3.77 \(\int \frac{1}{(-1+7 x^2-7 x^4+x^6)^2} \, dx\)

Optimal. Leaf size=91 \[ \frac{x}{32 \left (1-x^2\right )}+\frac{\left (99-17 x^2\right ) x}{128 \left (x^4-6 x^2+1\right )}+\frac{5}{32} \tanh ^{-1}(x)+\frac{1}{512} \left (3 \sqrt{2}-4\right ) \tanh ^{-1}\left (\left (\sqrt{2}-1\right ) x\right )+\frac{1}{512} \left (4+3 \sqrt{2}\right ) \tanh ^{-1}\left (\left (1+\sqrt{2}\right ) x\right ) \]

[Out]

x/(32*(1 - x^2)) + (x*(99 - 17*x^2))/(128*(1 - 6*x^2 + x^4)) + (5*ArcTanh[x])/32 + ((-4 + 3*Sqrt[2])*ArcTanh[(
-1 + Sqrt[2])*x])/512 + ((4 + 3*Sqrt[2])*ArcTanh[(1 + Sqrt[2])*x])/512

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Rubi [B]  time = 0.130008, antiderivative size = 205, normalized size of antiderivative = 2.25, number of steps used = 15, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {2057, 207, 638, 618, 206, 632, 31} \[ -\frac{41-17 x}{256 \left (-x^2+2 x+1\right )}+\frac{17 x+41}{256 \left (-x^2-2 x+1\right )}+\frac{1}{64 (1-x)}-\frac{1}{64 (x+1)}+\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (-x-\sqrt{2}+1\right )+\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )-\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )-\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right )-\frac{17 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{256 \sqrt{2}}+\frac{5}{32} \tanh ^{-1}(x)+\frac{17 \tanh ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{256 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 7*x^2 - 7*x^4 + x^6)^(-2),x]

[Out]

1/(64*(1 - x)) - 1/(64*(1 + x)) + (41 + 17*x)/(256*(1 - 2*x - x^2)) - (41 - 17*x)/(256*(1 + 2*x - x^2)) - (17*
ArcTanh[(1 - x)/Sqrt[2]])/(256*Sqrt[2]) + (5*ArcTanh[x])/32 + (17*ArcTanh[(1 + x)/Sqrt[2]])/(256*Sqrt[2]) + ((
2 - 7*Sqrt[2])*Log[1 - Sqrt[2] - x])/512 + ((2 + 7*Sqrt[2])*Log[1 + Sqrt[2] - x])/512 - ((2 - 7*Sqrt[2])*Log[1
- Sqrt[2] + x])/512 - ((2 + 7*Sqrt[2])*Log[1 + Sqrt[2] + x])/512

