3.21.73 \(\int \frac {1}{320} e^{-x} (-10000+10400 x+(400 x-204 x^2) \log (x^2)+(-3 x^2+x^3) \log ^2(x^2)) \, dx\)

Optimal. Leaf size=31 \[ 5+\frac {1}{4} \left (-4-\frac {1}{5} e^{-x} x \left (-25+\frac {1}{4} x \log \left (x^2\right )\right )^2\right ) \]

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Rubi [F]  time = 0.58, 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 {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10000 + 10400*x + (400*x - 204*x^2)*Log[x^2] + (-3*x^2 + x^3)*Log[x^2]^2)/(320*E^x),x]

[Out]

3/(40*E^x) - (1249*x)/(40*E^x) - ExpIntegralEi[-x]/20 + Log[x^2]/(40*E^x) + (x*Log[x^2])/(40*E^x) + (51*x^2*Lo
g[x^2])/(80*E^x) - (3*Defer[Int][(x^2*Log[x^2]^2)/E^x, x])/320 + Defer[Int][(x^3*Log[x^2]^2)/E^x, x]/320

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{320} \int e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx\\ &=\frac {1}{320} \int \left (-10000 e^{-x}+10400 e^{-x} x-4 e^{-x} x (-100+51 x) \log \left (x^2\right )+e^{-x} (-3+x) x^2 \log ^2\left (x^2\right )\right ) \, dx\\ &=\frac {1}{320} \int e^{-x} (-3+x) x^2 \log ^2\left (x^2\right ) \, dx-\frac {1}{80} \int e^{-x} x (-100+51 x) \log \left (x^2\right ) \, dx-\frac {125}{4} \int e^{-x} \, dx+\frac {65}{2} \int e^{-x} x \, dx\\ &=\frac {125 e^{-x}}{4}-\frac {65 e^{-x} x}{2}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int \left (-3 e^{-x} x^2 \log ^2\left (x^2\right )+e^{-x} x^3 \log ^2\left (x^2\right )\right ) \, dx+\frac {1}{80} \int \frac {2 e^{-x} \left (-2-2 x-51 x^2\right )}{x} \, dx+\frac {65}{2} \int e^{-x} \, dx\\ &=-\frac {5 e^{-x}}{4}-\frac {65 e^{-x} x}{2}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int e^{-x} x^3 \log ^2\left (x^2\right ) \, dx-\frac {3}{320} \int e^{-x} x^2 \log ^2\left (x^2\right ) \, dx+\frac {1}{40} \int \frac {e^{-x} \left (-2-2 x-51 x^2\right )}{x} \, dx\\ &=-\frac {5 e^{-x}}{4}-\frac {65 e^{-x} x}{2}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int e^{-x} x^3 \log ^2\left (x^2\right ) \, dx-\frac {3}{320} \int e^{-x} x^2 \log ^2\left (x^2\right ) \, dx+\frac {1}{40} \int \left (-2 e^{-x}-\frac {2 e^{-x}}{x}-51 e^{-x} x\right ) \, dx\\ &=-\frac {5 e^{-x}}{4}-\frac {65 e^{-x} x}{2}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int e^{-x} x^3 \log ^2\left (x^2\right ) \, dx-\frac {3}{320} \int e^{-x} x^2 \log ^2\left (x^2\right ) \, dx-\frac {1}{20} \int e^{-x} \, dx-\frac {1}{20} \int \frac {e^{-x}}{x} \, dx-\frac {51}{40} \int e^{-x} x \, dx\\ &=-\frac {6 e^{-x}}{5}-\frac {1249 e^{-x} x}{40}-\frac {\text {Ei}(-x)}{20}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int e^{-x} x^3 \log ^2\left (x^2\right ) \, dx-\frac {3}{320} \int e^{-x} x^2 \log ^2\left (x^2\right ) \, dx-\frac {51}{40} \int e^{-x} \, dx\\ &=\frac {3 e^{-x}}{40}-\frac {1249 e^{-x} x}{40}-\frac {\text {Ei}(-x)}{20}+\frac {1}{40} e^{-x} \log \left (x^2\right )+\frac {1}{40} e^{-x} x \log \left (x^2\right )+\frac {51}{80} e^{-x} x^2 \log \left (x^2\right )+\frac {1}{320} \int e^{-x} x^3 \log ^2\left (x^2\right ) \, dx-\frac {3}{320} \int e^{-x} x^2 \log ^2\left (x^2\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 20, normalized size = 0.65 \begin {gather*} -\frac {1}{320} e^{-x} x \left (-100+x \log \left (x^2\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10000 + 10400*x + (400*x - 204*x^2)*Log[x^2] + (-3*x^2 + x^3)*Log[x^2]^2)/(320*E^x),x]

