3.104.8 \(\int \frac {e^{-x} (24-8 e+8 x+e^x (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e (192+16 x-16 x^2))+(12-12 x-4 x^2+e (-4+4 x)+e^x (-144-16 e^2-96 x-16 x^2+e (96+32 x))) \log (x^2))}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx\)

Optimal. Leaf size=32 \[ 4 x \left (\frac {5}{x}+x-\left (4+\frac {e^{-x}}{-3+e-x}\right ) \log \left (x^2\right )\right ) \]

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Rubi [F]  time = 4.47, 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 (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24 - 8*E + 8*x + E^x*(-288 - 120*x + 16*x^2 + 8*x^3 + E^2*(-32 + 8*x) + E*(192 + 16*x - 16*x^2)) + (12 -
12*x - 4*x^2 + E*(-4 + 4*x) + E^x*(-144 - 16*E^2 - 96*x - 16*x^2 + E*(96 + 32*x)))*Log[x^2])/(E^x*(9 + E^2 + E
*(-6 - 2*x) + 6*x + x^2)),x]

[Out]

4*x^2 + 8*ExpIntegralEi[-x] + 8*(3 - E)*ExpIntegralEi[-x] - 8*(4 - E)*ExpIntegralEi[-x] + (4*Log[x^2])/E^x - 1
6*x*Log[x^2] - (4*(3 - E)*Log[x^2])/(E^x*(3 - E + x)) + 8*(3 - E)*E^(3 - E)*Defer[Int][ExpIntegralEi[-3 + E -
x]/x, x] + 8*(3 - E)*(4 - E)*E^(3 - E)*Defer[Int][ExpIntegralEi[-3 + E - x]/x, x] - 8*(3 - E)*(5 - E)*E^(3 - E
)*Defer[Int][ExpIntegralEi[-3 + E - x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (8 (-3+e-x) \left (-1+e^{1+x} (-4+x)+e^x \left (12+x-x^2\right )\right )-4 \left (-3+e+4 e^{2+x}+3 x-e x+x^2-8 e^{1+x} (3+x)+4 e^x (3+x)^2\right ) \log \left (x^2\right )\right )}{(3-e+x)^2} \, dx\\ &=\int \left (-\frac {8 e^{-x}}{-3+e-x}+8 \left (-4+x-2 \log \left (x^2\right )\right )+\frac {12 \left (1-\frac {e}{3}\right ) e^{-x} \log \left (x^2\right )}{(-3+e-x)^2}-\frac {12 \left (1-\frac {e}{3}\right ) e^{-x} x \log \left (x^2\right )}{(-3+e-x)^2}-\frac {4 e^{-x} x^2 \log \left (x^2\right )}{(-3+e-x)^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-x} x^2 \log \left (x^2\right )}{(-3+e-x)^2} \, dx\right )-8 \int \frac {e^{-x}}{-3+e-x} \, dx+8 \int \left (-4+x-2 \log \left (x^2\right )\right ) \, dx+(4 (3-e)) \int \frac {e^{-x} \log \left (x^2\right )}{(-3+e-x)^2} \, dx-(4 (3-e)) \int \frac {e^{-x} x \log \left (x^2\right )}{(-3+e-x)^2} \, dx\\ &=-32 x+4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+4 e^{-x} \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}+4 \int \frac {e^{-e-x} \left (-2 e^e \left (12-7 e+e^2+x\right )+2 e^{3+x} \left (15-8 e+e^2\right ) (-3+e-x) \text {Ei}(-3+e-x)\right )}{x (3-e+x)} \, dx-16 \int \log \left (x^2\right ) \, dx-(4 (3-e)) \int \frac {2 \left (\frac {e^{-x}}{-3+e-x}-e^{3-e} \text {Ei}(-3+e-x)\right )}{x} \, dx+(4 (3-e)) \int \frac {e^{-e-x} \left (-2 (-3+e) e^e+2 (-4+e) e^{3+x} (-3+e-x) \text {Ei}(-3+e-x)\right )}{x (3-e+x)} \, dx\\ &=4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}+4 \int \left (-\frac {2 e^{-x} \left (12-7 e+e^2+x\right )}{x (3-e+x)}-\frac {2 (-5+e) (-3+e) e^{3-e} \text {Ei}(-3+e-x)}{x}\right ) \, dx+(4 (3-e)) \int \left (\frac {2 (-3+e) e^{-x}}{(-3+e-x) x}-\frac {2 (-4+e) e^{3-e} \text {Ei}(-3+e-x)}{x}\right ) \, dx-(8 (3-e)) \int \frac {\frac {e^{-x}}{-3+e-x}-e^{3-e} \text {Ei}(-3+e-x)}{x} \, dx\\ &=4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}-8 \int \frac {e^{-x} \left (12-7 e+e^2+x\right )}{x (3-e+x)} \, dx-(8 (3-e)) \int \left (\frac {e^{-x}}{(-3+e-x) x}-\frac {e^{3-e} \text {Ei}(-3+e-x)}{x}\right ) \, dx-\left (8 (3-e)^2\right ) \int \frac {e^{-x}}{(-3+e-x) x} \, dx+\left (8 (3-e) (4-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx-\left (8 (3-e) (5-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx\\ &=4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}-8 \int \left (\frac {(3-e) e^{-x}}{-3+e-x}+\frac {(4-e) e^{-x}}{x}\right ) \, dx-(8 (3-e)) \int \frac {e^{-x}}{(-3+e-x) x} \, dx-\left (8 (3-e)^2\right ) \int \left (\frac {e^{-x}}{(-3+e) (-3+e-x)}+\frac {e^{-x}}{(-3+e) x}\right ) \, dx+\left (8 (3-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx+\left (8 (3-e) (4-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx-\left (8 (3-e) (5-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx\\ &=4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}-(8 (3-e)) \int \left (\frac {e^{-x}}{(-3+e) (-3+e-x)}+\frac {e^{-x}}{(-3+e) x}\right ) \, dx+(8 (3-e)) \int \frac {e^{-x}}{x} \, dx-(8 (4-e)) \int \frac {e^{-x}}{x} \, dx+\left (8 (3-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx+\left (8 (3-e) (4-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx-\left (8 (3-e) (5-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx\\ &=4 x^2+8 e^{3-e} \text {Ei}(-3+e-x)+8 (3-e) \text {Ei}(-x)-8 (4-e) \text {Ei}(-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}+8 \int \frac {e^{-x}}{-3+e-x} \, dx+8 \int \frac {e^{-x}}{x} \, dx+\left (8 (3-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx+\left (8 (3-e) (4-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx-\left (8 (3-e) (5-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx\\ &=4 x^2+8 \text {Ei}(-x)+8 (3-e) \text {Ei}(-x)-8 (4-e) \text {Ei}(-x)+4 e^{-x} \log \left (x^2\right )-16 x \log \left (x^2\right )-\frac {4 (3-e) e^{-x} \log \left (x^2\right )}{3-e+x}+\left (8 (3-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx+\left (8 (3-e) (4-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx-\left (8 (3-e) (5-e) e^{3-e}\right ) \int \frac {\text {Ei}(-3+e-x)}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 36, normalized size = 1.12 \begin {gather*} 4 x^2-\frac {4 x \left (12-4 e-e^{-x}+4 x\right ) \log \left (x^2\right )}{3-e+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 - 8*E + 8*x + E^x*(-288 - 120*x + 16*x^2 + 8*x^3 + E^2*(-32 + 8*x) + E*(192 + 16*x - 16*x^2)) +
(12 - 12*x - 4*x^2 + E*(-4 + 4*x) + E^x*(-144 - 16*E^2 - 96*x - 16*x^2 + E*(96 + 32*x)))*Log[x^2])/(E^x*(9 + E
^2 + E*(-6 - 2*x) + 6*x + x^2)),x]

