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

Optimal. Leaf size=91 \[ \frac{x}{16 \left (1-x^2\right )}+\frac{\left (29-5 x^2\right ) x}{32 \left (x^4-6 x^2+1\right )}+\frac{1}{4} \tanh ^{-1}(x)+\frac{1}{64} \left (\left (3-2 \sqrt{2}\right ) \tanh ^{-1}\left (\left (\sqrt{2}-1\right ) x\right )-\left (3+2 \sqrt{2}\right ) \tanh ^{-1}\left (\left (1+\sqrt{2}\right ) x\right )\right ) \]

[Out]

x/(16*(1 - x^2)) + (x*(29 - 5*x^2))/(32*(1 - 6*x^2 + x^4)) + ArcTanh[x]/4 + ((3 - 2*Sqrt[2])*ArcTanh[(-1 + Sqr
t[2])*x] - (3 + 2*Sqrt[2])*ArcTanh[(1 + Sqrt[2])*x])/64

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Rubi [B]  time = 0.148645, antiderivative size = 205, normalized size of antiderivative = 2.25, number of steps used = 15, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2073, 207, 638, 618, 206, 632, 31} \[ -\frac{12-5 x}{64 \left (-x^2+2 x+1\right )}+\frac{5 x+12}{64 \left (-x^2-2 x+1\right )}+\frac{1}{32 (1-x)}-\frac{1}{32 (x+1)}-\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (-x-\sqrt{2}+1\right )-\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )+\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right )-\frac{5 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{64 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x)+\frac{5 \tanh ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(32*(1 - x)) - 1/(32*(1 + x)) + (12 + 5*x)/(64*(1 - 2*x - x^2)) - (12 - 5*x)/(64*(1 + 2*x - x^2)) - (5*ArcTa
nh[(1 - x)/Sqrt[2]])/(64*Sqrt[2]) + ArcTanh[x]/4 + (5*ArcTanh[(1 + x)/Sqrt[2]])/(64*Sqrt[2]) - (3*(2 + 3*Sqrt[
2])*Log[1 - Sqrt[2] - x])/256 - (3*(2 - 3*Sqrt[2])*Log[1 + Sqrt[2] - x])/256 + (3*(2 + 3*Sqrt[2])*Log[1 - Sqrt
[2] + x])/256 + (3*(2 - 3*Sqrt[2])*Log[1 + Sqrt[2] + x])/256

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && 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+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx &=\int \left (\frac{1}{32 (-1+x)^2}+\frac{1}{32 (1+x)^2}-\frac{1}{4 \left (-1+x^2\right )}+\frac{17-7 x}{32 \left (-1-2 x+x^2\right )^2}-\frac{3 (-4+x)}{64 \left (-1-2 x+x^2\right )}+\frac{17+7 x}{32 \left (-1+2 x+x^2\right )^2}+\frac{3 (4+x)}{64 \left (-1+2 x+x^2\right )}\right ) \, dx\\ &=\frac{1}{32 (1-x)}-\frac{1}{32 (1+x)}+\frac{1}{32} \int \frac{17-7 x}{\left (-1-2 x+x^2\right )^2} \, dx+\frac{1}{32} \int \frac{17+7 x}{\left (-1+2 x+x^2\right )^2} \, dx-\frac{3}{64} \int \frac{-4+x}{-1-2 x+x^2} \, dx+\frac{3}{64} \int \frac{4+x}{-1+2 x+x^2} \, dx-\frac{1}{4} \int \frac{1}{-1+x^2} \, dx\\ &=\frac{1}{32 (1-x)}-\frac{1}{32 (1+x)}+\frac{12+5 x}{64 \left (1-2 x-x^2\right )}-\frac{12-5 x}{64 \left (1+2 x-x^2\right )}+\frac{1}{4} \tanh ^{-1}(x)-\frac{5}{64} \int \frac{1}{-1-2 x+x^2} \, dx-\frac{5}{64} \int \frac{1}{-1+2 x+x^2} \, dx-\frac{1}{256} \left (3 \left (2-3 \sqrt{2}\right )\right ) \int \frac{1}{-1-\sqrt{2}+x} \, dx+\frac{1}{256} \left (3 \left (2-3 \sqrt{2}\right )\right ) \int \frac{1}{1+\sqrt{2}+x} \, dx+\frac{1}{256} \left (3 \left (2+3 \sqrt{2}\right )\right ) \int \frac{1}{1-\sqrt{2}+x} \, dx-\frac{1}{256} \left (3 \left (2+3 \sqrt{2}\right )\right ) \int \frac{1}{-1+\sqrt{2}+x} \, dx\\ &=\frac{1}{32 (1-x)}-\frac{1}{32 (1+x)}+\frac{12+5 x}{64 \left (1-2 x-x^2\right )}-\frac{12-5 x}{64 \left (1+2 x-x^2\right )}+\frac{1}{4} \tanh ^{-1}(x)-\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (1-\sqrt{2}-x\right )-\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (1+\sqrt{2}-x\right )+\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (1-\sqrt{2}+x\right )+\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (1+\sqrt{2}+x\right )+\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,-2+2 x\right )+\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,2+2 x\right )\\ &=\frac{1}{32 (1-x)}-\frac{1}{32 (1+x)}+\frac{12+5 x}{64 \left (1-2 x-x^2\right )}-\frac{12-5 x}{64 \left (1+2 x-x^2\right )}-\frac{5 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{64 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x)+\frac{5 \tanh ^{-1}\left (\frac{1+x}{\sqrt{2}}\right )}{64 \sqrt{2}}-\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (1-\sqrt{2}-x\right )-\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (1+\sqrt{2}-x\right )+\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (1-\sqrt{2}+x\right )+\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (1+\sqrt{2}+x\right )\\ \end{align*}

