3.32 \(\int \frac{1+(1+\sqrt{3}) x^4}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=164 \[ -\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0947788, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1423, 1161, 618, 204, 1164, 628} \[ -\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + (1 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]

[Out]

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

Rule 1423

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), 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^(n/2))/(q - r*x^(n/2) + x^n), x], x] + D
ist[1/(2*c*q*r), Int[(d*r + (d - e*q)*x^(n/2))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*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+\left (1+\sqrt{3}\right ) x^4}{1-x^4+x^8} \, dx &=\frac{\int \frac{\sqrt{3}+\sqrt{3} x^2}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}+\frac{\int \frac{\sqrt{3}-\sqrt{3} x^2}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{1}{4} \int \frac{1}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx-\frac{1}{4} \sqrt{2+\sqrt{3}} \int \frac{\sqrt{2-\sqrt{3}}+2 x}{-1-\sqrt{2-\sqrt{3}} x-x^2} \, dx-\frac{1}{4} \sqrt{2+\sqrt{3}} \int \frac{\sqrt{2-\sqrt{3}}-2 x}{-1+\sqrt{2-\sqrt{3}} x-x^2} \, dx\\ &=-\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,-\sqrt{2+\sqrt{3}}+2 x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,\sqrt{2+\sqrt{3}}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{2-\sqrt{3}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}+2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{2-\sqrt{3}}}-\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.0373736, size = 72, normalized size = 0.44 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\sqrt{3} \text{$\#$1}^4 \log (x-\text{$\#$1})+\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (1 + Sqrt[3])*x^4)/(1 - x^4 + x^8),x]

[Out]

RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4 + Sqrt[3]*Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.039, size = 62, normalized size = 0.4 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{4}+2\,\sqrt{3}{{\it \_R}}^{4}+ \left ( 1+\sqrt{3} \right ) \left ( \sqrt{3}-1 \right ) \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^4*(1+3^(1/2)))/(x^8-x^4+1),x)

[Out]

1/8*sum(1/(2*_R^7-_R^3)*(2*_R^4+2*3^(1/2)*_R^4+(1+3^(1/2))*(3^(1/2)-1))*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^4*(1+3^(1/2)))/(x^8-x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38947, size = 355, normalized size = 2.16 \begin{align*} \frac{1}{2} \, \sqrt{\sqrt{3} + 2} \arctan \left (x^{3} \sqrt{\sqrt{3} + 2} - x \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 1\right )}\right ) + \frac{1}{2} \, \sqrt{\sqrt{3} + 2} \arctan \left (x \sqrt{\sqrt{3} + 2}\right ) + \frac{1}{4} \, \sqrt{\sqrt{3} + 2} \log \left (-\frac{x \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} - x^{2} - 1}{x \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} + x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^4*(1+3^(1/2)))/(x^8-x^4+1),x, algorithm="fricas")

[Out]

1/2*sqrt(sqrt(3) + 2)*arctan(x^3*sqrt(sqrt(3) + 2) - x*sqrt(sqrt(3) + 2)*(sqrt(3) - 1)) + 1/2*sqrt(sqrt(3) + 2
)*arctan(x*sqrt(sqrt(3) + 2)) + 1/4*sqrt(sqrt(3) + 2)*log(-(x*sqrt(sqrt(3) + 2)*(sqrt(3) - 2) - x^2 - 1)/(x*sq
rt(sqrt(3) + 2)*(sqrt(3) - 2) + x^2 + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**4*(1+3**(1/2)))/(x**8-x**4+1),x)

[Out]

Exception raised: PolynomialError

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Giac [A]  time = 1.14206, size = 166, normalized size = 1.01 \begin{align*} \frac{1}{4} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{4} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^4*(1+3^(1/2)))/(x^8-x^4+1),x, algorithm="giac")

[Out]

1/4*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) + 1/4*(sqrt(6) + sqrt(2))*arctan
((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2))
+ 1) - 1/8*(sqrt(6) + sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)