3.85.59 \(\int \frac {e^{-x} (-8 x+2 x^2+4 x^3-x^4-x^5+(4-4 x-4 x^2+4 x^3+x^4-x^5) \log ^2(5))}{4-4 x^2+x^4} \, dx\)

Optimal. Leaf size=38 \[ -3-e^{-x} \left (-3-x+\frac {4-x^2}{2-x^2}-x \log ^2(5)\right ) \]

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Rubi [F]  time = 0.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-x} \left (-8 x+2 x^2+4 x^3-x^4-x^5+\left (4-4 x-4 x^2+4 x^3+x^4-x^5\right ) \log ^2(5)\right )}{4-4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x + 2*x^2 + 4*x^3 - x^4 - x^5 + (4 - 4*x - 4*x^2 + 4*x^3 + x^4 - x^5)*Log[5]^2)/(E^x*(4 - 4*x^2 + x^4)
),x]

[Out]

(E^Sqrt[2]*ExpIntegralEi[-Sqrt[2] - x])/Sqrt[2] - ExpIntegralEi[Sqrt[2] - x]/(Sqrt[2]*E^Sqrt[2]) + (1 - Log[5]
^2)/E^x + (1 + Log[5]^2)/E^x + (x*(1 + Log[5]^2))/E^x - 4*Defer[Int][x/(E^x*(-2 + x^2)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-8 x+2 x^2+4 x^3-x^4-x^5+\left (4-4 x-4 x^2+4 x^3+x^4-x^5\right ) \log ^2(5)\right )}{\left (-2+x^2\right )^2} \, dx\\ &=\int \left (-\frac {4 e^{-x} x}{\left (-2+x^2\right )^2}-\frac {2 e^{-x}}{-2+x^2}+e^{-x} \left (-1+\log ^2(5)\right )-e^{-x} x \left (1+\log ^2(5)\right )\right ) \, dx\\ &=-\left (2 \int \frac {e^{-x}}{-2+x^2} \, dx\right )-4 \int \frac {e^{-x} x}{\left (-2+x^2\right )^2} \, dx+\left (-1-\log ^2(5)\right ) \int e^{-x} x \, dx+\left (-1+\log ^2(5)\right ) \int e^{-x} \, dx\\ &=e^{-x} \left (1-\log ^2(5)\right )+e^{-x} x \left (1+\log ^2(5)\right )-2 \int \left (-\frac {e^{-x}}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^{-x}}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx-4 \int \frac {e^{-x} x}{\left (-2+x^2\right )^2} \, dx+\left (-1-\log ^2(5)\right ) \int e^{-x} \, dx\\ &=e^{-x} \left (1-\log ^2(5)\right )+e^{-x} \left (1+\log ^2(5)\right )+e^{-x} x \left (1+\log ^2(5)\right )-4 \int \frac {e^{-x} x}{\left (-2+x^2\right )^2} \, dx+\frac {\int \frac {e^{-x}}{\sqrt {2}-x} \, dx}{\sqrt {2}}+\frac {\int \frac {e^{-x}}{\sqrt {2}+x} \, dx}{\sqrt {2}}\\ &=\frac {e^{\sqrt {2}} \text {Ei}\left (-\sqrt {2}-x\right )}{\sqrt {2}}-\frac {e^{-\sqrt {2}} \text {Ei}\left (\sqrt {2}-x\right )}{\sqrt {2}}+e^{-x} \left (1-\log ^2(5)\right )+e^{-x} \left (1+\log ^2(5)\right )+e^{-x} x \left (1+\log ^2(5)\right )-4 \int \frac {e^{-x} x}{\left (-2+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 25, normalized size = 0.66 \begin {gather*} e^{-x} \left (2+\frac {2}{-2+x^2}+x \left (1+\log ^2(5)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*x + 2*x^2 + 4*x^3 - x^4 - x^5 + (4 - 4*x - 4*x^2 + 4*x^3 + x^4 - x^5)*Log[5]^2)/(E^x*(4 - 4*x^2
+ x^4)),x]

[Out]

(2 + 2/(-2 + x^2) + x*(1 + Log[5]^2))/E^x

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fricas [A]  time = 0.66, size = 37, normalized size = 0.97 \begin {gather*} \frac {{\left (x^{3} + {\left (x^{3} - 2 \, x\right )} \log \relax (5)^{2} + 2 \, x^{2} - 2 \, x - 2\right )} e^{\left (-x\right )}}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+x^4+4*x^3-4*x^2-4*x+4)*log(5)^2-x^5-x^4+4*x^3+2*x^2-8*x)/(x^4-4*x^2+4)/exp(x),x, algorithm="f
ricas")

[Out]

