\(\int \frac {1+x^2}{(1-7 x^2+7 x^4-x^6)^2} \, dx\) [233]

Optimal result

Integrand size = 25, antiderivative size = 91 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {x}{16 \left (1-x^2\right )}+\frac {x \left (29-5 x^2\right )}{32 \left (1-6 x^2+x^4\right )}+\frac {\text {arctanh}(x)}{4}+\frac {1}{64} \left (\left (3-2 \sqrt {2}\right ) \text {arctanh}\left (\left (-1+\sqrt {2}\right ) x\right )-\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\left (1+\sqrt {2}\right ) x\right )\right ) \]

[Out]

1/16*x/(-x^2+1)+1/32*x*(-5*x^2+29)/(x^4-6*x^2+1)+1/4*arctanh(x)+1/64*arctanh(x*(2^(1/2)-1))*(3-2*2^(1/2))-1/64
*arctanh(x*(1+2^(1/2)))*(3+2*2^(1/2))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(91)=182\).

Time = 0.11 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.25, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2098, 213, 652, 632, 212, 646, 31} \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=-\frac {5 \text {arctanh}\left (\frac {1-x}{\sqrt {2}}\right )}{64 \sqrt {2}}+\frac {\text {arctanh}(x)}{4}+\frac {5 \text {arctanh}\left (\frac {x+1}{\sqrt {2}}\right )}{64 \sqrt {2}}-\frac {12-5 x}{64 \left (-x^2+2 x+1\right )}+\frac {5 x+12}{64 \left (-x^2-2 x+1\right )}+\frac {1}{32 (1-x)}-\frac {1}{32 (x+1)}-\frac {3}{256} \left (2+3 \sqrt {2}\right ) \log \left (-x-\sqrt {2}+1\right )-\frac {3}{256} \left (2-3 \sqrt {2}\right ) \log \left (-x+\sqrt {2}+1\right )+\frac {3}{256} \left (2+3 \sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )+\frac {3}{256} \left (2-3 \sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right ) \]

[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 31

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

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 213

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

Rule 632

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 646

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 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 2098

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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 x\right )+\frac {5}{32} \text {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] (verified)

Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{128} \left (-\frac {4 x \left (31-46 x^2+7 x^4\right )}{-1+7 x^2-7 x^4+x^6}-16 \log (1-x)+\left (3+2 \sqrt {2}\right ) \log \left (-1+\sqrt {2}-x\right )+\left (-3+2 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )+16 \log (1+x)-\left (3+2 \sqrt {2}\right ) \log \left (-1+\sqrt {2}+x\right )+\left (3-2 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )\right ) \]

[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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27

method	result	size
default	\(-\frac {1}{32 \left (x +1\right )}+\frac {\ln \left (x +1\right )}{8}+\frac {-5 x -12}{64 x^{2}+128 x -64}+\frac {3 \ln \left (x^{2}+2 x -1\right )}{128}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{32}-\frac {1}{32 \left (x -1\right )}-\frac {\ln \left (x -1\right )}{8}-\frac {5 x -12}{64 \left (x^{2}-2 x -1\right )}-\frac {3 \ln \left (x^{2}-2 x -1\right )}{128}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x -2\right ) \sqrt {2}}{4}\right )}{32}\)	\(116\)
risch	\(\frac {-\frac {7}{32} x^{5}+\frac {23}{16} x^{3}-\frac {31}{32} x}{x^{6}-7 x^{4}+7 x^{2}-1}-\frac {3 \ln \left (x -1-\sqrt {2}\right )}{128}+\frac {\ln \left (x -1-\sqrt {2}\right ) \sqrt {2}}{64}-\frac {3 \ln \left (x -1+\sqrt {2}\right )}{128}-\frac {\ln \left (x -1+\sqrt {2}\right ) \sqrt {2}}{64}+\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x -1\right )}{8}+\frac {3 \ln \left (2 x +2-2 \sqrt {2}\right )}{128}+\frac {\ln \left (2 x +2-2 \sqrt {2}\right ) \sqrt {2}}{64}+\frac {3 \ln \left (2 x +2+2 \sqrt {2}\right )}{128}-\frac {\ln \left (2 x +2+2 \sqrt {2}\right ) \sqrt {2}}{64}\)	\(150\)

[In]

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

[Out]

-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*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*x-2)*2^(1/
2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (71) = 142\).