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 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 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx &=\int \left (\frac{1}{64 (-1+x)^2}+\frac{1}{64 (1+x)^2}-\frac{5}{32 \left (-1+x^2\right )}+\frac{29-12 x}{64 \left (-1-2 x+x^2\right )^2}+\frac{6+x}{128 \left (-1-2 x+x^2\right )}+\frac{29+12 x}{64 \left (-1+2 x+x^2\right )^2}+\frac{6-x}{128 \left (-1+2 x+x^2\right )}\right ) \, dx\\ &=\frac{1}{64 (1-x)}-\frac{1}{64 (1+x)}+\frac{1}{128} \int \frac{6+x}{-1-2 x+x^2} \, dx+\frac{1}{128} \int \frac{6-x}{-1+2 x+x^2} \, dx+\frac{1}{64} \int \frac{29-12 x}{\left (-1-2 x+x^2\right )^2} \, dx+\frac{1}{64} \int \frac{29+12 x}{\left (-1+2 x+x^2\right )^2} \, dx-\frac{5}{32} \int \frac{1}{-1+x^2} \, dx\\ &=\frac{1}{64 (1-x)}-\frac{1}{64 (1+x)}+\frac{41+17 x}{256 \left (1-2 x-x^2\right )}-\frac{41-17 x}{256 \left (1+2 x-x^2\right )}+\frac{5}{32} \tanh ^{-1}(x)-\frac{17}{256} \int \frac{1}{-1-2 x+x^2} \, dx-\frac{17}{256} \int \frac{1}{-1+2 x+x^2} \, dx+\frac{1}{512} \left (2-7 \sqrt{2}\right ) \int \frac{1}{-1+\sqrt{2}+x} \, dx+\frac{1}{512} \left (-2+7 \sqrt{2}\right ) \int \frac{1}{1-\sqrt{2}+x} \, dx+\frac{1}{512} \left (2+7 \sqrt{2}\right ) \int \frac{1}{-1-\sqrt{2}+x} \, dx-\frac{1}{512} \left (2+7 \sqrt{2}\right ) \int \frac{1}{1+\sqrt{2}+x} \, dx\\ &=\frac{1}{64 (1-x)}-\frac{1}{64 (1+x)}+\frac{41+17 x}{256 \left (1-2 x-x^2\right )}-\frac{41-17 x}{256 \left (1+2 x-x^2\right )}+\frac{5}{32} \tanh ^{-1}(x)+\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (1-\sqrt{2}-x\right )+\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (1+\sqrt{2}-x\right )-\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (1-\sqrt{2}+x\right )-\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (1+\sqrt{2}+x\right )+\frac{17}{128} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 x\right )+\frac{17}{128} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 x\right )\\ &=\frac{1}{64 (1-x)}-\frac{1}{64 (1+x)}+\frac{41+17 x}{256 \left (1-2 x-x^2\right )}-\frac{41-17 x}{256 \left (1+2 x-x^2\right )}-\frac{17 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{256 \sqrt{2}}+\frac{5}{32} \tanh ^{-1}(x)+\frac{17 \tanh ^{-1}\left (\frac{1+x}{\sqrt{2}}\right )}{256 \sqrt{2}}+\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (1-\sqrt{2}-x\right )+\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (1+\sqrt{2}-x\right )-\frac{1}{512} \left (2-7 \sqrt{2}\right ) \log \left (1-\sqrt{2}+x\right )-\frac{1}{512} \left (2+7 \sqrt{2}\right ) \log \left (1+\sqrt{2}+x\right )\\ \end{align*}

Mathematica [A]  time = 0.0883109, size = 132, normalized size = 1.45 \[ \frac{-\frac{8 x \left (21 x^4-140 x^2+103\right )}{x^6-7 x^4+7 x^2-1}-80 \log (1-x)-\left (4+3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )+\left (4-3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )+80 \log (x+1)+\left (4+3 \sqrt{2}\right ) \log \left (x+\sqrt{2}-1\right )+\left (3 \sqrt{2}-4\right ) \log \left (x+\sqrt{2}+1\right )}{1024} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 7*x^2 - 7*x^4 + x^6)^(-2),x]

[Out]

((-8*x*(103 - 140*x^2 + 21*x^4))/(-1 + 7*x^2 - 7*x^4 + x^6) - 80*Log[1 - x] - (4 + 3*Sqrt[2])*Log[-1 + Sqrt[2]
- x] + (4 - 3*Sqrt[2])*Log[1 + Sqrt[2] - x] + 80*Log[1 + x] + (4 + 3*Sqrt[2])*Log[-1 + Sqrt[2] + x] + (-4 + 3
*Sqrt[2])*Log[1 + Sqrt[2] + x])/1024