Mathematica [A]  time = 0.0889908, size = 132, normalized size = 1.45 \[ \frac{1}{128} \left (-\frac{4 x \left (7 x^4-46 x^2+31\right )}{x^6-7 x^4+7 x^2-1}-16 \log (1-x)+\left (3+2 \sqrt{2}\right ) \log \left (-x+\sqrt{2}-1\right )+\left (2 \sqrt{2}-3\right ) \log \left (-x+\sqrt{2}+1\right )+16 \log (x+1)-\left (3+2 \sqrt{2}\right ) \log \left (x+\sqrt{2}-1\right )+\left (3-2 \sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*x*(31 - 46*x^2 + 7*x^4))/(-1 + 7*x^2 - 7*x^4 + x^6) - 16*Log[1 - x] + (3 + 2*Sqrt[2])*Log[-1 + Sqrt[2] -
x] + (-3 + 2*Sqrt[2])*Log[1 + Sqrt[2] - x] + 16*Log[1 + x] - (3 + 2*Sqrt[2])*Log[-1 + Sqrt[2] + x] + (3 - 2*Sq
rt[2])*Log[1 + Sqrt[2] + x])/128

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Maple [A]  time = 0.016, size = 116, normalized size = 1.3 \begin{align*} -{\frac{-12+5\,x}{64\,{x}^{2}-128\,x-64}}-{\frac{3\,\ln \left ({x}^{2}-2\,x-1 \right ) }{128}}-{\frac{\sqrt{2}}{32}{\it Artanh} \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{32\,x-32}}-{\frac{\ln \left ( x-1 \right ) }{8}}+{\frac{-5\,x-12}{64\,{x}^{2}+128\,x-64}}+{\frac{3\,\ln \left ({x}^{2}+2\,x-1 \right ) }{128}}-{\frac{\sqrt{2}}{32}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{32+32\,x}}+{\frac{\ln \left ( 1+x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(-12+5*x)/(x^2-2*x-1)-3/128*ln(x^2-2*x-1)-1/32*2^(1/2)*arctanh(1/4*(2*x-2)*2^(1/2))-1/32/(x-1)-1/8*ln(x-
1)+1/64*(-5*x-12)/(x^2+2*x-1)+3/128*ln(x^2+2*x-1)-1/32*2^(1/2)*arctanh(1/4*(2+2*x)*2^(1/2))-1/32/(1+x)+1/8*ln(
1+x)

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Maxima [A]  time = 1.5148, size = 154, normalized size = 1.69 \begin{align*} \frac{1}{64} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} + 1}{x + \sqrt{2} + 1}\right ) + \frac{1}{64} \, \sqrt{2} \log \left (\frac{x - \sqrt{2} - 1}{x + \sqrt{2} - 1}\right ) - \frac{7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac{3}{128} \, \log \left (x^{2} + 2 \, x - 1\right ) - \frac{3}{128} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*sqrt(2)*log((x - sqrt(2) + 1)/(x + sqrt(2) + 1)) + 1/64*sqrt(2)*log((x - sqrt(2) - 1)/(x + sqrt(2) - 1))
- 1/32*(7*x^5 - 46*x^3 + 31*x)/(x^6 - 7*x^4 + 7*x^2 - 1) + 3/128*log(x^2 + 2*x - 1) - 3/128*log(x^2 - 2*x - 1)
+ 1/8*log(x + 1) - 1/8*log(x - 1)

