3.120 \(\int \frac{1}{a+8 x-8 x^2+4 x^3-x^4} \, dx\)

Optimal. Leaf size=89 \[ \frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}}-\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}} \]

[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]])

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Rubi [A]  time = 0.17455, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}}-\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-1),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]])

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Rubi in Sympy [A]  time = 31.2683, size = 73, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{2 \sqrt{a + 4} \sqrt{\sqrt{a + 4} + 1}} - \frac{\operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{2 \sqrt{a + 4} \sqrt{- \sqrt{a + 4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**4+4*x**3-8*x**2+a+8*x),x)

[Out]

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 = 0.023153, size = 57, normalized size = 0.64 \[ -\frac{1}{4} \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-1),x]

[Out]

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

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Maple [C]  time = 0.023, size = 49, normalized size = 0.6 \[ -{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^4+4*x^3-8*x^2+a+8*x),x)

[Out]

-1/4*sum(1/(_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. \[ -\int \frac{1}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="maxima")

[Out]

-integrate(1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)

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Fricas [A]  time = 0.270787, size = 617, normalized size = 6.93 \[ \frac{1}{4} \, \sqrt{\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a - \frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt{\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac{1}{4} \, \sqrt{\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a - \frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt{\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) + \frac{1}{4} \, \sqrt{-\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a + \frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt{-\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac{1}{4} \, \sqrt{-\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a + \frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt{-\frac{\frac{a^{2} + 7 \, a + 12}{\sqrt{a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="fricas")

[Out]

1/4*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))
*log((a - (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(((a^2 + 7*a
+ 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12)) + x - 1) - 1/4*sqrt(
((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))*log(-(a
- (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(((a^2 + 7*a + 12)/sq
rt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12)) + x - 1) + 1/4*sqrt(-((a^2 +
7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12))*log((a + (a^2 +
7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3
+ 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12)) + x - 1) - 1/4*sqrt(-((a^2 + 7*a +
12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12))*log(-(a + (a^2 + 7*a +
12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a
^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12)) + x - 1)

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Sympy [A]  time = 2.27999, size = 66, normalized size = 0.74 \[ - \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} + 2816 a^{2} + 10240 a + 12288\right ) + t^{2} \left (- 32 a - 128\right ) - 1, \left ( t \mapsto t \log{\left (64 t^{3} a^{2} + 448 t^{3} a + 768 t^{3} - 4 t a - 20 t + x - 1 \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-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 - 128) -
1, Lambda(_t, _t*log(64*_t**3*a**2 + 448*_t**3*a + 768*_t**3 - 4*_t*a - 20*_t +
x - 1)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x),x, algorithm="giac")

[Out]

integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)