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Maple [A]  time = 0.019, size = 116, normalized size = 1.3 \begin{align*}{\frac{1}{128\,{x}^{2}-256\,x-128} \left ( -{\frac{17\,x}{2}}+{\frac{41}{2}} \right ) }+{\frac{\ln \left ({x}^{2}-2\,x-1 \right ) }{256}}+{\frac{3\,\sqrt{2}}{512}{\it Artanh} \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{64\,x-64}}-{\frac{5\,\ln \left ( x-1 \right ) }{64}}-{\frac{1}{128\,{x}^{2}+256\,x-128} \left ({\frac{17\,x}{2}}+{\frac{41}{2}} \right ) }-{\frac{\ln \left ({x}^{2}+2\,x-1 \right ) }{256}}+{\frac{3\,\sqrt{2}}{512}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{64+64\,x}}+{\frac{5\,\ln \left ( 1+x \right ) }{64}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6-7*x^4+7*x^2-1)^2,x)

[Out]

1/128*(-17/2*x+41/2)/(x^2-2*x-1)+1/256*ln(x^2-2*x-1)+3/512*2^(1/2)*arctanh(1/4*(2*x-2)*2^(1/2))-1/64/(x-1)-5/6
4*ln(x-1)-1/128*(17/2*x+41/2)/(x^2+2*x-1)-1/256*ln(x^2+2*x-1)+3/512*2^(1/2)*arctanh(1/4*(2+2*x)*2^(1/2))-1/64/
(1+x)+5/64*ln(1+x)

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Maxima [A]  time = 1.95656, size = 154, normalized size = 1.69 \begin{align*} -\frac{3}{1024} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} + 1}{x + \sqrt{2} + 1}\right ) - \frac{3}{1024} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} - 1}{x + \sqrt{2} - 1}\right ) - \frac{21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac{1}{256} \, \log \left (x^{2} + 2 \, x - 1\right ) + \frac{1}{256} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac{5}{64} \, \log \left (x + 1\right ) - \frac{5}{64} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-7*x^4+7*x^2-1)^2,x, algorithm="maxima")

[Out]

-3/1024*sqrt(2)*log((x - sqrt(2) + 1)/(x + sqrt(2) + 1)) - 3/1024*sqrt(2)*log((x - sqrt(2) - 1)/(x + sqrt(2) -
1)) - 1/128*(21*x^5 - 140*x^3 + 103*x)/(x^6 - 7*x^4 + 7*x^2 - 1) - 1/256*log(x^2 + 2*x - 1) + 1/256*log(x^2 -
2*x - 1) + 5/64*log(x + 1) - 5/64*log(x - 1)

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Fricas [B]  time = 1.78809, size = 589, normalized size = 6.47 \begin{align*} -\frac{168 \, x^{5} - 1120 \, x^{3} - 3 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{x^{2} + 2 \, \sqrt{2}{\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) - 3 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{x^{2} + 2 \, \sqrt{2}{\left (x - 1\right )} - 2 \, x + 3}{x^{2} - 2 \, x - 1}\right ) + 4 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) - 4 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 80 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 80 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 824 \, x}{1024 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-7*x^4+7*x^2-1)^2,x, algorithm="fricas")

[Out]

-1/1024*(168*x^5 - 1120*x^3 - 3*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log((x^2 + 2*sqrt(2)*(x + 1) + 2*x + 3)/(x^2
+ 2*x - 1)) - 3*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log((x^2 + 2*sqrt(2)*(x - 1) - 2*x + 3)/(x^2 - 2*x - 1)) +
4*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 + 2*x - 1) - 4*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 - 2*x - 1) - 80*(x^6 - 7*
x^4 + 7*x^2 - 1)*log(x + 1) + 80*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x - 1) + 824*x)/(x^6 - 7*x^4 + 7*x^2 - 1)

