - 48*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 16*x*(-sqrt(2) + 2)^(3/2) + 64*x^2))/((sqrt(2) + 2)^(3/2) + 3*
sqrt(sqrt(2) + 2)*(sqrt(2) - 2))) + 8*x*sqrt(sqrt(2) + 2)*arctan(((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(s
qrt(2) - 2) + 8*x - sqrt((sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) - 2)^3 + 16*x*(sqrt(2) +
2)^(3/2) + 64*x^2 - 3*((sqrt(2) + 2)^2 - 16*x*sqrt(sqrt(2) + 2))*(sqrt(2) - 2)))/(3*(sqrt(2) + 2)*sqrt(-sqrt(
2) + 2) - (-sqrt(2) + 2)^(3/2))) + 8*x*sqrt(sqrt(2) + 2)*arctan(-((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(s
qrt(2) - 2) - 8*x + sqrt((sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2) - 2)^3 - 16*x*(sqrt(2) +
2)^(3/2) + 64*x^2 - 3*((sqrt(2) + 2)^2 + 16*x*sqrt(sqrt(2) + 2))*(sqrt(2) - 2)))/(3*(sqrt(2) + 2)*sqrt(-sqrt(
2) + 2) - (-sqrt(2) + 2)^(3/2))) - 2*x*sqrt(sqrt(2) + 2)*log(1/64*(sqrt(2) + 2)^3 - 3/64*(sqrt(2) + 2)^2*(sqrt
(2) - 2) + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/4*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2)
- 1/4*x*(-sqrt(2) + 2)^(3/2) + x^2) + 2*x*sqrt(sqrt(2) + 2)*log(1/64*(sqrt(2) + 2)^3 - 3/64*(sqrt(2) + 2)^2*(s
qrt(2) - 2) + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/4*x*(sqrt(2) + 2)*sqrt(-sqrt(2) +
2) + 1/4*x*(-sqrt(2) + 2)^(3/2) + x^2) - 2*x*sqrt(-sqrt(2) + 2)*log(1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*
(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 1/4*x*(sqrt(2) + 2)^(3/2) + x^2 - 3/64*((sqrt(2) + 2)^2 - 16*x*sqrt(s
qrt(2) + 2))*(sqrt(2) - 2)) + 2*x*sqrt(-sqrt(2) + 2)*log(1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) -
2)^2 - 1/64*(sqrt(2) - 2)^3 - 1/4*x*(sqrt(2) + 2)^(3/2) + x^2 - 3/64*((sqrt(2) + 2)^2 + 16*x*sqrt(sqrt(2) + 2)
)*(sqrt(2) - 2)) + 4*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan((8*sqrt(2)*x + (sqrt(
2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2)
- sqrt(2)*sqrt(8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + (sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - (sqrt(2)
- 2)^3 - 24*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + 64*x^2 + 3*(8*sqrt
(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)^2)*(sqrt(2) - 2)))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2)
- 2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2))) + 4*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)
*x*sqrt(-sqrt(2) + 2))*arctan((8*sqrt(2)*x - (sqrt(2) + 2)^(3/2) - 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt
(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(2) + 2)^(3/2) - 8*sqrt(2)*sqrt(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*
(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt
(-sqrt(2) + 2) - 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 - 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2
)^2)*(sqrt(2) - 2)))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2)
+ 2) - (-sqrt(2) + 2)^(3/2))) + 4*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan(-(8*sqrt
(2)*x + (sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) + 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - (-sqrt(
2) + 2)^(3/2) - sqrt(2)*sqrt(8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + (sqrt(2) + 2)^3 + 3*(sqrt(2) + 2)*(sqrt(2) - 2)
^2 - (sqrt(2) - 2)^3 + 24*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + 64*x
^2 + 3*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)^2)*(sqrt(2) - 2)))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2)
+ 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2))) + 4*(sqrt(2)*x*sqrt(sqrt(2)
+ 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*arctan(-(8*sqrt(2)*x - (sqrt(2) + 2)^(3/2) - 3*sqrt(sqrt(2) + 2)*(sqrt(2)
- 2) - 3*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2) - 8*sqrt(2)*sqrt(-1/8*sqrt(2)*x*(sqrt(2) + 2
)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2)*x*(sq
rt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 - 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2)
+ (sqrt(2) + 2)^2)*(sqrt(2) - 2)))/((sqrt(2) + 2)^(3/2) + 3*sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 3*(sqrt(2) + 2)
*sqrt(-sqrt(2) + 2) + (-sqrt(2) + 2)^(3/2))) - (sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*lo
g(1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2
) - 2)^3 + 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 + 3/64*(8
*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)^2)*(sqrt(2) - 2)) + (sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt
(-sqrt(2) + 2))*log(1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2
)^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) + 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/
2) + x^2 + 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2) + 2)^2)*(sqrt(2) - 2)) - (sqrt(2)*x*sqrt(sqrt(2) + 2
) - sqrt(2)*x*sqrt(-sqrt(2) + 2))*log(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) + 2)^3 + 3/64*(sqrt(2
) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 + 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) + 2) - 1/8*sqrt(2)*x
*(-sqrt(2) + 2)^(3/2) + x^2 - 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(2) - 2)) + (sqrt(2)
*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2))*log(-1/8*sqrt(2)*x*(sqrt(2) + 2)^(3/2) + 1/64*(sqrt(2) +
2)^3 + 3/64*(sqrt(2) + 2)*(sqrt(2) - 2)^2 - 1/64*(sqrt(2) - 2)^3 - 3/8*sqrt(2)*x*(sqrt(2) + 2)*sqrt(-sqrt(2) +
2) + 1/8*sqrt(2)*x*(-sqrt(2) + 2)^(3/2) + x^2 - 3/64*(8*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2) + 2)^2)*(sqrt(
2) - 2)) + 64)/x

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Sympy [A]  time = 1.08924, size = 19, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} + 1, \left ( t \mapsto t \log{\left (- 2097152 t^{7} + x \right )} \right )\right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8 + 1, Lambda(_t, _t*log(-2097152*_t**7 + x))) - 1/x

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Giac [A]  time = 1.33336, size = 329, normalized size = 0.96 \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}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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*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/x