[Out]

-1/320*(x*(-100 + x*Log[x^2])^2)/E^x

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fricas [A]  time = 0.67, size = 36, normalized size = 1.16 \begin {gather*} -\frac {1}{320} \, x^{3} e^{\left (-x\right )} \log \left (x^{2}\right )^{2} + \frac {5}{8} \, x^{2} e^{\left (-x\right )} \log \left (x^{2}\right ) - \frac {125}{4} \, x e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x-10000)/exp(x),x, algorithm="fricas")

[Out]

-1/320*x^3*e^(-x)*log(x^2)^2 + 5/8*x^2*e^(-x)*log(x^2) - 125/4*x*e^(-x)

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giac [A]  time = 2.64, size = 36, normalized size = 1.16 \begin {gather*} -\frac {1}{320} \, x^{3} e^{\left (-x\right )} \log \left (x^{2}\right )^{2} + \frac {5}{8} \, x^{2} e^{\left (-x\right )} \log \left (x^{2}\right ) - \frac {125}{4} \, x e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x-10000)/exp(x),x, algorithm="giac")

[Out]

-1/320*x^3*e^(-x)*log(x^2)^2 + 5/8*x^2*e^(-x)*log(x^2) - 125/4*x*e^(-x)

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maple [C]  time = 0.08, size = 254, normalized size = 8.19

method	result	size
risch	\(-\frac {x^{3} {\mathrm e}^{-x} \ln \relax (x )^{2}}{80}+\frac {i x^{2} \left (x \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+x \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-200 i\right ) {\mathrm e}^{-x} \ln \relax (x )}{160}+\frac {\left (-40000 x -400 i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+800 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-400 i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi ^{2} x^{3} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} x^{3} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}+6 \pi ^{2} x^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}-4 \pi ^{2} x^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} x^{3} \mathrm {csgn}\left (i x^{2}\right )^{6}\right ) {\mathrm e}^{-x}}{1280}\)	\(254\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/320*((x^3-3*x^2)*ln(x^2)^2+(-204*x^2+400*x)*ln(x^2)+10400*x-10000)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-1/80*x^3*exp(-x)*ln(x)^2+1/160*I*x^2*(x*Pi*csgn(I*x)^2*csgn(I*x^2)-2*x*Pi*csgn(I*x)*csgn(I*x^2)^2+x*Pi*csgn(I
*x^2)^3-200*I)*exp(-x)*ln(x)+1/1280*(-40000*x-400*I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+800*I*Pi*x^2*csgn(I*x)*csgn
(I*x^2)^2-400*I*Pi*x^2*csgn(I*x^2)^3+Pi^2*x^3*csgn(I*x)^4*csgn(I*x^2)^2-4*Pi^2*x^3*csgn(I*x)^3*csgn(I*x^2)^3+6
*Pi^2*x^3*csgn(I*x)^2*csgn(I*x^2)^4-4*Pi^2*x^3*csgn(I*x)*csgn(I*x^2)^5+Pi^2*x^3*csgn(I*x^2)^6)*exp(-x)

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maxima [B]  time = 0.44, size = 42, normalized size = 1.35 \begin {gather*} -\frac {1}{80} \, {\left (x^{3} \log \relax (x)^{2} - 100 \, x^{2} \log \relax (x) - 100 \, x - 100\right )} e^{\left (-x\right )} - \frac {65}{2} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {125}{4} \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x-10000)/exp(x),x, algorithm="maxima")

[Out]

-1/80*(x^3*log(x)^2 - 100*x^2*log(x) - 100*x - 100)*e^(-x) - 65/2*(x + 1)*e^(-x) + 125/4*e^(-x)

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mupad [B]  time = 1.28, size = 17, normalized size = 0.55 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-x}\,{\left (x\,\ln \left (x^2\right )-100\right )}^2}{320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*((65*x)/2 + (log(x^2)*(400*x - 204*x^2))/320 - (log(x^2)^2*(3*x^2 - x^3))/320 - 125/4),x)

[Out]

-(x*exp(-x)*(x*log(x^2) - 100)^2)/320

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sympy [A]  time = 0.38, size = 27, normalized size = 0.87 \begin {gather*} \frac {\left (- x^{3} \log {\left (x^{2} \right )}^{2} + 200 x^{2} \log {\left (x^{2} \right )} - 10000 x\right ) e^{- x}}{320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/320*((x**3-3*x**2)*ln(x**2)**2+(-204*x**2+400*x)*ln(x**2)+10400*x-10000)/exp(x),x)

[Out]

(-x**3*log(x**2)**2 + 200*x**2*log(x**2) - 10000*x)*exp(-x)/320

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