3.368 \(\int \frac{1+x^3+x^6}{x+x^5} \, dx\)

Optimal. Leaf size=112 \[ -\frac{1}{4} \log \left (x^4+1\right )+\frac{x^2}{2}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{1}{2} \tan ^{-1}\left (x^2\right )+\log (x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.232911, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75 \[ -\frac{1}{4} \log \left (x^4+1\right )+\frac{x^2}{2}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{1}{2} \tan ^{-1}\left (x^2\right )+\log (x)-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^3 + x^6)/(x + x^5),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \log{\left (x \right )} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{8} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{8} - \frac{\operatorname{atan}{\left (x^{2} \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{4} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**6+x**3+1)/(x**5+x),x)

[Out]

log(x) - log(x**4 + 1)/4 + sqrt(2)*log(x**2 - sqrt(2)*x + 1)/8 - sqrt(2)*log(x**
2 + sqrt(2)*x + 1)/8 - atan(x**2)/2 + sqrt(2)*atan(sqrt(2)*x - 1)/4 + sqrt(2)*at
an(sqrt(2)*x + 1)/4 + Integral(x, x)

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Mathematica [A]  time = 0.0780835, size = 101, normalized size = 0.9 \[ \frac{1}{8} \left (-2 \log \left (x^4+1\right )+4 x^2+\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+8 \log (x)-2 \left (\sqrt{2}-2\right ) \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \left (2+\sqrt{2}\right ) \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^3 + x^6)/(x + x^5),x]

[Out]

(4*x^2 - 2*(-2 + Sqrt[2])*ArcTan[1 - Sqrt[2]*x] + 2*(2 + Sqrt[2])*ArcTan[1 + Sqr
t[2]*x] + 8*Log[x] + Sqrt[2]*Log[1 - Sqrt[2]*x + x^2] - Sqrt[2]*Log[1 + Sqrt[2]*
x + x^2] - 2*Log[1 + x^4])/8

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Maple [A]  time = 0.01, size = 79, normalized size = 0.7 \[{\frac{{x}^{2}}{2}}-{\frac{\arctan \left ({x}^{2} \right ) }{2}}+{\frac{\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{4}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-\sqrt{2}x}{1+{x}^{2}+\sqrt{2}x}} \right ) }+{\frac{\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{4}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}+\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^6+x^3+1)/(x^5+x),x)

[Out]

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

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Maxima [A]  time = 0.877967, size = 134, normalized size = 1.2 \[ \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} - 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2}{\left (\sqrt{2} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2}{\left (\sqrt{2} - 1\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{1}{2} \, x^{2} + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^6 + x^3 + 1)/(x^5 + x),x, algorithm="maxima")

[Out]

1/4*sqrt(2)*(sqrt(2) + 1)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) - 1/4*sqrt(2)*(sqr
t(2) - 1)*arctan(1/2*sqrt(2)*(2*x - sqrt(2))) - 1/8*sqrt(2)*(sqrt(2) + 1)*log(x^
2 + sqrt(2)*x + 1) - 1/8*sqrt(2)*(sqrt(2) - 1)*log(x^2 - sqrt(2)*x + 1) + 1/2*x^
2 + log(x)

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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((x^6 + x^3 + 1)/(x^5 + x),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.12875, size = 61, normalized size = 0.54 \[ \frac{x^{2}}{2} + \log{\left (x \right )} + \operatorname{RootSum}{\left (256 t^{4} + 256 t^{3} + 128 t^{2} + 16 t + 1, \left ( t \mapsto t \log{\left (\frac{1792 t^{4}}{73} + \frac{704 t^{3}}{219} - \frac{3152 t^{2}}{219} - \frac{2584 t}{219} + x - \frac{344}{219} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**6+x**3+1)/(x**5+x),x)

[Out]

x**2/2 + log(x) + RootSum(256*_t**4 + 256*_t**3 + 128*_t**2 + 16*_t + 1, Lambda(
_t, _t*log(1792*_t**4/73 + 704*_t**3/219 - 3152*_t**2/219 - 2584*_t/219 + x - 34
4/219)))

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GIAC/XCAS [A]  time = 0.261579, size = 124, normalized size = 1.11 \[ \frac{1}{2} \, x^{2} + \frac{1}{4} \,{\left (\sqrt{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \,{\left (\sqrt{2} - 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{8} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^6 + x^3 + 1)/(x^5 + x),x, algorithm="giac")

[Out]

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