3.1496 \(\int \frac{1}{x^7 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=100 \[ -\frac{1}{6 x^6}+\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]

[Out]

-1/(6*x^6) + ArcTan[1 - Sqrt[2]*x^2]/(4*Sqrt[2]) - ArcTan[1 + Sqrt[2]*x^2]/(4*Sq
rt[2]) + Log[1 - Sqrt[2]*x^2 + x^4]/(8*Sqrt[2]) - Log[1 + Sqrt[2]*x^2 + x^4]/(8*
Sqrt[2])

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Rubi [A]  time = 0.149932, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{1}{6 x^6}+\frac{\tan ^{-1}\left (1-\sqrt{2} x^2\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x^2+1\right )}{4 \sqrt{2}}+\frac{\log \left (x^4-\sqrt{2} x^2+1\right )}{8 \sqrt{2}}-\frac{\log \left (x^4+\sqrt{2} x^2+1\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(1 + x^8)),x]

[Out]

-1/(6*x^6) + ArcTan[1 - Sqrt[2]*x^2]/(4*Sqrt[2]) - ArcTan[1 + Sqrt[2]*x^2]/(4*Sq
rt[2]) + Log[1 - Sqrt[2]*x^2 + x^4]/(8*Sqrt[2]) - Log[1 + Sqrt[2]*x^2 + x^4]/(8*
Sqrt[2])

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Rubi in Sympy [A]  time = 17.2271, size = 87, normalized size = 0.87 \[ \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} - \frac{1}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**8+1),x)

[Out]

sqrt(2)*log(x**4 - sqrt(2)*x**2 + 1)/16 - sqrt(2)*log(x**4 + sqrt(2)*x**2 + 1)/1
6 - sqrt(2)*atan(sqrt(2)*x**2 - 1)/8 - sqrt(2)*atan(sqrt(2)*x**2 + 1)/8 - 1/(6*x
**6)

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Mathematica [A]  time = 0.18162, size = 193, normalized size = 1.93 \[ \frac{1}{48} \left (-\frac{8}{x^6}-3 \sqrt{2} \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )-3 \sqrt{2} \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+3 \sqrt{2} \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+3 \sqrt{2} \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )-6 \sqrt{2} \tan ^{-1}\left (x \sec \left (\frac{\pi }{8}\right )-\tan \left (\frac{\pi }{8}\right )\right )+6 \sqrt{2} \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+6 \sqrt{2} \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-x \csc \left (\frac{\pi }{8}\right )\right )+6 \sqrt{2} \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(1 + x^8)),x]

[Out]

(-8/x^6 + 6*Sqrt[2]*ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]] + 6*Sqrt[2]*ArcTan[Cot[Pi/
8] - x*Csc[Pi/8]] + 6*Sqrt[2]*ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])] - 6*Sqrt[2]*ArcT
an[x*Sec[Pi/8] - Tan[Pi/8]] + 3*Sqrt[2]*Log[1 + x^2 - 2*x*Cos[Pi/8]] + 3*Sqrt[2]
*Log[1 + x^2 + 2*x*Cos[Pi/8]] - 3*Sqrt[2]*Log[1 + x^2 - 2*x*Sin[Pi/8]] - 3*Sqrt[
2]*Log[1 + x^2 + 2*x*Sin[Pi/8]])/48

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Maple [A]  time = 0.006, size = 71, normalized size = 0.7 \[ -{\frac{1}{6\,{x}^{6}}}-{\frac{\arctan \left ( 1+{x}^{2}\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\arctan \left ({x}^{2}\sqrt{2}-1 \right ) \sqrt{2}}{8}}-{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{4}+{x}^{2}\sqrt{2}}{1+{x}^{4}-{x}^{2}\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^8+1),x)

[Out]

-1/6/x^6-1/8*arctan(1+x^2*2^(1/2))*2^(1/2)-1/8*arctan(x^2*2^(1/2)-1)*2^(1/2)-1/1
6*2^(1/2)*ln((1+x^4+x^2*2^(1/2))/(1+x^4-x^2*2^(1/2)))

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Maxima [A]  time = 1.57174, size = 115, normalized size = 1.15 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) - \frac{1}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^7),x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2))) - 1/8*sqrt(2)*arctan(1/2*sqrt
(2)*(2*x^2 - sqrt(2))) - 1/16*sqrt(2)*log(x^4 + sqrt(2)*x^2 + 1) + 1/16*sqrt(2)*
log(x^4 - sqrt(2)*x^2 + 1) - 1/6/x^6

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Fricas [A]  time = 0.230009, size = 171, normalized size = 1.71 \[ \frac{12 \, \sqrt{2} x^{6} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} + \sqrt{2} x^{2} + 1} + 1}\right ) + 12 \, \sqrt{2} x^{6} \arctan \left (\frac{1}{\sqrt{2} x^{2} + \sqrt{2} \sqrt{x^{4} - \sqrt{2} x^{2} + 1} - 1}\right ) - 3 \, \sqrt{2} x^{6} \log \left (x^{4} + \sqrt{2} x^{2} + 1\right ) + 3 \, \sqrt{2} x^{6} \log \left (x^{4} - \sqrt{2} x^{2} + 1\right ) - 8}{48 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^7),x, algorithm="fricas")

[Out]

1/48*(12*sqrt(2)*x^6*arctan(1/(sqrt(2)*x^2 + sqrt(2)*sqrt(x^4 + sqrt(2)*x^2 + 1)
+ 1)) + 12*sqrt(2)*x^6*arctan(1/(sqrt(2)*x^2 + sqrt(2)*sqrt(x^4 - sqrt(2)*x^2 +
1) - 1)) - 3*sqrt(2)*x^6*log(x^4 + sqrt(2)*x^2 + 1) + 3*sqrt(2)*x^6*log(x^4 - s
qrt(2)*x^2 + 1) - 8)/x^6

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Sympy [A]  time = 0.655528, size = 87, normalized size = 0.87 \[ \frac{\sqrt{2} \log{\left (x^{4} - \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \log{\left (x^{4} + \sqrt{2} x^{2} + 1 \right )}}{16} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x^{2} + 1 \right )}}{8} - \frac{1}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**8+1),x)

[Out]

sqrt(2)*log(x**4 - sqrt(2)*x**2 + 1)/16 - sqrt(2)*log(x**4 + sqrt(2)*x**2 + 1)/1
6 - sqrt(2)*atan(sqrt(2)*x**2 - 1)/8 - sqrt(2)*atan(sqrt(2)*x**2 + 1)/8 - 1/(6*x
**6)

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GIAC/XCAS [A]  time = 0.239926, size = 115, normalized size = 1.15 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x^{2} - \sqrt{2}\right )}\right ) - \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} + 1\right ) + \frac{1}{16} \, \sqrt{2}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} + 1\right ) - \frac{1}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^7),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2))) - 1/8*sqrt(2)*arctan(1/2*sqrt
(2)*(2*x^2 - sqrt(2))) - 1/16*sqrt(2)*ln(x^4 + sqrt(2)*x^2 + 1) + 1/16*sqrt(2)*l
n(x^4 - sqrt(2)*x^2 + 1) - 1/6/x^6