3.1.18 \(\int \frac {1+x^4}{1-5 x^4+x^8} \, dx\)

Optimal. Leaf size=171 \[ \frac {\tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {7}-\sqrt {3}}} x\right )}{\sqrt {6 \left (\sqrt {7}-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {7}-\sqrt {3}}} x\right )}{\sqrt {6 \left (\sqrt {7}-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}} \]

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Rubi [A]  time = 0.15, antiderivative size = 171, 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 (\sqrt {\frac {2}{\sqrt {7}-\sqrt {3}}} x\right )}{\sqrt {6 \left (\sqrt {7}-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {7}-\sqrt {3}}} x\right )}{\sqrt {6 \left (\sqrt {7}-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 5*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTan[Sqrt[2/(Sqrt[3] + Sqrt[7])]*x]/Sq
rt[6*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(-Sqrt[3] + Sqrt[7])] - ArcTanh[Sqr
t[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[6*(Sqrt[3] + Sqrt[7])]

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}{1-5 x^4+x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {7} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {7} x^2+x^4} \, dx\\ &=\frac {\int \frac {1}{-\frac {\sqrt {3}}{2}-\frac {\sqrt {7}}{2}+x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{\frac {\sqrt {3}}{2}-\frac {\sqrt {7}}{2}+x^2} \, dx}{2 \sqrt {3}}+\frac {\int \frac {1}{-\frac {\sqrt {3}}{2}+\frac {\sqrt {7}}{2}+x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{\frac {\sqrt {3}}{2}+\frac {\sqrt {7}}{2}+x^2} \, dx}{2 \sqrt {3}}\\ &=\frac {\tan ^{-1}\left (\sqrt {\frac {2}{-\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (-\sqrt {3}+\sqrt {7}\right )}}-\frac {\tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{-\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (-\sqrt {3}+\sqrt {7}\right )}}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {3}+\sqrt {7}}} x\right )}{\sqrt {6 \left (\sqrt {3}+\sqrt {7}\right )}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 55, normalized size = 0.32 \begin {gather*} \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-5 \text {$\#$1}^4+1\&,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-5 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(1 - 5*x^4 + x^8),x]

[Out]

RootSum[1 - 5*#1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-5*#1^3 + 2*#1^7) & ]/4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{1-5 x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x^4)/(1 - 5*x^4 + x^8),x]

[Out]

IntegrateAlgebraic[(1 + x^4)/(1 - 5*x^4 + x^8), x]

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fricas [B]  time = 1.65, size = 574, normalized size = 3.36 \begin {gather*} \frac {1}{6} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} \arctan \left (\frac {1}{48} \, {\left (\sqrt {7} \sqrt {6} \sqrt {3} \sqrt {2} + 3 \, \sqrt {6} \sqrt {2}\right )} \sqrt {4 \, x^{2} + {\left (\sqrt {7} \sqrt {3} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {7} \sqrt {3} + 5}} \sqrt {-\sqrt {7} \sqrt {3} + 5} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} - \frac {1}{24} \, {\left (\sqrt {7} \sqrt {6} \sqrt {3} \sqrt {2} x + 3 \, \sqrt {6} \sqrt {2} x\right )} \sqrt {-\sqrt {7} \sqrt {3} + 5} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}}\right ) - \frac {1}{6} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}} \arctan \left (\frac {1}{48} \, {\left ({\left (\sqrt {7} \sqrt {6} \sqrt {3} \sqrt {2} - 3 \, \sqrt {6} \sqrt {2}\right )} \sqrt {4 \, x^{2} - {\left (\sqrt {7} \sqrt {3} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {7} \sqrt {3} + 5}} \sqrt {\sqrt {7} \sqrt {3} + 5} - 2 \, {\left (\sqrt {7} \sqrt {6} \sqrt {3} \sqrt {2} x - 3 \, \sqrt {6} \sqrt {2} x\right )} \sqrt {\sqrt {7} \sqrt {3} + 5}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}}\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}} \log \left ({\left (\sqrt {7} \sqrt {6} \sqrt {3} - 3 \, \sqrt {6}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}} \log \left (-{\left (\sqrt {7} \sqrt {6} \sqrt {3} - 3 \, \sqrt {6}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {7} \sqrt {3} + 5}} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} \log \left ({\left (\sqrt {7} \sqrt {6} \sqrt {3} + 3 \, \sqrt {6}\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} \log \left (-{\left (\sqrt {7} \sqrt {6} \sqrt {3} + 3 \, \sqrt {6}\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {7} \sqrt {3} + 5}} + 12 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^8-5*x^4+1),x, algorithm="fricas")

