3.48.34 \(\int \frac {80 x^2-10 x^4+e^4 (-64+96 x-180 x^2+204 x^3-105 x^4+24 x^5-2 x^6)}{e^4 (128 x^2-192 x^3+104 x^4-24 x^5+2 x^6)} \, dx\)

Optimal. Leaf size=28 \[ -4+\frac {1}{2 x}-x+\frac {5 x}{e^4 (-4+x) (-2+x)} \]

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Rubi [A]  time = 0.07, antiderivative size = 35, normalized size of antiderivative = 1.25, number of steps used = 3, number of rules used = 2, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12, 2074} \begin {gather*} -x+\frac {5}{e^4 (2-x)}-\frac {10}{e^4 (4-x)}+\frac {1}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(80*x^2 - 10*x^4 + E^4*(-64 + 96*x - 180*x^2 + 204*x^3 - 105*x^4 + 24*x^5 - 2*x^6))/(E^4*(128*x^2 - 192*x^
3 + 104*x^4 - 24*x^5 + 2*x^6)),x]

[Out]

5/(E^4*(2 - x)) - 10/(E^4*(4 - x)) + 1/(2*x) - x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {80 x^2-10 x^4+e^4 \left (-64+96 x-180 x^2+204 x^3-105 x^4+24 x^5-2 x^6\right )}{128 x^2-192 x^3+104 x^4-24 x^5+2 x^6} \, dx}{e^4}\\ &=\frac {\int \left (-e^4-\frac {10}{(-4+x)^2}+\frac {5}{(-2+x)^2}-\frac {e^4}{2 x^2}\right ) \, dx}{e^4}\\ &=\frac {5}{e^4 (2-x)}-\frac {10}{e^4 (4-x)}+\frac {1}{2 x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 27, normalized size = 0.96 \begin {gather*} \frac {1}{2 x}-x+\frac {5 x}{e^4 \left (8-6 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80*x^2 - 10*x^4 + E^4*(-64 + 96*x - 180*x^2 + 204*x^3 - 105*x^4 + 24*x^5 - 2*x^6))/(E^4*(128*x^2 -
192*x^3 + 104*x^4 - 24*x^5 + 2*x^6)),x]

[Out]

1/(2*x) - x + (5*x)/(E^4*(8 - 6*x + x^2))

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fricas [A]  time = 0.52, size = 48, normalized size = 1.71 \begin {gather*} \frac {{\left (10 \, x^{2} - {\left (2 \, x^{4} - 12 \, x^{3} + 15 \, x^{2} + 6 \, x - 8\right )} e^{4}\right )} e^{\left (-4\right )}}{2 \, {\left (x^{3} - 6 \, x^{2} + 8 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+24*x^5-105*x^4+204*x^3-180*x^2+96*x-64)*exp(4)-10*x^4+80*x^2)/(2*x^6-24*x^5+104*x^4-192*x^3
+128*x^2)/exp(4),x, algorithm="fricas")

[Out]

1/2*(10*x^2 - (2*x^4 - 12*x^3 + 15*x^2 + 6*x - 8)*e^4)*e^(-4)/(x^3 - 6*x^2 + 8*x)

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giac [A]  time = 0.14, size = 47, normalized size = 1.68 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x e^{4} - \frac {x^{2} e^{4} + 10 \, x^{2} - 6 \, x e^{4} + 8 \, e^{4}}{x^{3} - 6 \, x^{2} + 8 \, x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+24*x^5-105*x^4+204*x^3-180*x^2+96*x-64)*exp(4)-10*x^4+80*x^2)/(2*x^6-24*x^5+104*x^4-192*x^3
+128*x^2)/exp(4),x, algorithm="giac")

[Out]

-1/2*(2*x*e^4 - (x^2*e^4 + 10*x^2 - 6*x*e^4 + 8*e^4)/(x^3 - 6*x^2 + 8*x))*e^(-4)

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maple [A]  time = 0.08, size = 33, normalized size = 1.18

method	result	size
default	\(\frac {{\mathrm e}^{-4} \left (-2 x \,{\mathrm e}^{4}-\frac {10}{x -2}+\frac {20}{x -4}+\frac {{\mathrm e}^{4}}{x}\right )}{2}\)	\(33\)
norman	\(\frac {4-51 x -x^{4}-\frac {\left (-57 \,{\mathrm e}^{4}-10\right ) {\mathrm e}^{-4} x^{2}}{2}}{x \left (x^{2}-6 x +8\right )}\)	\(40\)
risch	\(-x +\frac {{\mathrm e}^{-4} \left (\left (\frac {{\mathrm e}^{4}}{2}+5\right ) x^{2}-3 x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{4}\right )}{x \left (x^{2}-6 x +8\right )}\)	\(41\)
gosper	\(-\frac {\left (2 x^{4} {\mathrm e}^{4}-57 x^{2} {\mathrm e}^{4}+102 x \,{\mathrm e}^{4}-10 x^{2}-8 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{2 x \left (x^{2}-6 x +8\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^6+24*x^5-105*x^4+204*x^3-180*x^2+96*x-64)*exp(4)-10*x^4+80*x^2)/(2*x^6-24*x^5+104*x^4-192*x^3+128*x
^2)/exp(4),x,method=_RETURNVERBOSE)

[Out]

1/2/exp(4)*(-2*x*exp(4)-10/(x-2)+20/(x-4)+exp(4)/x)

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maxima [A]  time = 0.36, size = 44, normalized size = 1.57 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x e^{4} - \frac {x^{2} {\left (e^{4} + 10\right )} - 6 \, x e^{4} + 8 \, e^{4}}{x^{3} - 6 \, x^{2} + 8 \, x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^6+24*x^5-105*x^4+204*x^3-180*x^2+96*x-64)*exp(4)-10*x^4+80*x^2)/(2*x^6-24*x^5+104*x^4-192*x^3
+128*x^2)/exp(4),x, algorithm="maxima")

[Out]

-1/2*(2*x*e^4 - (x^2*(e^4 + 10) - 6*x*e^4 + 8*e^4)/(x^3 - 6*x^2 + 8*x))*e^(-4)

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mupad [B]  time = 3.33, size = 40, normalized size = 1.43 \begin {gather*} -x-\frac {\left (\frac {{\mathrm {e}}^{-4}\,\left (15\,{\mathrm {e}}^4-10\right )}{2}-8\right )\,x^2+3\,x-4}{x\,\left (x^2-6\,x+8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*(exp(4)*(180*x^2 - 96*x - 204*x^3 + 105*x^4 - 24*x^5 + 2*x^6 + 64) - 80*x^2 + 10*x^4))/(128*x^2
- 192*x^3 + 104*x^4 - 24*x^5 + 2*x^6),x)

[Out]

- x - (3*x + x^2*((exp(-4)*(15*exp(4) - 10))/2 - 8) - 4)/(x*(x^2 - 6*x + 8))

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sympy [B]  time = 0.88, size = 48, normalized size = 1.71 \begin {gather*} - x - \frac {x^{2} \left (- e^{4} - 10\right ) + 6 x e^{4} - 8 e^{4}}{2 x^{3} e^{4} - 12 x^{2} e^{4} + 16 x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**6+24*x**5-105*x**4+204*x**3-180*x**2+96*x-64)*exp(4)-10*x**4+80*x**2)/(2*x**6-24*x**5+104*x*
*4-192*x**3+128*x**2)/exp(4),x)

[Out]

-x - (x**2*(-exp(4) - 10) + 6*x*exp(4) - 8*exp(4))/(2*x**3*exp(4) - 12*x**2*exp(4) + 16*x*exp(4))

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