3.88.66 \(\int \frac {4 e^4+e^2 (-1-8 x)+4 x^2+e^4 (e^4-2 e^2 x+x^2)}{e^4-2 e^2 x+x^2} \, dx\)

Optimal. Leaf size=16 \[ x \left (4+e^4+\frac {1}{-e^2+x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {27, 1850} \begin {gather*} \left (4+e^4\right ) x-\frac {e^2}{e^2-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^4 + E^2*(-1 - 8*x) + 4*x^2 + E^4*(E^4 - 2*E^2*x + x^2))/(E^4 - 2*E^2*x + x^2),x]

[Out]

-(E^2/(E^2 - x)) + (4 + E^4)*x

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.81 \begin {gather*} -\frac {e^2}{e^2-x}-\left (4+e^4\right ) \left (e^2-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^4 + E^2*(-1 - 8*x) + 4*x^2 + E^4*(E^4 - 2*E^2*x + x^2))/(E^4 - 2*E^2*x + x^2),x]

[Out]

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

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fricas [B]  time = 0.54, size = 35, normalized size = 2.19 \begin {gather*} \frac {x^{2} e^{4} + 4 \, x^{2} - x e^{6} - {\left (4 \, x - 1\right )} e^{2}}{x - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(2)^2-2*exp(2)*x+x^2)*exp(4)+4*exp(2)^2+(-8*x-1)*exp(2)+4*x^2)/(exp(2)^2-2*exp(2)*x+x^2),x, alg
orithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(2)^2-2*exp(2)*x+x^2)*exp(4)+4*exp(2)^2+(-8*x-1)*exp(2)+4*x^2)/(exp(2)^2-2*exp(2)*x+x^2),x, alg
orithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: sageVARx*exp(4)+4*sageVARx-exp(2)*1/2/sq
rt(exp(2)^2-exp(4))*ln(sqrt((2*sageVARx-2*exp(2))^2+(-2*sqrt(-exp(2)^2+exp(4)))^2)/sqrt((2*sageVARx-2*exp(2))^
2+(2*sqrt(-exp(2)^2+e

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maple [A]  time = 0.52, size = 21, normalized size = 1.31

method	result	size
risch	\(x \,{\mathrm e}^{4}+4 x -\frac {{\mathrm e}^{2}}{{\mathrm e}^{2}-x}\)	\(21\)
norman	\(\frac {\left (-4-{\mathrm e}^{4}\right ) x^{2}+\left ({\mathrm e}^{4}\right )^{2}+4 \,{\mathrm e}^{4}-{\mathrm e}^{2}}{{\mathrm e}^{2}-x}\)	\(38\)
gosper	\(\frac {\left ({\mathrm e}^{4}\right )^{2}-x^{2} {\mathrm e}^{4}+4 \,{\mathrm e}^{4}-4 x^{2}-{\mathrm e}^{2}}{{\mathrm e}^{2}-x}\)	\(40\)
meijerg	\(\frac {4 x}{1-x \,{\mathrm e}^{-2}}+\left (-2 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{2}\right ) \left (\frac {x \,{\mathrm e}^{-2}}{1-x \,{\mathrm e}^{-2}}+\ln \left (1-x \,{\mathrm e}^{-2}\right )\right )-\left (4+{\mathrm e}^{4}\right ) {\mathrm e}^{2} \left (-\frac {x \,{\mathrm e}^{-2} \left (-3 x \,{\mathrm e}^{-2}+6\right )}{3 \left (1-x \,{\mathrm e}^{-2}\right )}-2 \ln \left (1-x \,{\mathrm e}^{-2}\right )\right )+\frac {{\mathrm e}^{4} x}{1-x \,{\mathrm e}^{-2}}-\frac {x \,{\mathrm e}^{-2}}{1-x \,{\mathrm e}^{-2}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(2)^2-2*exp(2)*x+x^2)*exp(4)+4*exp(2)^2+(-8*x-1)*exp(2)+4*x^2)/(exp(2)^2-2*exp(2)*x+x^2),x,method=_RE
TURNVERBOSE)

[Out]

x*exp(4)+4*x-exp(2)/(exp(2)-x)

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maxima [A]  time = 0.35, size = 18, normalized size = 1.12 \begin {gather*} x {\left (e^{4} + 4\right )} + \frac {e^{2}}{x - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(2)^2-2*exp(2)*x+x^2)*exp(4)+4*exp(2)^2+(-8*x-1)*exp(2)+4*x^2)/(exp(2)^2-2*exp(2)*x+x^2),x, alg
orithm="maxima")

[Out]

x*(e^4 + 4) + e^2/(x - e^2)

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mupad [B]  time = 0.13, size = 18, normalized size = 1.12 \begin {gather*} x\,\left ({\mathrm {e}}^4+4\right )+\frac {{\mathrm {e}}^2}{x-{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(4) + exp(4)*(exp(4) - 2*x*exp(2) + x^2) + 4*x^2 - exp(2)*(8*x + 1))/(exp(4) - 2*x*exp(2) + x^2),x)

[Out]

x*(exp(4) + 4) + exp(2)/(x - exp(2))

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sympy [A]  time = 0.12, size = 14, normalized size = 0.88 \begin {gather*} x \left (4 + e^{4}\right ) + \frac {e^{2}}{x - e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(2)**2-2*exp(2)*x+x**2)*exp(4)+4*exp(2)**2+(-8*x-1)*exp(2)+4*x**2)/(exp(2)**2-2*exp(2)*x+x**2),
x)

[Out]

x*(4 + exp(4)) + exp(2)/(x - exp(2))

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