3.6 \(\int \log (\frac {1}{x^4}+x^4) \, dx\)

Optimal. Leaf size=334 \[ x \log \left (x^4+\frac {1}{x^4}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-4 x-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.35, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2523, 12, 388, 213, 1169, 634, 618, 204, 628} \[ x \log \left (x^4+\frac {1}{x^4}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-4 x-\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[x^(-4) + x^4],x]

[Out]

-4*x - Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]] - Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[
2 + Sqrt[2]] - 2*x)/Sqrt[2 - Sqrt[2]]] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]
+ Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]] - (Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 -
Sqrt[2]]*x + x^2])/2 + (Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/2 - (Sqrt[2 + Sqrt[2]]*Log[1 -
Sqrt[2 + Sqrt[2]]*x + x^2])/2 + (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2])/2 + x*Log[x^(-4) + x^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 213

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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 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 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 30, normalized size = 0.09 \[ 8 x \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-x^8\right )+x \log \left (x^4+\frac {1}{x^4}\right )-4 x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x^(-4) + x^4],x]

[Out]

-4*x + 8*x*Hypergeometric2F1[1/8, 1, 9/8, -x^8] + x*Log[x^(-4) + x^4]

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fricas [B]  time = 0.45, size = 1058, normalized size = 3.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(-(2*sqrt(2)*x - sqrt(2)*sqrt(4*x^2 + 2*sq
rt(2)*x*sqrt(sqrt(2) + 2) - 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqr
t(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 1/2*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*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))) + 1/2*(sqrt(2)*sqrt(sqrt(2) + 2)
- sqrt(2)*sqrt(-sqrt(2) + 2))*arctan((2*sqrt(2)*x - sqrt(2)*sqrt(4*x^2 + 2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 2*sqr
t(2)*x*sqrt(-sqrt(2) + 2) + 4) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) +
2))) + 1/2*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*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))) + 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*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) + 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) -
sqrt(2)*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) -
1/8*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*log(4*x^2 - 2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 2*sq
rt(2)*x*sqrt(-sqrt(2) + 2) + 4) - 1/8*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*log(4*x^2 - 2*s
qrt(2)*x*sqrt(sqrt(2) + 2) - 2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 4) + x*log((x^8 + 1)/x^4) - sqrt(sqrt(2) + 2)*ar
ctan(-(2*x - 2*sqrt(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 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)) - 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)) - 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)) +
1/4*sqrt(sqrt(2) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/4*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2)
+ 1) + 1/4*sqrt(-sqrt(2) + 2)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) - 1/4*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(-
sqrt(2) + 2) + 1) - 4*x

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giac [A]  time = 0.20, size = 248, normalized size = 0.74 \[ x \log \left (x^{4} + \frac {1}{x^{4}}\right ) + \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) - 4 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^4 + 1/x^4) + sqrt(sqrt(2) + 2)*arctan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + sqrt(sqrt(2) + 2
)*arctan((2*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + sqrt(-sqrt(2) + 2)*arctan((2*x + sqrt(sqrt(2) + 2))/s
qrt(-sqrt(2) + 2)) + sqrt(-sqrt(2) + 2)*arctan((2*x - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 1/2*sqrt(sqrt(2
) + 2)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/2*sqrt(sqrt(2) + 2)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1) + 1/2*sqr
t(-sqrt(2) + 2)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) - 1/2*sqrt(-sqrt(2) + 2)*log(x^2 - x*sqrt(-sqrt(2) + 2) +
1) - 4*x

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maple [C]  time = 0.03, size = 36, normalized size = 0.11 \[ x \ln \left (\frac {x^{8}+1}{x^{4}}\right )-4 x +\frac {\ln \left (-\RootOf \left (\textit {\_Z}^{8}+1\right )+x \right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln((x^8+1)/x^4)-4*x+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 \[ x \log \left (x^{8} + 1\right ) - 4 \, x \log \relax (x) - 4 \, x + 8 \, \int \frac {1}{x^{8} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.26, size = 313, normalized size = 0.94 \[ x\,\ln \left (\frac {1}{x^4}+x^4\right )-4\,x+\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+2097152\,\sqrt {2}}\right )\,\left (\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}-\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,2097152{}\mathrm {i}}{2097152\,\sqrt {2}+2097152\,\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}+\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )+\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{2}-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}-\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{2}-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(1/x^4 + x^4),x)

[Out]

x*log(1/x^4 + x^4) - 4*x + atan((x*(- 2^(1/2) - 2)^(1/2)*2097152i)/(2097152*(2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2
)^(1/2) + 2097152*2^(1/2)) - (x*(2 - 2^(1/2))^(1/2)*2097152i)/(2097152*(2 - 2^(1/2))^(1/2)*(- 2^(1/2) - 2)^(1/
2) + 2097152*2^(1/2)))*((- 2^(1/2) - 2)^(1/2)*1i - (2 - 2^(1/2))^(1/2)*1i) - atan((x*(2^(1/2) - 2)^(1/2)*20971
52i)/(2097152*2^(1/2) + 2097152*(2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)) + (x*(2^(1/2) + 2)^(1/2)*2097152i)/(2
097152*2^(1/2) + 2097152*(2^(1/2) - 2)^(1/2)*(2^(1/2) + 2)^(1/2)))*((2^(1/2) - 2)^(1/2)*1i + (2^(1/2) + 2)^(1/
2)*1i) + 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)/2 - (1/2 +
1i/2))*(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)/2 - (1/2 - 1i/2))*(2^(1/2) + 2)^(1/2)*2i

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sympy [A]  time = 2.81, size = 26, normalized size = 0.08 \[ x \log {\left (x^{4} + \frac {1}{x^{4}} \right )} - 4 x - \operatorname {RootSum} {\left (t^{8} + 1, \left (t \mapsto t \log {\left (- t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x**4 + x**(-4)) - 4*x - RootSum(_t**8 + 1, Lambda(_t, _t*log(-_t + x)))

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