[Out]

1/6*sqrt(6)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))*arctan(1/48*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2) + 3*sqrt(6)*
sqrt(2))*sqrt(4*x^2 + (sqrt(7)*sqrt(3)*sqrt(2) + 5*sqrt(2))*sqrt(-sqrt(7)*sqrt(3) + 5))*sqrt(-sqrt(7)*sqrt(3)
+ 5)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)) - 1/24*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2)*x + 3*sqrt(6)*sqrt(2)*x)
*sqrt(-sqrt(7)*sqrt(3) + 5)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))) - 1/6*sqrt(6)*sqrt(sqrt(2)*sqrt(sqrt(7)*
sqrt(3) + 5))*arctan(1/48*((sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2) - 3*sqrt(6)*sqrt(2))*sqrt(4*x^2 - (sqrt(7)*sqrt(3)
*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(7)*sqrt(3) + 5))*sqrt(sqrt(7)*sqrt(3) + 5) - 2*(sqrt(7)*sqrt(6)*sqrt(3)*sqrt(2
)*x - 3*sqrt(6)*sqrt(2)*x)*sqrt(sqrt(7)*sqrt(3) + 5))*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))) + 1/24*sqrt(6)*
sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))*log((sqrt(7)*sqrt(6)*sqrt(3) - 3*sqrt(6))*sqrt(sqrt(2)*sqrt(sqrt(7)*sq
rt(3) + 5)) + 12*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5))*log(-(sqrt(7)*sqrt(6)*sqrt(3) - 3*s
qrt(6))*sqrt(sqrt(2)*sqrt(sqrt(7)*sqrt(3) + 5)) + 12*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)
)*log((sqrt(7)*sqrt(6)*sqrt(3) + 3*sqrt(6))*sqrt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5)) + 12*x) + 1/24*sqrt(6)*sq
rt(sqrt(2)*sqrt(-sqrt(7)*sqrt(3) + 5))*log(-(sqrt(7)*sqrt(6)*sqrt(3) + 3*sqrt(6))*sqrt(sqrt(2)*sqrt(-sqrt(7)*s
qrt(3) + 5)) + 12*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^8-5*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 = 42, normalized size = 0.25 \begin {gather*} \frac {\left (\RootOf \left (\textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )^{7}-20 \RootOf \left (\textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^8-5*x^4+1),x)

[Out]

1/4*sum((_R^4+1)/(2*_R^7-5*_R^3)*ln(-_R+x),_R=RootOf(_Z^8-5*_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} - 5 \, x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^8-5*x^4+1),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/(x^8 - 5*x^4 + 1), x)

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mupad [B]  time = 1.76, size = 483, normalized size = 2.82 \begin {gather*} \frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {12005\,2^{3/4}\,\sqrt {3}\,x\,{\left (5-\sqrt {21}\right )}^{1/4}}{2\,\left (4802\,\sqrt {2}\,\sqrt {5-\sqrt {21}}-1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {5-\sqrt {21}}\right )}-\frac {7889\,2^{3/4}\,\sqrt {3}\,\sqrt {21}\,x\,{\left (5-\sqrt {21}\right )}^{1/4}}{6\,\left (4802\,\sqrt {2}\,\sqrt {5-\sqrt {21}}-1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {5-\sqrt {21}}\right )}\right )\,{\left (5-\sqrt {21}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (5-\sqrt {21}\right )}^{1/4}\,12005{}\mathrm {i}}{2\,\left (4802\,\sqrt {2}\,\sqrt {5-\sqrt {21}}-1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {5-\sqrt {21}}\right )}-\frac {2^{3/4}\,\sqrt {3}\,\sqrt {21}\,x\,{\left (5-\sqrt {21}\right )}^{1/4}\,7889{}\mathrm {i}}{6\,\left (4802\,\sqrt {2}\,\sqrt {5-\sqrt {21}}-1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {5-\sqrt {21}}\right )}\right )\,{\left (5-\sqrt {21}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {12005\,2^{3/4}\,\sqrt {3}\,x\,{\left (\sqrt {21}+5\right )}^{1/4}}{2\,\left (4802\,\sqrt {2}\,\sqrt {\sqrt {21}+5}+1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {\sqrt {21}+5}\right )}+\frac {7889\,2^{3/4}\,\sqrt {3}\,\sqrt {21}\,x\,{\left (\sqrt {21}+5\right )}^{1/4}}{6\,\left (4802\,\sqrt {2}\,\sqrt {\sqrt {21}+5}+1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {\sqrt {21}+5}\right )}\right )\,{\left (\sqrt {21}+5\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (\sqrt {21}+5\right )}^{1/4}\,12005{}\mathrm {i}}{2\,\left (4802\,\sqrt {2}\,\sqrt {\sqrt {21}+5}+1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {\sqrt {21}+5}\right )}+\frac {2^{3/4}\,\sqrt {3}\,\sqrt {21}\,x\,{\left (\sqrt {21}+5\right )}^{1/4}\,7889{}\mathrm {i}}{6\,\left (4802\,\sqrt {2}\,\sqrt {\sqrt {21}+5}+1029\,\sqrt {2}\,\sqrt {21}\,\sqrt {\sqrt {21}+5}\right )}\right )\,{\left (\sqrt {21}+5\right )}^{1/4}\,1{}\mathrm {i}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/(x^8 - 5*x^4 + 1),x)