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Fricas [B]  time = 1.3181, size = 585, normalized size = 6.43 \begin{align*} -\frac{28 \, x^{5} - 184 \, x^{3} - 2 \, \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 ) - 2 \, \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 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) + 3 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 16 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 16 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 124 \, x}{128 \,{\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((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x, algorithm="fricas")

[Out]

-1/128*(28*x^5 - 184*x^3 - 2*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)) - 2*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*(
x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 + 2*x - 1) + 3*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 - 2*x - 1) - 16*(x^6 - 7*x^4
+ 7*x^2 - 1)*log(x + 1) + 16*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x - 1) + 124*x)/(x^6 - 7*x^4 + 7*x^2 - 1)

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Sympy [B]  time = 1.04127, size = 272, normalized size = 2.99 \begin{align*} - \frac{7 x^{5} - 46 x^{3} + 31 x}{32 x^{6} - 224 x^{4} + 224 x^{2} - 32} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} + \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right ) \log{\left (x - \frac{38423555}{909328} - \frac{38423555 \sqrt{2}}{1363992} + \frac{9549859782656 \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{5}}{170499} - \frac{56267374592 \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{3}}{56833} \right )} + \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right ) \log{\left (x - \frac{38423555}{909328} + \frac{9549859782656 \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right )^{5}}{170499} - \frac{56267374592 \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right )^{3}}{56833} + \frac{38423555 \sqrt{2}}{1363992} \right )} + \left (\frac{3}{128} - \frac{\sqrt{2}}{64}\right ) \log{\left (x - \frac{38423555 \sqrt{2}}{1363992} - \frac{56267374592 \left (\frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{3}}{56833} + \frac{9549859782656 \left (\frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{5}}{170499} + \frac{38423555}{909328} \right )} + \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right ) \log{\left (x - \frac{56267374592 \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{3}}{56833} + \frac{9549859782656 \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{5}}{170499} + \frac{38423555 \sqrt{2}}{1363992} + \frac{38423555}{909328} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(7*x**5 - 46*x**3 + 31*x)/(32*x**6 - 224*x**4 + 224*x**2 - 32) - log(x - 1)/8 + log(x + 1)/8 + (-3/128 - sqrt
(2)/64)*log(x - 38423555/909328 - 38423555*sqrt(2)/1363992 + 9549859782656*(-3/128 - sqrt(2)/64)**5/170499 - 5
6267374592*(-3/128 - sqrt(2)/64)**3/56833) + (-3/128 + sqrt(2)/64)*log(x - 38423555/909328 + 9549859782656*(-3
/128 + sqrt(2)/64)**5/170499 - 56267374592*(-3/128 + sqrt(2)/64)**3/56833 + 38423555*sqrt(2)/1363992) + (3/128
- sqrt(2)/64)*log(x - 38423555*sqrt(2)/1363992 - 56267374592*(3/128 - sqrt(2)/64)**3/56833 + 9549859782656*(3
/128 - sqrt(2)/64)**5/170499 + 38423555/909328) + (sqrt(2)/64 + 3/128)*log(x - 56267374592*(sqrt(2)/64 + 3/128
)**3/56833 + 9549859782656*(sqrt(2)/64 + 3/128)**5/170499 + 38423555*sqrt(2)/1363992 + 38423555/909328)

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Giac [A]  time = 1.23768, size = 181, normalized size = 1.99 \begin{align*} \frac{1}{64} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) + \frac{1}{64} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} - 2 \right |}}\right ) - \frac{7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac{3}{128} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) - \frac{3}{128} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac{1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*sqrt(2)*log(abs(2*x - 2*sqrt(2) + 2)/abs(2*x + 2*sqrt(2) + 2)) + 1/64*sqrt(2)*log(abs(2*x - 2*sqrt(2) - 2
)/abs(2*x + 2*sqrt(2) - 2)) - 1/32*(7*x^5 - 46*x^3 + 31*x)/(x^6 - 7*x^4 + 7*x^2 - 1) + 3/128*log(abs(x^2 + 2*x
- 1)) - 3/128*log(abs(x^2 - 2*x - 1)) + 1/8*log(abs(x + 1)) - 1/8*log(abs(x - 1))