3.19 \(\int \frac {1+x^4}{1-6 x^4+x^8} \, dx\)

Optimal. Leaf size=117 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}} \]

[Out]

1/4*arctan(x/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)+1/4*arctanh(x/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)-1/4*arcta
n(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)-1/4*arctanh(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 117, 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} \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 6*x^4 + x^8),x]

[Out]

ArcTan[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTan[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]]) + ArcT
anh[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTanh[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]])

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

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Mathematica [A]  time = 0.05, size = 111, normalized size = 0.95 \[ \frac {1}{4} \left (\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(1 - 6*x^4 + x^8),x]

[Out]

(Sqrt[1 + Sqrt[2]]*ArcTan[x/Sqrt[-1 + Sqrt[2]]] - Sqrt[-1 + Sqrt[2]]*ArcTan[x/Sqrt[1 + Sqrt[2]]] + Sqrt[1 + Sq
rt[2]]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[x/Sqrt[1 + Sqrt[2]]])/4

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fricas [B]  time = 0.76, size = 181, normalized size = 1.55 \[ -\frac {1}{2} \, \sqrt {\sqrt {2} + 1} \arctan \left (-x \sqrt {\sqrt {2} + 1} + \sqrt {x^{2} + \sqrt {2} - 1} \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \arctan \left (-x \sqrt {\sqrt {2} - 1} + \sqrt {x^{2} + \sqrt {2} + 1} \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(sqrt(2) + 1)*arctan(-x*sqrt(sqrt(2) + 1) + sqrt(x^2 + sqrt(2) - 1)*sqrt(sqrt(2) + 1)) + 1/2*sqrt(sqr
t(2) - 1)*arctan(-x*sqrt(sqrt(2) - 1) + sqrt(x^2 + sqrt(2) + 1)*sqrt(sqrt(2) - 1)) - 1/8*sqrt(sqrt(2) - 1)*log
((sqrt(2) + 1)*sqrt(sqrt(2) - 1) + x) + 1/8*sqrt(sqrt(2) - 1)*log(-(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + x) + 1/8*
sqrt(sqrt(2) + 1)*log(sqrt(sqrt(2) + 1)*(sqrt(2) - 1) + x) - 1/8*sqrt(sqrt(2) + 1)*log(-sqrt(sqrt(2) + 1)*(sqr
t(2) - 1) + x)

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giac [A]  time = 0.91, size = 123, normalized size = 1.05 \[ -\frac {1}{4} \, \sqrt {\sqrt {2} - 1} \arctan \left (\frac {x}{\sqrt {\sqrt {2} + 1}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \arctan \left (\frac {x}{\sqrt {\sqrt {2} - 1}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left ({\left | x + \sqrt {\sqrt {2} + 1} \right |}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} - 1} \log \left ({\left | x - \sqrt {\sqrt {2} + 1} \right |}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left ({\left | x + \sqrt {\sqrt {2} - 1} \right |}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left ({\left | x - \sqrt {\sqrt {2} - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(sqrt(2) - 1)*arctan(x/sqrt(sqrt(2) + 1)) + 1/4*sqrt(sqrt(2) + 1)*arctan(x/sqrt(sqrt(2) - 1)) - 1/8*s
qrt(sqrt(2) - 1)*log(abs(x + sqrt(sqrt(2) + 1))) + 1/8*sqrt(sqrt(2) - 1)*log(abs(x - sqrt(sqrt(2) + 1))) + 1/8
*sqrt(sqrt(2) + 1)*log(abs(x + sqrt(sqrt(2) - 1))) - 1/8*sqrt(sqrt(2) + 1)*log(abs(x - sqrt(sqrt(2) - 1)))

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maple [A]  time = 0.06, size = 78, normalized size = 0.67 \[ -\frac {\arctanh \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\arctanh \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}}-\frac {\arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {\sqrt {2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^8-6*x^4+1),x)

[Out]

1/4*arctan(x/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)+1/4*arctanh(x/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)-1/4*arcta
n(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)-1/4*arctanh(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} + 1}{x^{8} - 6 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.19, size = 233, normalized size = 1.99 \[ -\frac {\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-1}\,49152{}\mathrm {i}}{34816\,\sqrt {2}-49152}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}-1}\,34816{}\mathrm {i}}{34816\,\sqrt {2}-49152}\right )\,\sqrt {\sqrt {2}-1}\,1{}\mathrm {i}}{4}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}+1}\,49152{}\mathrm {i}}{34816\,\sqrt {2}+49152}+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+1}\,34816{}\mathrm {i}}{34816\,\sqrt {2}+49152}\right )\,\sqrt {\sqrt {2}+1}\,1{}\mathrm {i}}{4}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {1-\sqrt {2}}\,49152{}\mathrm {i}}{34816\,\sqrt {2}-49152}-\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {2}}\,34816{}\mathrm {i}}{34816\,\sqrt {2}-49152}\right )\,\sqrt {1-\sqrt {2}}\,1{}\mathrm {i}}{4}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-1}\,49152{}\mathrm {i}}{34816\,\sqrt {2}+49152}+\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {2}-1}\,34816{}\mathrm {i}}{34816\,\sqrt {2}+49152}\right )\,\sqrt {-\sqrt {2}-1}\,1{}\mathrm {i}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/(x^8 - 6*x^4 + 1),x)

[Out]

(atan((x*(1 - 2^(1/2))^(1/2)*49152i)/(34816*2^(1/2) - 49152) - (2^(1/2)*x*(1 - 2^(1/2))^(1/2)*34816i)/(34816*2
^(1/2) - 49152))*(1 - 2^(1/2))^(1/2)*1i)/4 - (atan((x*(2^(1/2) + 1)^(1/2)*49152i)/(34816*2^(1/2) + 49152) + (2
^(1/2)*x*(2^(1/2) + 1)^(1/2)*34816i)/(34816*2^(1/2) + 49152))*(2^(1/2) + 1)^(1/2)*1i)/4 - (atan((x*(2^(1/2) -
1)^(1/2)*49152i)/(34816*2^(1/2) - 49152) - (2^(1/2)*x*(2^(1/2) - 1)^(1/2)*34816i)/(34816*2^(1/2) - 49152))*(2^
(1/2) - 1)^(1/2)*1i)/4 + (atan((x*(- 2^(1/2) - 1)^(1/2)*49152i)/(34816*2^(1/2) + 49152) + (2^(1/2)*x*(- 2^(1/2
) - 1)^(1/2)*34816i)/(34816*2^(1/2) + 49152))*(- 2^(1/2) - 1)^(1/2)*1i)/4

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sympy [A]  time = 1.16, size = 49, normalized size = 0.42 \[ \operatorname {RootSum} {\left (4096 t^{4} - 128 t^{2} - 1, \left (t \mapsto t \log {\left (16384 t^{5} - 20 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (4096 t^{4} + 128 t^{2} - 1, \left (t \mapsto t \log {\left (16384 t^{5} - 20 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(4096*_t**4 - 128*_t**2 - 1, Lambda(_t, _t*log(16384*_t**5 - 20*_t + x))) + RootSum(4096*_t**4 + 128*_t
**2 - 1, Lambda(_t, _t*log(16384*_t**5 - 20*_t + x)))

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