3.4.97 \(\int \frac {x^6}{1-3 x^4+x^8} \, dx\) [397]

Optimal. Leaf size=167 \[ \frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{144-64 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{144-64 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]

[Out]

-1/20*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(144-64*5^(1/2))^(1/4)*5^(1/2)+1/20*arctanh(1/2*x*(3+5^(1/2))^(1
/4)*2^(3/4))*(144-64*5^(1/2))^(1/4)*5^(1/2)+1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(3+5^(1/2))^(3/4)*2^(
1/4)*5^(1/2)-1/20*arctanh(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(3+5^(1/2))^(3/4)*2^(1/4)*5^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1388, 304, 209, 212} \begin {gather*} \frac {\left (3+\sqrt {5}\right )^{3/4} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{144-64 \sqrt {5}} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{144-64 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(1 - 3*x^4 + x^8),x]

[Out]

((3 + Sqrt[5])^(3/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ((144 - 64*Sqrt[5])^(1/4)*ArcTan
[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - ((3 + Sqrt[5])^(3/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*2^(3/4)
*Sqrt[5]) + ((144 - 64*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

Rule 209

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

Rule 212

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

Rule 304

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

Rule 1388

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

Rubi steps

\begin {align*} \int \frac {x^6}{1-3 x^4+x^8} \, dx &=\frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=\frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}\\ &=\frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\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 {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\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 = 0.10, size = 160, normalized size = 0.96 \begin {gather*} \frac {\frac {\left (-3+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}+\frac {\left (3+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}-\frac {\left (-3+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(1 - 3*x^4 + x^8),x]

[Out]

(((-3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] + ((3 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt
[5])]*x])/Sqrt[1 + Sqrt[5]] - ((-3 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((3 + Sq
rt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])

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Maple [A]
time = 0.07, size = 130, normalized size = 0.78

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}-7 \textit {\_R} +2 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}+7 \textit {\_R} +2 x \right )\right )}{4}\)	\(68\)
default	\(-\frac {\sqrt {5}\, \left (\sqrt {5}-3\right ) \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (\sqrt {5}-3\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

integrate(x^6/(x^8 - 3*x^4 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (113) = 226\).
time = 0.39, size = 243, normalized size = 1.46 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \arctan \left (\frac {1}{4} \, \sqrt {4 \, x^{2} + 2 \, \sqrt {5} - 2} {\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} - \frac {1}{2} \, {\left (\sqrt {5} x + 3 \, x\right )} \sqrt {\sqrt {5} - 2}\right ) + \frac {1}{5} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \arctan \left (\frac {1}{4} \, \sqrt {4 \, x^{2} + 2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} - \frac {1}{2} \, {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {\sqrt {5} + 2}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.73, size = 53, normalized size = 0.32 \begin {gather*} \operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(-1792000*_t**7 + 4920*_t**3 + x))) + RootSum(6400*_t**4
+ 320*_t**2 - 1, Lambda(_t, _t*log(-1792000*_t**7 + 4920*_t**3 + x)))

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Giac [A]
time = 4.10, size = 147, normalized size = 0.88 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(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*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*sqrt(5*sqrt(5) - 10)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/20*sqr
t(5*sqrt(5) - 10)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))

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Mupad [B]
time = 0.19, size = 147, normalized size = 0.88 \begin {gather*} \frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {2-\sqrt {5}}}{8\,\sqrt {5}-24}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {-\sqrt {5}-2}}{8\,\sqrt {5}+24}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,16{}\mathrm {i}}{8\,\sqrt {5}-24}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,16{}\mathrm {i}}{8\,\sqrt {5}+24}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^8 - 3*x^4 + 1),x)

[Out]

(5^(1/2)*atan((16*x*(2 - 5^(1/2))^(1/2))/(8*5^(1/2) - 24))*(5^(1/2) - 2)^(1/2)*1i)/10 + (5^(1/2)*atan((16*x*(-
5^(1/2) - 2)^(1/2))/(8*5^(1/2) + 24))*(5^(1/2) + 2)^(1/2)*1i)/10 + (5^(1/2)*atan((x*(2 - 5^(1/2))^(1/2)*16i)/
(8*5^(1/2) - 24))*(2 - 5^(1/2))^(1/2)*1i)/10 + (5^(1/2)*atan((x*(- 5^(1/2) - 2)^(1/2)*16i)/(8*5^(1/2) + 24))*(
- 5^(1/2) - 2)^(1/2)*1i)/10

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