3.1304 \(\int \frac{1}{x^3 \left (1+x^5\right )} \, dx\)

Optimal. Leaf size=192 \[ -\frac{1}{2 x^2}+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

[Out]

-1/(2*x^2) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + 2*Sqrt[2/(5
+ Sqrt[5])]*x])/5 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - Sqr
t[(2*(5 + Sqrt[5]))/5]*x])/5 - Log[1 + x]/5 + ((1 + Sqrt[5])*Log[1 - ((1 - Sqrt[
5])*x)/2 + x^2])/20 + ((1 - Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2])/20

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Rubi [A]  time = 0.622379, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{2 x^2}+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 + x^5)),x]

[Out]

-1/(2*x^2) + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + 2*Sqrt[2/(5
+ Sqrt[5])]*x])/5 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - Sqr
t[(2*(5 + Sqrt[5]))/5]*x])/5 - Log[1 + x]/5 + ((1 + Sqrt[5])*Log[1 - ((1 - Sqrt[
5])*x)/2 + x^2])/20 + ((1 - Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2])/20

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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(1/x**3/(x**5+1),x)

[Out]

Timed out

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Mathematica [A]  time = 0.275905, size = 149, normalized size = 0.78 \[ \frac{1}{20} \left (-\frac{10}{x^2}+\left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )-\left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-4 \log (x+1)+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )+2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*(1 + x^5)),x]

[Out]

(-10/x^2 + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5
]]] + 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]]
- 4*Log[1 + x] + (1 + Sqrt[5])*Log[1 + ((-1 + Sqrt[5])*x)/2 + x^2] - (-1 + Sqrt[
5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2])/20

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Maple [A]  time = 0.023, size = 161, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+x \right ) }{5}}-{\frac{1}{2\,{x}^{2}}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}+{\frac{2\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}-{\frac{2\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^5+1),x)

[Out]

-1/5*ln(1+x)-1/2/x^2+1/20*ln(x*5^(1/2)+2*x^2-x+2)*5^(1/2)+1/20*ln(x*5^(1/2)+2*x^
2-x+2)+2/5/(10+2*5^(1/2))^(1/2)*arctan((5^(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))*5^(
1/2)-1/20*ln(-x*5^(1/2)+2*x^2-x+2)*5^(1/2)+1/20*ln(-x*5^(1/2)+2*x^2-x+2)-2/5/(10
-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2))*5^(1/2)

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Maxima [A]  time = 1.6077, size = 174, normalized size = 0.91 \[ \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{2 \, \sqrt{5} + 10}} - \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} + 10}} - \frac{\log \left (2 \, x^{2} - x{\left (\sqrt{5} + 1\right )} + 2\right )}{5 \, \sqrt{5} + 5} + \frac{\log \left (2 \, x^{2} + x{\left (\sqrt{5} - 1\right )} + 2\right )}{5 \,{\left (\sqrt{5} - 1\right )}} - \frac{1}{2 \, x^{2}} - \frac{1}{5} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^3),x, algorithm="maxima")

[Out]

2/5*sqrt(5)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10))/sqrt(2*sqrt(5) + 10
) - 2/5*sqrt(5)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10))/sqrt(-2*sqrt(5
) + 10) - log(2*x^2 - x*(sqrt(5) + 1) + 2)/((5*sqrt(5)) + 5) + 1/5*log(2*x^2 + x
*(sqrt(5) - 1) + 2)/(sqrt(5) - 1) - 1/2/x^2 - 1/5*log(x + 1)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^3),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.82015, size = 42, normalized size = 0.22 \[ - \frac{\log{\left (x + 1 \right )}}{5} + \operatorname{RootSum}{\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log{\left (25 t^{2} + x \right )} \right )\right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**5+1),x)

[Out]

-log(x + 1)/5 + RootSum(625*_t**4 - 125*_t**3 + 25*_t**2 - 5*_t + 1, Lambda(_t,
_t*log(25*_t**2 + x))) - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.234659, size = 178, normalized size = 0.93 \[ \frac{1}{10} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{1}{20} \, \sqrt{5}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{20} \, \sqrt{5}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) - \frac{1}{2 \, x^{2}} + \frac{1}{20} \,{\rm ln}\left (x^{4} - x^{3} + x^{2} - x + 1\right ) - \frac{1}{5} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^3),x, algorithm="giac")

[Out]

1/10*sqrt(-2*sqrt(5) + 10)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) - 1/
10*sqrt(2*sqrt(5) + 10)*arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10)) - 1/20
*sqrt(5)*ln(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/20*sqrt(5)*ln(x^2 + 1/2*x*(sqrt(5
) - 1) + 1) - 1/2/x^2 + 1/20*ln(x^4 - x^3 + x^2 - x + 1) - 1/5*ln(abs(x + 1))