Time = 0.34 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.45 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=-\frac {28 \, x^{5} - 184 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) - 2 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x - 1\right )} - 2 \, x + 3}{x^{2} - 2 \, x - 1}\right ) - 3 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) + 3 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 16 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 16 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 124 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (73) = 146\).

Time = 0.80 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.99 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {- 7 x^{5} + 46 x^{3} - 31 x}{32 x^{6} - 224 x^{4} + 224 x^{2} - 32} - \frac {\log {\left (x - 1 \right )}}{8} + \frac {\log {\left (x + 1 \right )}}{8} + \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555}{909328} - \frac {38423555 \sqrt {2}}{1363992} + \frac {9549859782656 \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{5}}{170499} - \frac {56267374592 \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{3}}{56833} \right )} + \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555}{909328} + \frac {9549859782656 \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right )^{5}}{170499} - \frac {56267374592 \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right )^{3}}{56833} + \frac {38423555 \sqrt {2}}{1363992} \right )} + \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555 \sqrt {2}}{1363992} - \frac {56267374592 \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{3}}{56833} + \frac {9549859782656 \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{5}}{170499} + \frac {38423555}{909328} \right )} + \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right ) \log {\left (x - \frac {56267374592 \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right )^{3}}{56833} + \frac {9549859782656 \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right )^{5}}{170499} + \frac {38423555 \sqrt {2}}{1363992} + \frac {38423555}{909328} \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.25 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{64} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} + 1}{x + \sqrt {2} + 1}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} - 1}{x + \sqrt {2} - 1}\right ) - \frac {7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac {3}{128} \, \log \left (x^{2} + 2 \, x - 1\right ) - \frac {3}{128} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{64} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + 2 \right |}}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} - 2 \right |}}\right ) - \frac {7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac {3}{128} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) - \frac {3}{128} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.36 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\frac {7\,x^5}{32}-\frac {23\,x^3}{16}+\frac {31\,x}{32}}{x^6-7\,x^4+7\,x^2-1}+\mathrm {atan}\left (\frac {x\,23313{}\mathrm {i}}{8192\,\left (\frac {27309\,\sqrt {2}}{32768}-\frac {19317}{16384}\right )}-\frac {\sqrt {2}\,x\,65943{}\mathrm {i}}{32768\,\left (\frac {27309\,\sqrt {2}}{32768}-\frac {19317}{16384}\right )}\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{32}-\frac {3}{64}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {x\,23313{}\mathrm {i}}{8192\,\left (\frac {27309\,\sqrt {2}}{32768}+\frac {19317}{16384}\right )}+\frac {\sqrt {2}\,x\,65943{}\mathrm {i}}{32768\,\left (\frac {27309\,\sqrt {2}}{32768}+\frac {19317}{16384}\right )}\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{32}+\frac {3}{64}{}\mathrm {i}\right ) \]

[In]

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

[Out]

atan((x*23313i)/(8192*((27309*2^(1/2))/32768 - 19317/16384)) - (2^(1/2)*x*65943i)/(32768*((27309*2^(1/2))/3276
8 - 19317/16384)))*((2^(1/2)*1i)/32 - 3i/64) - ((31*x)/32 - (23*x^3)/16 + (7*x^5)/32)/(7*x^2 - 7*x^4 + x^6 - 1
) - (atan(x*1i)*1i)/4 + atan((x*23313i)/(8192*((27309*2^(1/2))/32768 + 19317/16384)) + (2^(1/2)*x*65943i)/(327
68*((27309*2^(1/2))/32768 + 19317/16384)))*((2^(1/2)*1i)/32 + 3i/64)