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

Optimal. Leaf size=165 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1419, 1093, 207, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {\sqrt {3}-1}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (\sqrt {3}-1\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}} \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[3*(-1 + Sqrt[3])]) + ArcTan[(2^(1/4)*x)/Sqrt[1 + Sqrt[3
]]]/(2*2^(1/4)*Sqrt[3*(1 + Sqrt[3])]) + ArcTanh[(2^(1/4)*x)/Sqrt[-1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(-1 + Sqrt[3
])]) + ArcTanh[(2^(1/4)*x)/Sqrt[1 + Sqrt[3]]]/(2*2^(1/4)*Sqrt[3*(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 &=-\left (\frac {1}{2} \int \frac {1}{-1-\sqrt {2} x^2+x^4} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+\sqrt {2} x^2+x^4} \, dx\\ &=-\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {3}{2}}-\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}-\frac {\int \frac {1}{-\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}+\frac {\int \frac {1}{\sqrt {\frac {3}{2}}+\frac {1}{\sqrt {2}}+x^2} \, dx}{2 \sqrt {6}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (-1+\sqrt {3}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+\sqrt {3}}}\right )}{2 \sqrt [4]{2} \sqrt {3 \left (1+\sqrt {3}\right )}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 55, normalized size = 0.33 \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]

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

________________________________________________________________________________________

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]

________________________________________________________________________________________

fricas [B]  time = 1.58, size = 302, normalized size = 1.83 \begin {gather*} -\frac {1}{6} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{6} \, \sqrt {6} \sqrt {x^{2} + {\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2}} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}} - \frac {1}{6} \, \sqrt {6} {\left (\sqrt {3} x + 3 \, x\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {3}{4}}\right ) + \frac {1}{6} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \arctan \left (\frac {1}{6} \, {\left (\sqrt {6} \sqrt {x^{2} - \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )}} \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 3\right )} - \sqrt {6} {\left (\sqrt {3} x - 3 \, x\right )} \sqrt {\sqrt {3} + 2}\right )} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} {\left (\sqrt {3} - 3\right )} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\sqrt {6} {\left (\sqrt {3} + 2\right )}^{\frac {1}{4}} {\left (\sqrt {3} - 3\right )} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\sqrt {6} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 6 \, x\right ) - \frac {1}{24} \, \sqrt {6} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (-\sqrt {6} {\left (\sqrt {3} + 3\right )} {\left (-\sqrt {3} + 2\right )}^{\frac {1}{4}} + 6 \, 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]

-1/6*sqrt(6)*(-sqrt(3) + 2)^(1/4)*arctan(1/6*sqrt(6)*sqrt(x^2 + (sqrt(3) + 2)*sqrt(-sqrt(3) + 2))*(sqrt(3) + 3
)*(-sqrt(3) + 2)^(3/4) - 1/6*sqrt(6)*(sqrt(3)*x + 3*x)*(-sqrt(3) + 2)^(3/4)) + 1/6*sqrt(6)*(sqrt(3) + 2)^(1/4)
*arctan(1/6*(sqrt(6)*sqrt(x^2 - sqrt(sqrt(3) + 2)*(sqrt(3) - 2))*sqrt(sqrt(3) + 2)*(sqrt(3) - 3) - sqrt(6)*(sq
rt(3)*x - 3*x)*sqrt(sqrt(3) + 2))*(sqrt(3) + 2)^(1/4)) - 1/24*sqrt(6)*(sqrt(3) + 2)^(1/4)*log(sqrt(6)*(sqrt(3)
+ 2)^(1/4)*(sqrt(3) - 3) + 6*x) + 1/24*sqrt(6)*(sqrt(3) + 2)^(1/4)*log(-sqrt(6)*(sqrt(3) + 2)^(1/4)*(sqrt(3)
- 3) + 6*x) + 1/24*sqrt(6)*(-sqrt(3) + 2)^(1/4)*log(sqrt(6)*(sqrt(3) + 3)*(-sqrt(3) + 2)^(1/4) + 6*x) - 1/24*s
qrt(6)*(-sqrt(3) + 2)^(1/4)*log(-sqrt(6)*(sqrt(3) + 3)*(-sqrt(3) + 2)^(1/4) + 6*x)