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Sympy [B]  time = 1.11242, size = 296, normalized size = 3.25 \begin{align*} - \frac{21 x^{5} - 140 x^{3} + 103 x}{128 x^{6} - 896 x^{4} + 896 x^{2} - 128} - \frac{5 \log{\left (x - 1 \right )}}{64} + \frac{5 \log{\left (x + 1 \right )}}{64} + \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right ) \log{\left (x - \frac{8071264001}{202624020} - \frac{471550901878784 \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{3}}{2979765} + \frac{1299552375287054336 \left (- \frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{5}}{50656005} + \frac{8071264001 \sqrt{2}}{270165360} \right )} + \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right ) \log{\left (x - \frac{8071264001 \sqrt{2}}{270165360} - \frac{8071264001}{202624020} + \frac{1299552375287054336 \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right )^{5}}{50656005} - \frac{471550901878784 \left (- \frac{3 \sqrt{2}}{1024} - \frac{1}{256}\right )^{3}}{2979765} \right )} + \left (\frac{1}{256} - \frac{3 \sqrt{2}}{1024}\right ) \log{\left (x - \frac{8071264001 \sqrt{2}}{270165360} + \frac{1299552375287054336 \left (\frac{1}{256} - \frac{3 \sqrt{2}}{1024}\right )^{5}}{50656005} - \frac{471550901878784 \left (\frac{1}{256} - \frac{3 \sqrt{2}}{1024}\right )^{3}}{2979765} + \frac{8071264001}{202624020} \right )} + \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right ) \log{\left (x - \frac{471550901878784 \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{3}}{2979765} + \frac{1299552375287054336 \left (\frac{1}{256} + \frac{3 \sqrt{2}}{1024}\right )^{5}}{50656005} + \frac{8071264001}{202624020} + \frac{8071264001 \sqrt{2}}{270165360} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**6-7*x**4+7*x**2-1)**2,x)

[Out]

-(21*x**5 - 140*x**3 + 103*x)/(128*x**6 - 896*x**4 + 896*x**2 - 128) - 5*log(x - 1)/64 + 5*log(x + 1)/64 + (-1
/256 + 3*sqrt(2)/1024)*log(x - 8071264001/202624020 - 471550901878784*(-1/256 + 3*sqrt(2)/1024)**3/2979765 + 1
299552375287054336*(-1/256 + 3*sqrt(2)/1024)**5/50656005 + 8071264001*sqrt(2)/270165360) + (-3*sqrt(2)/1024 -
1/256)*log(x - 8071264001*sqrt(2)/270165360 - 8071264001/202624020 + 1299552375287054336*(-3*sqrt(2)/1024 - 1/
256)**5/50656005 - 471550901878784*(-3*sqrt(2)/1024 - 1/256)**3/2979765) + (1/256 - 3*sqrt(2)/1024)*log(x - 80
71264001*sqrt(2)/270165360 + 1299552375287054336*(1/256 - 3*sqrt(2)/1024)**5/50656005 - 471550901878784*(1/256
- 3*sqrt(2)/1024)**3/2979765 + 8071264001/202624020) + (1/256 + 3*sqrt(2)/1024)*log(x - 471550901878784*(1/25
6 + 3*sqrt(2)/1024)**3/2979765 + 1299552375287054336*(1/256 + 3*sqrt(2)/1024)**5/50656005 + 8071264001/2026240
20 + 8071264001*sqrt(2)/270165360)

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Giac [A]  time = 1.11241, size = 181, normalized size = 1.99 \begin{align*} -\frac{3}{1024} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) - \frac{3}{1024} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} - 2 \right |}}\right ) - \frac{21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac{1}{256} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) + \frac{1}{256} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac{5}{64} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{5}{64} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6-7*x^4+7*x^2-1)^2,x, algorithm="giac")

[Out]

-3/1024*sqrt(2)*log(abs(2*x - 2*sqrt(2) + 2)/abs(2*x + 2*sqrt(2) + 2)) - 3/1024*sqrt(2)*log(abs(2*x - 2*sqrt(2
) - 2)/abs(2*x + 2*sqrt(2) - 2)) - 1/128*(21*x^5 - 140*x^3 + 103*x)/(x^6 - 7*x^4 + 7*x^2 - 1) - 1/256*log(abs(
x^2 + 2*x - 1)) + 1/256*log(abs(x^2 - 2*x - 1)) + 5/64*log(abs(x + 1)) - 5/64*log(abs(x - 1))