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

Optimal. Leaf size=169 \[ -\frac {\tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]

[Out]

-1/10*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(3/4)-1/10*arctanh(2^(1/4)*x*(1/(3+5
^(1/2)))^(1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(3/4)+1/10*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(9+4*5^(1/2))^(
1/4)*5^(1/2)+1/10*arctanh(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(9+4*5^(1/2))^(1/4)*5^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1347, 212, 206, 203} \[ -\frac {\tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x^4 + x^8)^(-1),x]

[Out]

-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4))) + ((3 + Sqrt[5])^(3/4)*ArcTan[((3 +
Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])
^(3/4)) + ((3 + Sqrt[5])^(3/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5])

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 206

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 1347

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 160, normalized size = 0.95 \[ \frac {\frac {\left (1+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {\sqrt {5}-1}}-\frac {\left (\sqrt {5}-1\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (1+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {\sqrt {5}-1}}-\frac {\left (\sqrt {5}-1\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x^4 + x^8)^(-1),x]

[Out]

(((1 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt
[5])]*x])/Sqrt[1 + Sqrt[5]] + ((1 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sq
rt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])

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fricas [B]  time = 0.90, size = 251, normalized size = 1.49 \[ -\frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 2} - \frac {1}{2} \, {\left (\sqrt {5} x - x\right )} \sqrt {\sqrt {5} + 2}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 2} - \frac {1}{2} \, {\left (\sqrt {5} x + x\right )} \sqrt {\sqrt {5} - 2}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(5)*sqrt(sqrt(5) + 2)*arctan(1/4*sqrt(2*x^2 + sqrt(5) - 1)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) +
2) - 1/2*(sqrt(5)*x - x)*sqrt(sqrt(5) + 2)) + 1/5*sqrt(5)*sqrt(sqrt(5) - 2)*arctan(1/4*sqrt(2*x^2 + sqrt(5) +
1)*(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 2) - 1/2*(sqrt(5)*x + x)*sqrt(sqrt(5) - 2)) - 1/20*sqrt(5)*sqrt
(sqrt(5) - 2)*log((sqrt(5) + 3)*sqrt(sqrt(5) - 2) + 2*x) + 1/20*sqrt(5)*sqrt(sqrt(5) - 2)*log(-(sqrt(5) + 3)*s
qrt(sqrt(5) - 2) + 2*x) - 1/20*sqrt(5)*sqrt(sqrt(5) + 2)*log(sqrt(sqrt(5) + 2)*(sqrt(5) - 3) + 2*x) + 1/20*sqr
t(5)*sqrt(sqrt(5) + 2)*log(-sqrt(sqrt(5) + 2)*(sqrt(5) - 3) + 2*x)

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giac [A]  time = 0.48, size = 147, normalized size = 0.87 \[ -\frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(5*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/10*sqrt(5*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqr
t(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) - 10)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*log
(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/20*sq
rt(5*sqrt(5) + 10)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))

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maple [A]  time = 0.03, size = 206, normalized size = 1.22 \[ \frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(2+2*5^(1/2))^(1/2)*arctanh(2/(2+2*5^(1/2))^(1/2)*x)+1/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2/(2+2*5^(1
/2))^(1/2)*x)+1/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2/(-2+2*5^(1/2))^(1/2)*x)+1/2/(-2+2*5^(1/2))^(1/2)*arct
an(2/(-2+2*5^(1/2))^(1/2)*x)+1/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2/(-2+2*5^(1/2))^(1/2)*x)+1/2/(-2+2*5^(
1/2))^(1/2)*arctanh(2/(-2+2*5^(1/2))^(1/2)*x)-1/2/(2+2*5^(1/2))^(1/2)*arctan(2/(2+2*5^(1/2))^(1/2)*x)+1/10*5^(
1/2)/(2+2*5^(1/2))^(1/2)*arctan(2/(2+2*5^(1/2))^(1/2)*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^8 - 3*x^4 + 1), x)

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mupad [B]  time = 0.08, size = 245, normalized size = 1.45 \[ -\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {2-\sqrt {5}}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {-\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}+2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}+2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8 - 3*x^4 + 1),x)

[Out]

(5^(1/2)*atan((x*(- 5^(1/2) - 2)^(1/2)*144i)/(104*5^(1/2) + 232) + (5^(1/2)*x*(- 5^(1/2) - 2)^(1/2)*64i)/(104*
5^(1/2) + 232))*(- 5^(1/2) - 2)^(1/2)*1i)/10 - (5^(1/2)*atan((x*(2 - 5^(1/2))^(1/2)*144i)/(104*5^(1/2) - 232)
- (5^(1/2)*x*(2 - 5^(1/2))^(1/2)*64i)/(104*5^(1/2) - 232))*(2 - 5^(1/2))^(1/2)*1i)/10 + (5^(1/2)*atan((x*(5^(1
/2) - 2)^(1/2)*144i)/(104*5^(1/2) - 232) - (5^(1/2)*x*(5^(1/2) - 2)^(1/2)*64i)/(104*5^(1/2) - 232))*(5^(1/2) -
2)^(1/2)*1i)/10 - (5^(1/2)*atan((x*(5^(1/2) + 2)^(1/2)*144i)/(104*5^(1/2) + 232) + (5^(1/2)*x*(5^(1/2) + 2)^(
1/2)*64i)/(104*5^(1/2) + 232))*(5^(1/2) + 2)^(1/2)*1i)/10

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sympy [A]  time = 1.21, size = 53, normalized size = 0.31 \[ \operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left (t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left (t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 + x))) + RootSum(6400*_t**4 + 320*_
t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 + x)))

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