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

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Rubi [A]  time = 0.0830145, 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, Rules used = {1680, 1673, 1093, 204, 1107, 618, 206} \[ -\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{2 \sqrt{a+4} \sqrt{1-\sqrt{a+4}}}+\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{2 \sqrt{a+4} \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{2 \sqrt{a+4}} \]

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

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(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[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

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])

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 618

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

Rule 206

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

Rubi steps

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

Mathematica [C]  time = 0.0156602, 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 = 0.003, size = 50, normalized size = 0.4 \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{{\it \_R}\,\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(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. \begin{align*} -\int \frac{x}{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(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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Sympy [A]  time = 2.57714, size = 155, normalized size = 1.34 \begin{align*} - \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 )} \end{align*}

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

[Out]

Exception raised: TypeError