3.89.20 \(\int \frac {-71+96 e^2-80 e^4+32 e^6-8 e^8-3 x+3 x^2-x^3}{-1+3 x-3 x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ 25+\frac {4 \left (2+\left (-1+e^2\right )^2\right )^2}{(1-x)^2}-x \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2074} \begin {gather*} \frac {4 \left (3-2 e^2+e^4\right )^2}{(1-x)^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-71 + 96*E^2 - 80*E^4 + 32*E^6 - 8*E^8 - 3*x + 3*x^2 - x^3)/(-1 + 3*x - 3*x^2 + x^3),x]

[Out]

(4*(3 - 2*E^2 + E^4)^2)/(1 - x)^2 - x

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {8 \left (3-2 e^2+e^4\right )^2}{(-1+x)^3}\right ) \, dx\\ &=\frac {4 \left (3-2 e^2+e^4\right )^2}{(1-x)^2}-x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 23, normalized size = 0.92 \begin {gather*} \frac {4 \left (3-2 e^2+e^4\right )^2}{(-1+x)^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-71 + 96*E^2 - 80*E^4 + 32*E^6 - 8*E^8 - 3*x + 3*x^2 - x^3)/(-1 + 3*x - 3*x^2 + x^3),x]

[Out]

(4*(3 - 2*E^2 + E^4)^2)/(-1 + x)^2 - x

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 39, normalized size = 1.56 \begin {gather*} -\frac {x^{3} - 2 \, x^{2} + x - 4 \, e^{8} + 16 \, e^{6} - 40 \, e^{4} + 48 \, e^{2} - 36}{x^{2} - 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)^4+32*exp(2)^3-80*exp(2)^2+96*exp(2)-x^3+3*x^2-3*x-71)/(x^3-3*x^2+3*x-1),x, algorithm="fri
cas")

[Out]

-(x^3 - 2*x^2 + x - 4*e^8 + 16*e^6 - 40*e^4 + 48*e^2 - 36)/(x^2 - 2*x + 1)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 27, normalized size = 1.08 \begin {gather*} -x + \frac {4 \, {\left (e^{8} - 4 \, e^{6} + 10 \, e^{4} - 12 \, e^{2} + 9\right )}}{{\left (x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)^4+32*exp(2)^3-80*exp(2)^2+96*exp(2)-x^3+3*x^2-3*x-71)/(x^3-3*x^2+3*x-1),x, algorithm="gia
c")

[Out]

-x + 4*(e^8 - 4*e^6 + 10*e^4 - 12*e^2 + 9)/(x - 1)^2

________________________________________________________________________________________

maple [A]  time = 0.06, size = 30, normalized size = 1.20

method	result	size
default	\(-x -\frac {-72+96 \,{\mathrm e}^{2}-80 \,{\mathrm e}^{4}-8 \,{\mathrm e}^{8}+32 \,{\mathrm e}^{6}}{2 \left (x -1\right )^{2}}\)	\(30\)
gosper	\(\frac {4 \,{\mathrm e}^{8}-16 \,{\mathrm e}^{6}-x^{3}+40 \,{\mathrm e}^{4}-48 \,{\mathrm e}^{2}+3 x +34}{x^{2}-2 x +1}\)	\(44\)
risch	\(-x +\frac {4 \,{\mathrm e}^{8}}{x^{2}-2 x +1}-\frac {16 \,{\mathrm e}^{6}}{x^{2}-2 x +1}+\frac {40 \,{\mathrm e}^{4}}{x^{2}-2 x +1}-\frac {48 \,{\mathrm e}^{2}}{x^{2}-2 x +1}+\frac {36}{x^{2}-2 x +1}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(2)^4+32*exp(2)^3-80*exp(2)^2+96*exp(2)-x^3+3*x^2-3*x-71)/(x^3-3*x^2+3*x-1),x,method=_RETURNVERBOSE
)

[Out]

-x-1/2*(-72+96*exp(2)-80*exp(4)-8*exp(8)+32*exp(6))/(x-1)^2

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 32, normalized size = 1.28 \begin {gather*} -x + \frac {4 \, {\left (e^{8} - 4 \, e^{6} + 10 \, e^{4} - 12 \, e^{2} + 9\right )}}{x^{2} - 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)^4+32*exp(2)^3-80*exp(2)^2+96*exp(2)-x^3+3*x^2-3*x-71)/(x^3-3*x^2+3*x-1),x, algorithm="max
ima")

[Out]

-x + 4*(e^8 - 4*e^6 + 10*e^4 - 12*e^2 + 9)/(x^2 - 2*x + 1)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 28, normalized size = 1.12 \begin {gather*} \frac {40\,{\mathrm {e}}^4-48\,{\mathrm {e}}^2-16\,{\mathrm {e}}^6+4\,{\mathrm {e}}^8+36}{{\left (x-1\right )}^2}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - 96*exp(2) + 80*exp(4) - 32*exp(6) + 8*exp(8) - 3*x^2 + x^3 + 71)/(3*x - 3*x^2 + x^3 - 1),x)

[Out]

(40*exp(4) - 48*exp(2) - 16*exp(6) + 4*exp(8) + 36)/(x - 1)^2 - x

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 32, normalized size = 1.28 \begin {gather*} - x - \frac {- 4 e^{8} - 40 e^{4} - 36 + 48 e^{2} + 16 e^{6}}{x^{2} - 2 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)**4+32*exp(2)**3-80*exp(2)**2+96*exp(2)-x**3+3*x**2-3*x-71)/(x**3-3*x**2+3*x-1),x)

[Out]

-x - (-4*exp(8) - 40*exp(4) - 36 + 48*exp(2) + 16*exp(6))/(x**2 - 2*x + 1)

________________________________________________________________________________________