3.66.33 \(\int \frac {-384 x+96 x^2-12 x^3+e^{\frac {1}{4} (4 x+e^{-5+x} x-x^2)} (-48 x^2+60 x^3-36 x^4+6 x^5+e^{-5+x} (12 x^3+9 x^4-3 x^5))+(-48 x^2+12 x^3) \log (-4+x)}{-256+64 x+e^{\frac {1}{2} (4 x+e^{-5+x} x-x^2)} (-16 x^2+4 x^3)+(-128 x+32 x^2) \log (-4+x)+(-16 x^2+4 x^3) \log ^2(-4+x)+e^{\frac {1}{4} (4 x+e^{-5+x} x-x^2)} (-128 x+32 x^2+(-32 x^2+8 x^3) \log (-4+x))} \, dx\)

Optimal. Leaf size=34 \[ -1+\frac {3 x}{e^{\frac {1}{4} \left (4+e^{-5+x}-x\right ) x}+\frac {4}{x}+\log (-4+x)} \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-384*x + 96*x^2 - 12*x^3 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-48*x^2 + 60*x^3 - 36*x^4 + 6*x^5 + E^(-5 +
x)*(12*x^3 + 9*x^4 - 3*x^5)) + (-48*x^2 + 12*x^3)*Log[-4 + x])/(-256 + 64*x + E^((4*x + E^(-5 + x)*x - x^2)/2)
*(-16*x^2 + 4*x^3) + (-128*x + 32*x^2)*Log[-4 + x] + (-16*x^2 + 4*x^3)*Log[-4 + x]^2 + E^((4*x + E^(-5 + x)*x
- x^2)/4)*(-128*x + 32*x^2 + (-32*x^2 + 8*x^3)*Log[-4 + x])),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.49, size = 59, normalized size = 1.74 \begin {gather*} \frac {3 e^{\frac {x^2}{4}} x^2}{4 e^{\frac {x^2}{4}}+e^{x+\frac {1}{4} e^{-5+x} x} x+e^{\frac {x^2}{4}} x \log (-4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-384*x + 96*x^2 - 12*x^3 + E^((4*x + E^(-5 + x)*x - x^2)/4)*(-48*x^2 + 60*x^3 - 36*x^4 + 6*x^5 + E^
(-5 + x)*(12*x^3 + 9*x^4 - 3*x^5)) + (-48*x^2 + 12*x^3)*Log[-4 + x])/(-256 + 64*x + E^((4*x + E^(-5 + x)*x - x
^2)/2)*(-16*x^2 + 4*x^3) + (-128*x + 32*x^2)*Log[-4 + x] + (-16*x^2 + 4*x^3)*Log[-4 + x]^2 + E^((4*x + E^(-5 +
x)*x - x^2)/4)*(-128*x + 32*x^2 + (-32*x^2 + 8*x^3)*Log[-4 + x])),x]

[Out]

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

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fricas [A]  time = 0.78, size = 32, normalized size = 0.94 \begin {gather*} \frac {3 \, x^{2}}{x e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} + x \log \left (x - 4\right ) + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^5+9*x^4+12*x^3)*exp(x-5)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(12*x^3-4
8*x^2)*log(x-4)-12*x^3+96*x^2-384*x)/((4*x^3-16*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(x-4)+
32*x^2-128*x)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(4*x^3-16*x^2)*log(x-4)^2+(32*x^2-128*x)*log(x-4)+64*x-256),x, alg
orithm="fricas")

[Out]

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

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giac [B]  time = 26.58, size = 617, normalized size = 18.15 \begin {gather*} \frac {3 \, {\left (2 \, x^{6} e^{5} \log \left (x - 4\right ) - x^{6} e^{x} \log \left (x - 4\right ) - 12 \, x^{5} e^{5} \log \left (x - 4\right ) + 3 \, x^{5} e^{x} \log \left (x - 4\right ) + 8 \, x^{5} e^{5} - 4 \, x^{5} e^{x} + 16 \, x^{4} e^{5} \log \left (x - 4\right ) + 4 \, x^{4} e^{x} \log \left (x - 4\right ) - 44 \, x^{4} e^{5} + 12 \, x^{4} e^{x} + 48 \, x^{3} e^{5} + 16 \, x^{3} e^{x} + 64 \, x^{2} e^{5}\right )}}{2 \, x^{5} e^{5} \log \left (x - 4\right )^{2} - x^{5} e^{x} \log \left (x - 4\right )^{2} - x^{5} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} \log \left (x - 4\right ) + 2 \, x^{5} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} \log \left (x - 4\right ) - 12 \, x^{4} e^{5} \log \left (x - 4\right )^{2} + 3 \, x^{4} e^{x} \log \left (x - 4\right )^{2} + 16 \, x^{4} e^{5} \log \left (x - 4\right ) + 3 \, x^{4} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} \log \left (x - 4\right ) - 12 \, x^{4} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} \log \left (x - 4\right ) - 8 \, x^{4} e^{x} \log \left (x - 4\right ) + 16 \, x^{3} e^{5} \log \left (x - 4\right )^{2} + 4 \, x^{3} e^{x} \log \left (x - 4\right )^{2} - 4 \, x^{4} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} + 8 \, x^{4} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} - 92 \, x^{3} e^{5} \log \left (x - 4\right ) + 4 \, x^{3} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} \log \left (x - 4\right ) + 16 \, x^{3} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} \log \left (x - 4\right ) + 24 \, x^{3} e^{x} \log \left (x - 4\right ) + 32 \, x^{3} e^{5} + 12 \, x^{3} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} - 44 \, x^{3} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} - 16 \, x^{3} e^{x} + 112 \, x^{2} e^{5} \log \left (x - 4\right ) + 32 \, x^{2} e^{x} \log \left (x - 4\right ) - 176 \, x^{2} e^{5} + 16 \, x^{2} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + 2 \, x\right )} + 48 \, x^{2} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} + 48 \, x^{2} e^{x} + 64 \, x e^{5} \log \left (x - 4\right ) + 192 \, x e^{5} + 64 \, x e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x + 5\right )} + 64 \, x e^{x} + 256 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^5+9*x^4+12*x^3)*exp(x-5)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(12*x^3-4
8*x^2)*log(x-4)-12*x^3+96*x^2-384*x)/((4*x^3-16*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(x-4)+
32*x^2-128*x)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(4*x^3-16*x^2)*log(x-4)^2+(32*x^2-128*x)*log(x-4)+64*x-256),x, alg
orithm="giac")