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [C]  time = 0.01, size = 42, 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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 399, normalized size = 2.42 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {64\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {112\,\sqrt {3}\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}}{3\,\left (80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}\right )}\right )\,{\left (\sqrt {3}+2\right )}^{1/4}}{12}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,64{}\mathrm {i}}{48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}}-\frac {\sqrt {3}\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}\,112{}\mathrm {i}}{3\,\left (48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}\right )}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {64\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}}-\frac {112\,\sqrt {3}\,\sqrt {6}\,x\,{\left (2-\sqrt {3}\right )}^{1/4}}{3\,\left (48\,\sqrt {3}\,\sqrt {2-\sqrt {3}}-80\,\sqrt {2-\sqrt {3}}\right )}\right )\,{\left (2-\sqrt {3}\right )}^{1/4}}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,64{}\mathrm {i}}{80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}}+\frac {\sqrt {3}\,\sqrt {6}\,x\,{\left (\sqrt {3}+2\right )}^{1/4}\,112{}\mathrm {i}}{3\,\left (80\,\sqrt {\sqrt {3}+2}+48\,\sqrt {3}\,\sqrt {\sqrt {3}+2}\right )}\right )\,{\left (\sqrt {3}+2\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 - 4*x^4 + 1),x)

[Out]

(6^(1/2)*atan((6^(1/2)*x*(2 - 3^(1/2))^(1/4)*64i)/(48*3^(1/2)*(2 - 3^(1/2))^(1/2) - 80*(2 - 3^(1/2))^(1/2)) -
(3^(1/2)*6^(1/2)*x*(2 - 3^(1/2))^(1/4)*112i)/(3*(48*3^(1/2)*(2 - 3^(1/2))^(1/2) - 80*(2 - 3^(1/2))^(1/2))))*(2
- 3^(1/2))^(1/4)*1i)/12 - (6^(1/2)*atan((64*6^(1/2)*x*(2 - 3^(1/2))^(1/4))/(48*3^(1/2)*(2 - 3^(1/2))^(1/2) -
80*(2 - 3^(1/2))^(1/2)) - (112*3^(1/2)*6^(1/2)*x*(2 - 3^(1/2))^(1/4))/(3*(48*3^(1/2)*(2 - 3^(1/2))^(1/2) - 80*
(2 - 3^(1/2))^(1/2))))*(2 - 3^(1/2))^(1/4))/12 + (6^(1/2)*atan((64*6^(1/2)*x*(3^(1/2) + 2)^(1/4))/(80*(3^(1/2)
+ 2)^(1/2) + 48*3^(1/2)*(3^(1/2) + 2)^(1/2)) + (112*3^(1/2)*6^(1/2)*x*(3^(1/2) + 2)^(1/4))/(3*(80*(3^(1/2) +
2)^(1/2) + 48*3^(1/2)*(3^(1/2) + 2)^(1/2))))*(3^(1/2) + 2)^(1/4))/12 - (6^(1/2)*atan((6^(1/2)*x*(3^(1/2) + 2)^
(1/4)*64i)/(80*(3^(1/2) + 2)^(1/2) + 48*3^(1/2)*(3^(1/2) + 2)^(1/2)) + (3^(1/2)*6^(1/2)*x*(3^(1/2) + 2)^(1/4)*
112i)/(3*(80*(3^(1/2) + 2)^(1/2) + 48*3^(1/2)*(3^(1/2) + 2)^(1/2))))*(3^(1/2) + 2)^(1/4)*1i)/12

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 26, normalized size = 0.16 \begin {gather*} - \operatorname {RootSum} {\left (84934656 t^{8} - 36864 t^{4} + 1, \left (t \mapsto t \log {\left (36864 t^{5} - 20 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(84934656*_t**8 - 36864*_t**4 + 1, Lambda(_t, _t*log(36864*_t**5 - 20*_t + x)))

________________________________________________________________________________________