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 - 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]

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Rubi [A]  time = 0.347468, antiderivative size = 334, normalized size of antiderivative = 1., 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 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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 \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*}

Mathematica [C]  time = 0.0041239, size = 30, normalized size = 0.09 \[ 8 x \text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},-x^8\right )+x \log \left (x^4+\frac{1}{x^4}\right )-4 x \]

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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Maple [C]  time = 0.015, size = 36, normalized size = 0.1 \begin{align*} x\ln \left ({\frac{{x}^{8}+1}{{x}^{4}}} \right ) -4\,x+\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} x \log \left (x^{8} + 1\right ) - 4 \, x \log \left (x\right ) - 4 \, x + 8 \, \int \frac{1}{x^{8} + 1}\,{d x} \end{align*}

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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Fricas [B]  time = 1.21965, size = 3152, normalized size = 9.44 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 1.19951, size = 26, normalized size = 0.08 \begin{align*} 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 )} \end{align*}

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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Giac [A]  time = 1.12418, size = 335, normalized size = 1. \begin{align*} 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 \end{align*}

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