3.16.8 \(\int \frac {1}{x^8 (1+x^8)} \, dx\) [1508]

Optimal. Leaf size=346 \[ -\frac {1}{7 x^7}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right ) \]

[Out]

-1/7/x^7+1/16*ln(1+x^2-x*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)-1/16*ln(1+x^2+x*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(
1/2)+1/4*arctan((-2*x+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/4*arctan((2*x+(2-2^(1/2))^(1
/2))/(2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)+1/16*ln(1+x^2-x*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)-1/16*ln(1+x^2
+x*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/4*arctan((-2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(
1/2)-1/4*arctan((2*x+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {331, 219, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {1}{7 x^7}+\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 + x^8)),x]

[Out]

-1/7*1/x^7 + 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])])
+ (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 + S
qrt[2]]*x + x^2])/16

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 219

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

Rule 331

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 632

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 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 648

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 1183

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]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \left (1+x^8\right )} \, dx &=-\frac {1}{7 x^7}-\int \frac {1}{1+x^8} \, dx\\ &=-\frac {1}{7 x^7}-\frac {\int \frac {\sqrt {2}-x^2}{1-\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}+x^2}{1+\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}\\ &=-\frac {1}{7 x^7}-\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\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 \left (2-\sqrt {2}\right )}+\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 \left (2+\sqrt {2}\right )}-\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 \left (2+\sqrt {2}\right )}+\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}{7 x^7}-\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}{7 x^7}+\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 )} \text {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 )} \text {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 )} \text {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 )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )\\ &=-\frac {1}{7 x^7}+\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*}

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Mathematica [A]
time = 0.01, size = 216, normalized size = 0.62 \begin {gather*} -\frac {1}{7 x^7}-\frac {1}{4} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x-\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{4} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )+\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right )-\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right )-\frac {1}{4} \tan ^{-1}\left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{4} \tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{8} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{8} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(1 + x^8)),x]

[Out]

-1/7*1/x^7 - (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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 28, normalized size = 0.08

method	result	size
risch	\(-\frac {1}{7 x^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (x -\textit {\_R} \right )\right )}{8}\)	\(26\)
default	\(-\frac {1}{7 x^{7}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8}\)	\(28\)
meijerg	\(-\frac {1}{7 x^{7}}-\frac {x \left (-\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}-\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}\right )}{8}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/x^7 - integrate(1/(x^8 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1121 vs. \(2 (250) = 500\).
time = 0.40, size = 1121, normalized size = 3.24 \begin {gather*} \frac {56 \, x^{7} \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} + x \sqrt {-\sqrt {2} + 2} + 1} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + 56 \, x^{7} \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} - x \sqrt {-\sqrt {2} + 2} + 1} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + 56 \, x^{7} \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} + x \sqrt {\sqrt {2} + 2} + 1} + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + 56 \, x^{7} \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} - x \sqrt {\sqrt {2} + 2} + 1} - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - 14 \, x^{7} \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) + 14 \, x^{7} \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) - 14 \, x^{7} \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) + 14 \, x^{7} \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) + 28 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} + \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} + \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) + 28 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} + \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} - \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) - 28 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} - \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} + \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) - 28 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} - \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} - \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) - 7 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} + \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) - 7 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} - \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) + 7 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} - \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) + 7 \, {\left (\sqrt {2} x^{7} \sqrt {\sqrt {2} + 2} + \sqrt {2} x^{7} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) - 64}{448 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/448*(56*x^7*sqrt(sqrt(2) + 2)*arctan(-(2*x - 2*sqrt(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + sqrt(-sqrt(2) + 2))/sq
rt(sqrt(2) + 2)) + 56*x^7*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)) + 56*x^7*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)) + 56*x^7*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)) - 14*x^7*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2)
+ 1) + 14*x^7*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) - 14*x^7*sqrt(-sqrt(2) + 2)*log(x^2 + x*sqr
t(-sqrt(2) + 2) + 1) + 14*x^7*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2) + 2) + 1) + 28*(sqrt(2)*x^7*sqrt(sq
rt(2) + 2) + sqrt(2)*x^7*sqrt(-sqrt(2) + 2))*arctan(-(2*sqrt(2)*x - sqrt(2)*sqrt(4*x^2 + 2*sqrt(2)*x*sqrt(sqrt
(2) + 2) - 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) +
sqrt(-sqrt(2) + 2))) + 28*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) + sqrt(2)*x^7*sqrt(-sqrt(2) + 2))*arctan(-(2*sqrt(2)*
x - sqrt(2)*sqrt(4*x^2 - 2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) - sqrt(sqrt(2) +
2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 28*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) - sqrt(
2)*x^7*sqrt(-sqrt(2) + 2))*arctan((2*sqrt(2)*x - sqrt(2)*sqrt(4*x^2 + 2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 2*sqrt(2
)*x*sqrt(-sqrt(2) + 2) + 4) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))
) - 28*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) - sqrt(2)*x^7*sqrt(-sqrt(2) + 2))*arctan((2*sqrt(2)*x - sqrt(2)*sqrt(4*x
^2 - 2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) - sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) +
2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 7*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) + sqrt(2)*x^7*sqrt(-sqrt(2)
+ 2))*log(4*x^2 + 2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) - 7*(sqrt(2)*x^7*sqrt(sq
rt(2) + 2) - sqrt(2)*x^7*sqrt(-sqrt(2) + 2))*log(4*x^2 + 2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 2*sqrt(2)*x*sqrt(-sqr
t(2) + 2) + 4) + 7*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) - sqrt(2)*x^7*sqrt(-sqrt(2) + 2))*log(4*x^2 - 2*sqrt(2)*x*sq
rt(sqrt(2) + 2) + 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) + 7*(sqrt(2)*x^7*sqrt(sqrt(2) + 2) + sqrt(2)*x^7*sqrt(-s
qrt(2) + 2))*log(4*x^2 - 2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) - 64)/x^7