[Out]

3*(2*x^6*e^5*log(x - 4) - x^6*e^x*log(x - 4) - 12*x^5*e^5*log(x - 4) + 3*x^5*e^x*log(x - 4) + 8*x^5*e^5 - 4*x^
5*e^x + 16*x^4*e^5*log(x - 4) + 4*x^4*e^x*log(x - 4) - 44*x^4*e^5 + 12*x^4*e^x + 48*x^3*e^5 + 16*x^3*e^x + 64*
x^2*e^5)/(2*x^5*e^5*log(x - 4)^2 - x^5*e^x*log(x - 4)^2 - x^5*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x)*log(x - 4)
+ 2*x^5*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x + 5)*log(x - 4) - 12*x^4*e^5*log(x - 4)^2 + 3*x^4*e^x*log(x - 4)^2 +
16*x^4*e^5*log(x - 4) + 3*x^4*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x)*log(x - 4) - 12*x^4*e^(-1/4*x^2 + 1/4*x*e^
(x - 5) + x + 5)*log(x - 4) - 8*x^4*e^x*log(x - 4) + 16*x^3*e^5*log(x - 4)^2 + 4*x^3*e^x*log(x - 4)^2 - 4*x^4*
e^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x) + 8*x^4*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x + 5) - 92*x^3*e^5*log(x - 4) +
4*x^3*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x)*log(x - 4) + 16*x^3*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x + 5)*log(x -
4) + 24*x^3*e^x*log(x - 4) + 32*x^3*e^5 + 12*x^3*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x) - 44*x^3*e^(-1/4*x^2 + 1
/4*x*e^(x - 5) + x + 5) - 16*x^3*e^x + 112*x^2*e^5*log(x - 4) + 32*x^2*e^x*log(x - 4) - 176*x^2*e^5 + 16*x^2*e
^(-1/4*x^2 + 1/4*x*e^(x - 5) + 2*x) + 48*x^2*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x + 5) + 48*x^2*e^x + 64*x*e^5*lo
g(x - 4) + 192*x*e^5 + 64*x*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x + 5) + 64*x*e^x + 256*e^5)

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maple [A]  time = 0.10, size = 31, normalized size = 0.91

method	result	size
risch	\(\frac {3 x^{2}}{{\mathrm e}^{-\frac {\left (-4-{\mathrm e}^{x -5}+x \right ) x}{4}} x +x \ln \left (x -4\right )+4}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^5+9*x^4+12*x^3)*exp(x-5)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(12*x^3-48*x^2)
*ln(x-4)-12*x^3+96*x^2-384*x)/((4*x^3-16*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)^2+((8*x^3-32*x^2)*ln(x-4)+32*x^2-1
28*x)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(4*x^3-16*x^2)*ln(x-4)^2+(32*x^2-128*x)*ln(x-4)+64*x-256),x,method=_RETURN
VERBOSE)

[Out]