[Out]

4*x^2 - (4*x*(12 - 4*E - E^(-x) + 4*x)*Log[x^2])/(3 - E + x)

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fricas [A]  time = 0.70, size = 61, normalized size = 1.91 \begin {gather*} \frac {4 \, {\left ({\left (x^{3} - x^{2} e + 3 \, x^{2}\right )} e^{x} - {\left (4 \, {\left (x^{2} - x e + 3 \, x\right )} e^{x} - x\right )} \log \left (x^{2}\right )\right )} e^{\left (-x\right )}}{x - e + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(4*x-4)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*
x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*ex
p(1)+x^2+6*x+9)/exp(x),x, algorithm="fricas")

[Out]

4*((x^3 - x^2*e + 3*x^2)*e^x - (4*(x^2 - x*e + 3*x)*e^x - x)*log(x^2))*e^(-x)/(x - e + 3)

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giac [A]  time = 0.20, size = 62, normalized size = 1.94 \begin {gather*} \frac {4 \, {\left (x^{3} - x^{2} e - 4 \, x^{2} \log \left (x^{2}\right ) + 4 \, x e \log \left (x^{2}\right ) + x e^{\left (-x\right )} \log \left (x^{2}\right ) + 3 \, x^{2} - 12 \, x \log \left (x^{2}\right )\right )}}{x - e + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(4*x-4)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*
x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*ex
p(1)+x^2+6*x+9)/exp(x),x, algorithm="giac")

[Out]

4*(x^3 - x^2*e - 4*x^2*log(x^2) + 4*x*e*log(x^2) + x*e^(-x)*log(x^2) + 3*x^2 - 12*x*log(x^2))/(x - e + 3)