(x^3 + (x^3 - 2*x)*log(5)^2 + 2*x^2 - 2*x - 2)*e^(-x)/(x^2 - 2)

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giac [B]  time = 0.14, size = 62, normalized size = 1.63 \begin {gather*} \frac {x^{3} e^{\left (-x\right )} \log \relax (5)^{2} + x^{3} e^{\left (-x\right )} - 2 \, x e^{\left (-x\right )} \log \relax (5)^{2} + 2 \, x^{2} e^{\left (-x\right )} - 2 \, x e^{\left (-x\right )} - 2 \, e^{\left (-x\right )}}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+x^4+4*x^3-4*x^2-4*x+4)*log(5)^2-x^5-x^4+4*x^3+2*x^2-8*x)/(x^4-4*x^2+4)/exp(x),x, algorithm="g
iac")

[Out]

(x^3*e^(-x)*log(5)^2 + x^3*e^(-x) - 2*x*e^(-x)*log(5)^2 + 2*x^2*e^(-x) - 2*x*e^(-x) - 2*e^(-x))/(x^2 - 2)

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maple [A]  time = 0.12, size = 37, normalized size = 0.97

method	result	size
default	\({\mathrm e}^{-x}+\left (x +1\right ) {\mathrm e}^{-x}+\frac {2 \,{\mathrm e}^{-x}}{x^{2}-2}+\ln \relax (5)^{2} {\mathrm e}^{-x} x\)	\(37\)
norman	\(\frac {\left (-2+\left (-2 \ln \relax (5)^{2}-2\right ) x +\left (\ln \relax (5)^{2}+1\right ) x^{3}+2 x^{2}\right ) {\mathrm e}^{-x}}{x^{2}-2}\)	\(40\)
gosper	\(\frac {\left (x^{3} \ln \relax (5)^{2}-2 x \ln \relax (5)^{2}+x^{3}+2 x^{2}-2 x -2\right ) {\mathrm e}^{-x}}{x^{2}-2}\)	\(41\)
risch	\(\frac {\left (x^{3} \ln \relax (5)^{2}-2 x \ln \relax (5)^{2}+x^{3}+2 x^{2}-2 x -2\right ) {\mathrm e}^{-x}}{x^{2}-2}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^5+x^4+4*x^3-4*x^2-4*x+4)*ln(5)^2-x^5-x^4+4*x^3+2*x^2-8*x)/(x^4-4*x^2+4)/exp(x),x,method=_RETURNVERBOS
E)

[Out]

exp(-x)+(x+1)*exp(-x)+2*exp(-x)/(x^2-2)+ln(5)^2*exp(-x)*x

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maxima [A]  time = 0.70, size = 38, normalized size = 1.00 \begin {gather*} \frac {{\left ({\left (\log \relax (5)^{2} + 1\right )} x^{3} - 2 \, {\left (\log \relax (5)^{2} + 1\right )} x + 2 \, x^{2} - 2\right )} e^{\left (-x\right )}}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+x^4+4*x^3-4*x^2-4*x+4)*log(5)^2-x^5-x^4+4*x^3+2*x^2-8*x)/(x^4-4*x^2+4)/exp(x),x, algorithm="m
axima")

[Out]

((log(5)^2 + 1)*x^3 - 2*(log(5)^2 + 1)*x + 2*x^2 - 2)*e^(-x)/(x^2 - 2)

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mupad [B]  time = 5.29, size = 28, normalized size = 0.74 \begin {gather*} {\mathrm {e}}^{-x}\,\left (x+x\,{\ln \relax (5)}^2+2\right )+\frac {2\,{\mathrm {e}}^{-x}}{x^2-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(8*x - 2*x^2 - 4*x^3 + x^4 + x^5 + log(5)^2*(4*x + 4*x^2 - 4*x^3 - x^4 + x^5 - 4)))/(x^4 - 4*x^2
+ 4),x)

[Out]

exp(-x)*(x + x*log(5)^2 + 2) + (2*exp(-x))/(x^2 - 2)

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sympy [A]  time = 0.17, size = 37, normalized size = 0.97 \begin {gather*} \frac {\left (x^{3} + x^{3} \log {\relax (5 )}^{2} + 2 x^{2} - 2 x \log {\relax (5 )}^{2} - 2 x - 2\right ) e^{- x}}{x^{2} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**5+x**4+4*x**3-4*x**2-4*x+4)*ln(5)**2-x**5-x**4+4*x**3+2*x**2-8*x)/(x**4-4*x**2+4)/exp(x),x)

[Out]

(x**3 + x**3*log(5)**2 + 2*x**2 - 2*x*log(5)**2 - 2*x - 2)*exp(-x)/(x**2 - 2)

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