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

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 + Sqrt[2])*x] - (3 + 2*Sqrt[2])*ArcTanh[(1 + Sqrt[2])*x
])/64

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Rubi [B]  time = 0.34491, 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 \[ -\frac{12-5 x}{64 \left (-x^2+2 x+1\right )}+\frac{5 x+12}{64 \left (-x^2-2 x+1\right )}+\frac{1}{32 (1-x)}-\frac{1}{32 (x+1)}-\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (-x-\sqrt{2}+1\right )-\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (-x+\sqrt{2}+1\right )+\frac{3}{256} \left (2+3 \sqrt{2}\right ) \log \left (x-\sqrt{2}+1\right )+\frac{3}{256} \left (2-3 \sqrt{2}\right ) \log \left (x+\sqrt{2}+1\right )-\frac{5 \tanh ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{64 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x)+\frac{5 \tanh ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{64 \sqrt{2}} \]

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*ArcTanh[(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])*Lo
g[1 - Sqrt[2] + x])/256 + (3*(2 - 3*Sqrt[2])*Log[1 + Sqrt[2] + x])/256

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)/(-x**6+7*x**4-7*x**2+1)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.153365, 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*Lo
g[1 + x] - (3 + 2*Sqrt[2])*Log[-1 + Sqrt[2] + x] + (3 - 2*Sqrt[2])*Log[1 + Sqrt[
2] + x])/128

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Maple [A]  time = 0.026, size = 116, normalized size = 1.3 \[ -{\frac{-12+5\,x}{64\,{x}^{2}-128\,x-64}}-{\frac{3\,\ln \left ({x}^{2}-2\,x-1 \right ) }{128}}-{\frac{\sqrt{2}}{32}{\it Artanh} \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{-32+32\,x}}-{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{1}{32+32\,x}}+{\frac{\ln \left ( 1+x \right ) }{8}}+{\frac{-5\,x-12}{64\,{x}^{2}+128\,x-64}}+{\frac{3\,\ln \left ({x}^{2}+2\,x-1 \right ) }{128}}-{\frac{\sqrt{2}}{32}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{2}}{4}} \right ) } \]

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/(-1+x)-1/8*ln(-1+x)-1/32/(1+x)+1/8*ln(1+x)+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))

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Maxima [A]  time = 0.89445, size = 162, normalized size = 1.78 \[ \frac{1}{64} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2} + 1\right )}}{2 \, x + 2 \, \sqrt{2} + 2}\right ) + \frac{1}{64} \, \sqrt{2} \log \left (\frac{2 \,{\left (x - \sqrt{2} - 1\right )}}{2 \, x + 2 \, \sqrt{2} - 2}\right ) - \frac{7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac{3}{128} \, \log \left (x^{2} + 2 \, x - 1\right ) - \frac{3}{128} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]

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(2*(x - sqrt(2) + 1)/((2*sqrt(2)) + 2*x + 2)) + 1/64*sqrt(2)*log
(2*(x - sqrt(2) - 1)/((2*sqrt(2)) + 2*x - 2)) - 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 [A]  time = 0.275451, size = 324, normalized size = 3.56 \[ \frac{\sqrt{2}{\left (3 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) - 3 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) + 16 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) - 16 \, \sqrt{2}{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 4 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{\sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} - 4 \, x - 4}{x^{2} + 2 \, x - 1}\right ) + 4 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac{\sqrt{2}{\left (x^{2} - 2 \, x + 3\right )} - 4 \, x + 4}{x^{2} - 2 \, x - 1}\right ) - 4 \, \sqrt{2}{\left (7 \, x^{5} - 46 \, x^{3} + 31 \, x\right )}\right )}}{256 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \]

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]

1/256*sqrt(2)*(3*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 + 2*x - 1) - 3*sqrt(2
)*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 - 2*x - 1) + 16*sqrt(2)*(x^6 - 7*x^4 + 7*x^2
- 1)*log(x + 1) - 16*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x - 1) + 4*(x^6 - 7*
x^4 + 7*x^2 - 1)*log((sqrt(2)*(x^2 + 2*x + 3) - 4*x - 4)/(x^2 + 2*x - 1)) + 4*(x
^6 - 7*x^4 + 7*x^2 - 1)*log((sqrt(2)*(x^2 - 2*x + 3) - 4*x + 4)/(x^2 - 2*x - 1))
- 4*sqrt(2)*(7*x^5 - 46*x^3 + 31*x))/(x^6 - 7*x^4 + 7*x^2 - 1)

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Sympy [A]  time = 3.89062, size = 272, normalized size = 2.99 \[ - \frac{7 x^{5} - 46 x^{3} + 31 x}{32 x^{6} - 224 x^{4} + 224 x^{2} - 32} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} + \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right ) \log{\left (x - \frac{38423555}{909328} - \frac{38423555 \sqrt{2}}{1363992} + \frac{9549859782656 \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{5}}{170499} - \frac{56267374592 \left (- \frac{3}{128} - \frac{\sqrt{2}}{64}\right )^{3}}{56833} \right )} + \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right ) \log{\left (x - \frac{38423555}{909328} + \frac{9549859782656 \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right )^{5}}{170499} - \frac{56267374592 \left (- \frac{3}{128} + \frac{\sqrt{2}}{64}\right )^{3}}{56833} + \frac{38423555 \sqrt{2}}{1363992} \right )} + \left (- \frac{\sqrt{2}}{64} + \frac{3}{128}\right ) \log{\left (x - \frac{38423555 \sqrt{2}}{1363992} - \frac{56267374592 \left (- \frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{3}}{56833} + \frac{9549859782656 \left (- \frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{5}}{170499} + \frac{38423555}{909328} \right )} + \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right ) \log{\left (x - \frac{56267374592 \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{3}}{56833} + \frac{9549859782656 \left (\frac{\sqrt{2}}{64} + \frac{3}{128}\right )^{5}}{170499} + \frac{38423555 \sqrt{2}}{1363992} + \frac{38423555}{909328} \right )} \]

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 - 56267374592*(-3/128 -
sqrt(2)/64)**3/56833) + (-3/128 + sqrt(2)/64)*log(x - 38423555/909328 + 9549859
782656*(-3/128 + sqrt(2)/64)**5/170499 - 56267374592*(-3/128 + sqrt(2)/64)**3/56
833 + 38423555*sqrt(2)/1363992) + (-sqrt(2)/64 + 3/128)*log(x - 38423555*sqrt(2)
/1363992 - 56267374592*(-sqrt(2)/64 + 3/128)**3/56833 + 9549859782656*(-sqrt(2)/
64 + 3/128)**5/170499 + 38423555/909328) + (sqrt(2)/64 + 3/128)*log(x - 56267374
592*(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/XCAS [A]  time = 0.266122, size = 181, normalized size = 1.99 \[ \frac{1}{64} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} + 2 \right |}}\right ) + \frac{1}{64} \, \sqrt{2}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} - 2 \right |}}\right ) - \frac{7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \,{\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac{3}{128} \,{\rm ln}\left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) - \frac{3}{128} \,{\rm ln}\left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

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)*ln(abs(2*x - 2*sqrt(2) + 2)/abs(2*x + 2*sqrt(2) + 2)) + 1/64*sqrt(2
)*ln(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*ln(abs(x^2 + 2*x - 1)) - 3/128*ln(abs(x
^2 - 2*x - 1)) + 1/8*ln(abs(x + 1)) - 1/8*ln(abs(x - 1))