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Sympy [A]
time = 1.16, size = 20, normalized size = 0.06 \begin {gather*} \operatorname {RootSum} {\left (16777216 t^{8} + 1, \left ( t \mapsto t \log {\left (- 8 t + x \right )} \right )\right )} - \frac {1}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.57, size = 244, normalized size = 0.71 \begin {gather*} -\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}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(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) - 1/7/x^7

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Mupad [B]
time = 0.04, size = 293, normalized size = 0.85 \begin {gather*} -\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+\sqrt {2}}\right )\,\left (\frac {\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )-\frac {1}{7\,x^7}+\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2}+\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{\sqrt {2}+\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\frac {\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )-\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{16}-\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}+\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(x^8 + 1)),x)

[Out]

atan((x*(2^(1/2) - 2)^(1/2)*1i)/(2^(1/2) + (2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)) + (x*(2^(1/2) + 2)^(1/2)*1
i)/(2^(1/2) + (2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)))*(((2^(1/2) - 2)^(1/2)*1i)/8 + ((2^(1/2) + 2)^(1/2)*1i)
/8) - 1/(7*x^7) - atan((x*(- 2^(1/2) - 2)^(1/2)*1i)/((2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2)^(1/2) + 2^(1/2)) - (x
*(2 - 2^(1/2))^(1/2)*1i)/((2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2)^(1/2) + 2^(1/2)))*(((- 2^(1/2) - 2)^(1/2)*1i)/8
- ((2 - 2^(1/2))^(1/2)*1i)/8) - atan(x*(2^(1/2) + 2)^(1/2)*(1/2 + 1i/2) - (2^(1/2)*x*(2^(1/2) + 2)^(1/2))/2)*(
(2^(1/2)*1i)/16 - (1/16 + 1i/16))*(2^(1/2) + 2)^(1/2)*2i + atan(x*(2^(1/2) + 2)^(1/2)*(1/2 - 1i/2) + (2^(1/2)*
x*(2^(1/2) + 2)^(1/2)*1i)/2)*(2^(1/2)/16 - (1/16 - 1i/16))*(2^(1/2) + 2)^(1/2)*2i

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