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 = 0.086782, 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, Rules used = {1106, 1093, 204} \[ \frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}}-\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}} \]

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

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{a+8 x-8 x^2+4 x^3-x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{2 \sqrt{4+a}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{4+a}-x^2} \, dx,x,-1+x\right )}{2 \sqrt{4+a}}\\ &=\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{4+a}}}\right )}{2 \sqrt{4+a} \sqrt{1-\sqrt{4+a}}}-\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{1+\sqrt{4+a}}}\right )}{2 \sqrt{4+a} \sqrt{1+\sqrt{4+a}}}\\ \end{align*}

Mathematica [C]  time = 0.0142267, 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 = 0.013, size = 49, normalized size = 0.6 \begin{align*} -{\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}}} \end{align*}

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. \begin{align*} -\int \frac{1}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x} \end{align*}

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 [B]  time = 1.54701, size = 1254, normalized size = 14.09 \begin{align*} \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 ) \end{align*}

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)/sqr
t(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) + 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 = 0.705436, size = 66, normalized size = 0.74 \begin{align*} - \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 )} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError