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

Optimal. Leaf size=157 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}} \]

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Rubi [A]  time = 0.09, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1419, 1093, 203, 207} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {\sqrt {3}-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 4*x^4 + x^8),x]

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1419

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[(2*d)/e -
b/c, 2]}, Dist[e/(2*c), Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x^(n/2
) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2,
0] && IGtQ[n/2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !LtQ[(2*d)/e - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))

Rubi steps

\begin {align*} \int \frac {1+x^4}{1-4 x^4+x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {6} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {6} x^2+x^4} \, dx\\ &=\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {2}}+\frac {\int \frac {1}{\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1}{\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {-1+\sqrt {3}}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {-1+\sqrt {3}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {1+\sqrt {3}}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 53, normalized size = 0.34 \begin {gather*} \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^4+1\&,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{\text {$\#$1}^7-2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{1-4 x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x^4)/(1 - 4*x^4 + x^8),x]

[Out]

IntegrateAlgebraic[(1 + x^4)/(1 - 4*x^4 + x^8), x]

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fricas [B]  time = 1.26, size = 331, normalized size = 2.11 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \, \sqrt {x^{2} + {\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2}} {\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}} - \frac {1}{2} \, {\left (\sqrt {3} \sqrt {2} x + \sqrt {2} x\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {x^{2} - \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )}} {\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} - {\left (\sqrt {3} \sqrt {2} x - \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left ({\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-{\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) - \frac {1}{8} \, \sqrt {2} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left ({\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) + \frac {1}{8} \, \sqrt {2} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-{\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*(-sqrt(3) + 2)^(1/4)*arctan(1/2*sqrt(x^2 + (sqrt(3) + 2)*sqrt(-sqrt(3) + 2))*(sqrt(3)*sqrt(2) + sq
rt(2))*(-sqrt(3) + 2)^(3/4) - 1/2*(sqrt(3)*sqrt(2)*x + sqrt(2)*x)*(-sqrt(3) + 2)^(3/4)) - 1/2*sqrt(2)*(sqrt(3)
+ 2)^(1/4)*arctan(1/2*(sqrt(x^2 - sqrt(sqrt(3) + 2)*(sqrt(3) - 2))*(sqrt(3)*sqrt(2) - sqrt(2))*sqrt(sqrt(3) +
2) - (sqrt(3)*sqrt(2)*x - sqrt(2)*x)*sqrt(sqrt(3) + 2))*(sqrt(3) + 2)^(1/4)) + 1/8*sqrt(2)*(sqrt(3) + 2)^(1/4
)*log((sqrt(3)*sqrt(2) - sqrt(2))*(sqrt(3) + 2)^(1/4) + 2*x) - 1/8*sqrt(2)*(sqrt(3) + 2)^(1/4)*log(-(sqrt(3)*s
qrt(2) - sqrt(2))*(sqrt(3) + 2)^(1/4) + 2*x) - 1/8*sqrt(2)*(-sqrt(3) + 2)^(1/4)*log((sqrt(3)*sqrt(2) + sqrt(2)
)*(-sqrt(3) + 2)^(1/4) + 2*x) + 1/8*sqrt(2)*(-sqrt(3) + 2)^(1/4)*log(-(sqrt(3)*sqrt(2) + sqrt(2))*(-sqrt(3) +
2)^(1/4) + 2*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [C]  time = 0.01, size = 40, normalized size = 0.25 \begin {gather*} \frac {\left (\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )^{7}-16 \RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{x^{8} - 4 \, x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.72, size = 399, normalized size = 2.54 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {5184\,\sqrt {2}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {3024\,\sqrt {2}\,\sqrt {3}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,5184{}\mathrm {i}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}-\frac {\sqrt {2}\,\sqrt {3}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,3024{}\mathrm {i}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}\,1{}\mathrm {i}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {5184\,\sqrt {2}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}-\frac {3024\,\sqrt {2}\,\sqrt {3}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{2160\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-3888\,\sqrt {2-\sqrt {3}}}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,5184{}\mathrm {i}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {\sqrt {2}\,\sqrt {3}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,3024{}\mathrm {i}}{3888\,\sqrt {\sqrt {3}+2}+2160\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}\,1{}\mathrm {i}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/(x^8 - 4*x^4 + 1),x)

[Out]

(2^(1/2)*atan((2^(1/2)*x*(2 - 3^(1/2))^(1/4)*5184i)/(2160*3^(1/2)*(2 - 3^(1/2))^(1/2) - 3888*(2 - 3^(1/2))^(1/
2)) - (2^(1/2)*3^(1/2)*x*(2 - 3^(1/2))^(1/4)*3024i)/(2160*3^(1/2)*(2 - 3^(1/2))^(1/2) - 3888*(2 - 3^(1/2))^(1/
2)))*(2 - 3^(1/2))^(1/4)*1i)/4 - (2^(1/2)*atan((5184*2^(1/2)*x*(2 - 3^(1/2))^(1/4))/(2160*3^(1/2)*(2 - 3^(1/2)
)^(1/2) - 3888*(2 - 3^(1/2))^(1/2)) - (3024*2^(1/2)*3^(1/2)*x*(2 - 3^(1/2))^(1/4))/(2160*3^(1/2)*(2 - 3^(1/2))
^(1/2) - 3888*(2 - 3^(1/2))^(1/2)))*(2 - 3^(1/2))^(1/4))/4 + (2^(1/2)*atan((5184*2^(1/2)*x*(3^(1/2) + 2)^(1/4)
)/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)) + (3024*2^(1/2)*3^(1/2)*x*(3^(1/2) + 2)^(1/4))
/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)))*(3^(1/2) + 2)^(1/4))/4 - (2^(1/2)*atan((2^(1/2
)*x*(3^(1/2) + 2)^(1/4)*5184i)/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)) + (2^(1/2)*3^(1/2
)*x*(3^(1/2) + 2)^(1/4)*3024i)/(3888*(3^(1/2) + 2)^(1/2) + 2160*3^(1/2)*(3^(1/2) + 2)^(1/2)))*(3^(1/2) + 2)^(1
/4)*1i)/4

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sympy [A]  time = 0.19, size = 24, normalized size = 0.15 \begin {gather*} \operatorname {RootSum} {\left (1048576 t^{8} - 4096 t^{4} + 1, \left (t \mapsto t \log {\left (4096 t^{5} - 12 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(1048576*_t**8 - 4096*_t**4 + 1, Lambda(_t, _t*log(4096*_t**5 - 12*_t + x)))

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