3.127 \(\int \frac{x}{a+8 x-8 x^2+4 x^3-x^4} \, dx\)

Optimal. Leaf size=116 \[ -\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}}+\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{2 \sqrt{a+4}} \]

[Out]

-ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 - Sqrt[4 + a]]) +
ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 + Sqrt[4 + a]]) + A
rcTanh[(1 + (-1 + x)^2)/Sqrt[4 + a]]/(2*Sqrt[4 + a])

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Rubi [A]  time = 0.240668, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}}-\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{2 \sqrt{a+4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4),x]

[Out]

ArcTan[(1 - x)/Sqrt[1 - Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 - Sqrt[4 + a]]) - Ar
cTan[(1 - x)/Sqrt[1 + Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 + Sqrt[4 + a]]) + ArcT
anh[(1 + (-1 + x)^2)/Sqrt[4 + a]]/(2*Sqrt[4 + a])

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Rubi in Sympy [A]  time = 43.0483, size = 99, normalized size = 0.85 \[ - \frac{\operatorname{atanh}{\left (\frac{- \left (x - 1\right )^{2} - 1}{\sqrt{a + 4}} \right )}}{2 \sqrt{a + 4}} + \frac{\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{2 \sqrt{a + 4} \sqrt{\sqrt{a + 4} + 1}} - \frac{\operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{2 \sqrt{a + 4} \sqrt{- \sqrt{a + 4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**4+4*x**3-8*x**2+a+8*x),x)

[Out]

-atanh((-(x - 1)**2 - 1)/sqrt(a + 4))/(2*sqrt(a + 4)) + atan((x - 1)/sqrt(sqrt(a
+ 4) + 1))/(2*sqrt(a + 4)*sqrt(sqrt(a + 4) + 1)) - atan((x - 1)/sqrt(-sqrt(a +
4) + 1))/(2*sqrt(a + 4)*sqrt(-sqrt(a + 4) + 1))

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Mathematica [C]  time = 0.0283873, size = 59, normalized size = 0.51 \[ -\frac{1}{4} \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4),x]

[Out]

-RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (Log[x - #1]*#1)/(-2 + 4*#1 - 3*#
1^2 + #1^3) & ]/4

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Maple [C]  time = 0.004, size = 50, normalized size = 0.4 \[ -{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^4+4*x^3-8*x^2+a+8*x),x)

[Out]

-1/4*sum(_R/(_R^3-3*_R^2+4*_R-2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="maxima")

[Out]

-integrate(x/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), 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/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 8.24331, size = 155, normalized size = 1.34 \[ - \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} + 2816 a^{2} + 10240 a + 12288\right ) + t^{2} \left (- 32 a^{2} - 256 a - 512\right ) + t \left (- 16 a - 64\right ) + a, \left ( t \mapsto t \log{\left (x + \frac{- 128 t^{3} a^{4} - 1728 t^{3} a^{3} - 8640 t^{3} a^{2} - 18944 t^{3} a - 15360 t^{3} + 48 t^{2} a^{3} + 464 t^{2} a^{2} + 1472 t^{2} a + 1536 t^{2} + 8 t a^{3} + 88 t a^{2} + 312 t a + 352 t - a^{2} - 2 a}{4 a^{2} + 21 a + 28} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**4+4*x**3-8*x**2+a+8*x),x)

[Out]

-RootSum(_t**4*(256*a**3 + 2816*a**2 + 10240*a + 12288) + _t**2*(-32*a**2 - 256*
a - 512) + _t*(-16*a - 64) + a, Lambda(_t, _t*log(x + (-128*_t**3*a**4 - 1728*_t
**3*a**3 - 8640*_t**3*a**2 - 18944*_t**3*a - 15360*_t**3 + 48*_t**2*a**3 + 464*_
t**2*a**2 + 1472*_t**2*a + 1536*_t**2 + 8*_t*a**3 + 88*_t*a**2 + 312*_t*a + 352*
_t - a**2 - 2*a)/(4*a**2 + 21*a + 28))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="giac")

[Out]

integrate(-x/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)