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maple [B]  time = 0.43, size = 67, normalized size = 2.09

method	result	size
norman	\(\frac {\left (\left (4 \,{\mathrm e}-12\right ) x^{2} {\mathrm e}^{x}+\left (-16 \,{\mathrm e}+48\right ) x \,{\mathrm e}^{x} \ln \left (x^{2}\right )-4 x \ln \left (x^{2}\right )-4 \,{\mathrm e}^{x} x^{3}+16 x^{2} {\mathrm e}^{x} \ln \left (x^{2}\right )\right ) {\mathrm e}^{-x}}{{\mathrm e}-3-x}\)	\(67\)
default	\(\frac {4 \left (x^{2} \left (\ln \left (x^{2}\right )-2 \ln \relax (x )\right )+\left (-{\mathrm e} \left (\ln \left (x^{2}\right )-2 \ln \relax (x )\right )+3 \ln \left (x^{2}\right )-6 \ln \relax (x )\right ) x +\left (-2 \,{\mathrm e}+6\right ) x \ln \relax (x )+2 x^{2} \ln \relax (x )\right ) {\mathrm e}^{-x}}{\left ({\mathrm e}-3-x \right )^{2}}+4 x^{2}-16 x \ln \left (x^{2}\right )\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(4*x-4)*exp(1)-4*x^2-12*x+12)*ln(x^2)+((8*x-32)*e
xp(1)^2+(-16*x^2+16*x+192)*exp(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*exp(1)+x^
2+6*x+9)/exp(x),x,method=_RETURNVERBOSE)

[Out]

((4*exp(1)-12)*x^2*exp(x)+(-16*exp(1)+48)*x*exp(x)*ln(x^2)-4*x*ln(x^2)-4*exp(x)*x^3+16*x^2*exp(x)*ln(x^2))/(ex
p(1)-3-x)/exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 8 \, {\left (e - 3\right )} \int \frac {e^{\left (-x\right )}}{x^{2} - 2 \, x {\left (e - 3\right )} + e^{2} - 6 \, e + 9}\,{d x} + \frac {8 \, e^{\left (-e + 4\right )} E_{2}\left (x - e + 3\right )}{x - e + 3} - \frac {24 \, e^{\left (-e + 3\right )} E_{2}\left (x - e + 3\right )}{x - e + 3} + \frac {4 \, {\left (x^{3} - x^{2} {\left (e - 3\right )} + 2 \, x e^{\left (-x\right )} \log \relax (x) - 8 \, {\left (x^{2} - x {\left (e - 3\right )}\right )} \log \relax (x)\right )}}{x - e + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(4*x-4)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*
x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*ex
p(1)+x^2+6*x+9)/exp(x),x, algorithm="maxima")

[Out]

8*(e - 3)*integrate(e^(-x)/(x^2 - 2*x*(e - 3) + e^2 - 6*e + 9), x) + 8*e^(-e + 4)*exp_integral_e(2, x - e + 3)
/(x - e + 3) - 24*e^(-e + 3)*exp_integral_e(2, x - e + 3)/(x - e + 3) + 4*(x^3 - x^2*(e - 3) + 2*x*e^(-x)*log(
x) - 8*(x^2 - x*(e - 3))*log(x))/(x - e + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-x}\,\left (8\,x-8\,\mathrm {e}+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (-16\,x^2+16\,x+192\right )-120\,x+16\,x^2+8\,x^3+{\mathrm {e}}^2\,\left (8\,x-32\right )-288\right )-\ln \left (x^2\right )\,\left (12\,x+{\mathrm {e}}^x\,\left (96\,x+16\,{\mathrm {e}}^2+16\,x^2-\mathrm {e}\,\left (32\,x+96\right )+144\right )+4\,x^2-\mathrm {e}\,\left (4\,x-4\right )-12\right )+24\right )}{6\,x+{\mathrm {e}}^2+x^2-\mathrm {e}\,\left (2\,x+6\right )+9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(8*x - 8*exp(1) + exp(x)*(exp(1)*(16*x - 16*x^2 + 192) - 120*x + 16*x^2 + 8*x^3 + exp(2)*(8*x - 3
2) - 288) - log(x^2)*(12*x + exp(x)*(96*x + 16*exp(2) + 16*x^2 - exp(1)*(32*x + 96) + 144) + 4*x^2 - exp(1)*(4
*x - 4) - 12) + 24))/(6*x + exp(2) + x^2 - exp(1)*(2*x + 6) + 9),x)

[Out]

int((exp(-x)*(8*x - 8*exp(1) + exp(x)*(exp(1)*(16*x - 16*x^2 + 192) - 120*x + 16*x^2 + 8*x^3 + exp(2)*(8*x - 3
2) - 288) - log(x^2)*(12*x + exp(x)*(96*x + 16*exp(2) + 16*x^2 - exp(1)*(32*x + 96) + 144) + 4*x^2 - exp(1)*(4
*x - 4) - 12) + 24))/(6*x + exp(2) + x^2 - exp(1)*(2*x + 6) + 9), x)

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sympy [A]  time = 0.48, size = 31, normalized size = 0.97 \begin {gather*} 4 x^{2} - 16 x \log {\left (x^{2} \right )} + \frac {4 x e^{- x} \log {\left (x^{2} \right )}}{x - e + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*exp(1)**2+(32*x+96)*exp(1)-16*x**2-96*x-144)*exp(x)+(4*x-4)*exp(1)-4*x**2-12*x+12)*ln(x**2)+(
(8*x-32)*exp(1)**2+(-16*x**2+16*x+192)*exp(1)+8*x**3+16*x**2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)**2+(-2
*x-6)*exp(1)+x**2+6*x+9)/exp(x),x)

[Out]

4*x**2 - 16*x*log(x**2) + 4*x*exp(-x)*log(x**2)/(x - E + 3)

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