[Out]

(2^(3/4)*3^(1/2)*atan((12005*2^(3/4)*3^(1/2)*x*(5 - 21^(1/2))^(1/4))/(2*(4802*2^(1/2)*(5 - 21^(1/2))^(1/2) - 1
029*2^(1/2)*21^(1/2)*(5 - 21^(1/2))^(1/2))) - (7889*2^(3/4)*3^(1/2)*21^(1/2)*x*(5 - 21^(1/2))^(1/4))/(6*(4802*
2^(1/2)*(5 - 21^(1/2))^(1/2) - 1029*2^(1/2)*21^(1/2)*(5 - 21^(1/2))^(1/2))))*(5 - 21^(1/2))^(1/4))/12 - (2^(3/
4)*3^(1/2)*atan((2^(3/4)*3^(1/2)*x*(5 - 21^(1/2))^(1/4)*12005i)/(2*(4802*2^(1/2)*(5 - 21^(1/2))^(1/2) - 1029*2
^(1/2)*21^(1/2)*(5 - 21^(1/2))^(1/2))) - (2^(3/4)*3^(1/2)*21^(1/2)*x*(5 - 21^(1/2))^(1/4)*7889i)/(6*(4802*2^(1
/2)*(5 - 21^(1/2))^(1/2) - 1029*2^(1/2)*21^(1/2)*(5 - 21^(1/2))^(1/2))))*(5 - 21^(1/2))^(1/4)*1i)/12 + (2^(3/4
)*3^(1/2)*atan((12005*2^(3/4)*3^(1/2)*x*(21^(1/2) + 5)^(1/4))/(2*(4802*2^(1/2)*(21^(1/2) + 5)^(1/2) + 1029*2^(
1/2)*21^(1/2)*(21^(1/2) + 5)^(1/2))) + (7889*2^(3/4)*3^(1/2)*21^(1/2)*x*(21^(1/2) + 5)^(1/4))/(6*(4802*2^(1/2)
*(21^(1/2) + 5)^(1/2) + 1029*2^(1/2)*21^(1/2)*(21^(1/2) + 5)^(1/2))))*(21^(1/2) + 5)^(1/4))/12 - (2^(3/4)*3^(1
/2)*atan((2^(3/4)*3^(1/2)*x*(21^(1/2) + 5)^(1/4)*12005i)/(2*(4802*2^(1/2)*(21^(1/2) + 5)^(1/2) + 1029*2^(1/2)*
21^(1/2)*(21^(1/2) + 5)^(1/2))) + (2^(3/4)*3^(1/2)*21^(1/2)*x*(21^(1/2) + 5)^(1/4)*7889i)/(6*(4802*2^(1/2)*(21
^(1/2) + 5)^(1/2) + 1029*2^(1/2)*21^(1/2)*(21^(1/2) + 5)^(1/2))))*(21^(1/2) + 5)^(1/4)*1i)/12

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sympy [A]  time = 0.19, size = 24, normalized size = 0.14 \begin {gather*} \operatorname {RootSum} {\left (5308416 t^{8} - 11520 t^{4} + 1, \left (t \mapsto t \log {\left (9216 t^{5} - 16 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**8-5*x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 11520*_t**4 + 1, Lambda(_t, _t*log(9216*_t**5 - 16*_t + x)))

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