3*x^2/(exp(-1/4*(-4-exp(x-5)+x)*x)*x+x*ln(x-4)+4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3}{4} \, \int \frac {4 \, x^{3} - 32 \, x^{2} - {\left (2 \, x^{5} - 12 \, x^{4} + 20 \, x^{3} - 16 \, x^{2} - {\left (x^{5} - 3 \, x^{4} - 4 \, x^{3}\right )} e^{\left (x - 5\right )}\right )} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} - 4 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right ) + 128 \, x}{{\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right )^{2} + 2 \, {\left (4 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x - 4\right ) - 16 \, x\right )} e^{\left (-\frac {1}{4} \, x^{2} + \frac {1}{4} \, x e^{\left (x - 5\right )} + x\right )} + {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (-\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (x - 5\right )} + 2 \, x\right )} + 8 \, {\left (x^{2} - 4 \, x\right )} \log \left (x - 4\right ) + 16 \, x - 64}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^5+9*x^4+12*x^3)*exp(x-5)+6*x^5-36*x^4+60*x^3-48*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(12*x^3-4
8*x^2)*log(x-4)-12*x^3+96*x^2-384*x)/((4*x^3-16*x^2)*exp(1/4*x*exp(x-5)-1/4*x^2+x)^2+((8*x^3-32*x^2)*log(x-4)+
32*x^2-128*x)*exp(1/4*x*exp(x-5)-1/4*x^2+x)+(4*x^3-16*x^2)*log(x-4)^2+(32*x^2-128*x)*log(x-4)+64*x-256),x, alg
orithm="maxima")

[Out]

-3/4*integrate((4*x^3 - 32*x^2 - (2*x^5 - 12*x^4 + 20*x^3 - 16*x^2 - (x^5 - 3*x^4 - 4*x^3)*e^(x - 5))*e^(-1/4*
x^2 + 1/4*x*e^(x - 5) + x) - 4*(x^3 - 4*x^2)*log(x - 4) + 128*x)/((x^3 - 4*x^2)*log(x - 4)^2 + 2*(4*x^2 + (x^3
- 4*x^2)*log(x - 4) - 16*x)*e^(-1/4*x^2 + 1/4*x*e^(x - 5) + x) + (x^3 - 4*x^2)*e^(-1/2*x^2 + 1/2*x*e^(x - 5)
+ 2*x) + 8*(x^2 - 4*x)*log(x - 4) + 16*x - 64), x)

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mupad [B]  time = 4.63, size = 295, normalized size = 8.68 \begin {gather*} \frac {192\,x^3\,{\mathrm {e}}^{x-5}+96\,x^4\,{\mathrm {e}}^{x-5}-84\,x^5\,{\mathrm {e}}^{x-5}+12\,x^6\,{\mathrm {e}}^{x-5}+\ln \left (x-4\right )\,\left (48\,x^4\,{\mathrm {e}}^{x-5}+24\,x^5\,{\mathrm {e}}^{x-5}-21\,x^6\,{\mathrm {e}}^{x-5}+3\,x^7\,{\mathrm {e}}^{x-5}+192\,x^4-192\,x^5+60\,x^6-6\,x^7\right )+768\,x^2+384\,x^3-672\,x^4+228\,x^5-24\,x^6}{\left (x\,\ln \left (x-4\right )+x\,{\mathrm {e}}^{x-\frac {x^2}{4}+\frac {x\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^x}{4}}+4\right )\,\left (128\,x+64\,x\,{\mathrm {e}}^{x-5}+32\,x^2\,{\mathrm {e}}^{x-5}-28\,x^3\,{\mathrm {e}}^{x-5}+4\,x^4\,{\mathrm {e}}^{x-5}+64\,x^2\,\ln \left (x-4\right )-64\,x^3\,\ln \left (x-4\right )+20\,x^4\,\ln \left (x-4\right )-2\,x^5\,\ln \left (x-4\right )-224\,x^2+76\,x^3-8\,x^4+16\,x^2\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+8\,x^3\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}-7\,x^4\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+x^5\,\ln \left (x-4\right )\,{\mathrm {e}}^{x-5}+256\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((384*x + log(x - 4)*(48*x^2 - 12*x^3) - exp(x + (x*exp(x - 5))/4 - x^2/4)*(exp(x - 5)*(12*x^3 + 9*x^4 - 3*
x^5) - 48*x^2 + 60*x^3 - 36*x^4 + 6*x^5) - 96*x^2 + 12*x^3)/(log(x - 4)*(128*x - 32*x^2) - 64*x + exp(2*x + (x
*exp(x - 5))/2 - x^2/2)*(16*x^2 - 4*x^3) + exp(x + (x*exp(x - 5))/4 - x^2/4)*(128*x + log(x - 4)*(32*x^2 - 8*x
^3) - 32*x^2) + log(x - 4)^2*(16*x^2 - 4*x^3) + 256),x)

[Out]

(192*x^3*exp(x - 5) + 96*x^4*exp(x - 5) - 84*x^5*exp(x - 5) + 12*x^6*exp(x - 5) + log(x - 4)*(48*x^4*exp(x - 5
) + 24*x^5*exp(x - 5) - 21*x^6*exp(x - 5) + 3*x^7*exp(x - 5) + 192*x^4 - 192*x^5 + 60*x^6 - 6*x^7) + 768*x^2 +
384*x^3 - 672*x^4 + 228*x^5 - 24*x^6)/((x*log(x - 4) + x*exp(x - x^2/4 + (x*exp(-5)*exp(x))/4) + 4)*(128*x +
64*x*exp(x - 5) + 32*x^2*exp(x - 5) - 28*x^3*exp(x - 5) + 4*x^4*exp(x - 5) + 64*x^2*log(x - 4) - 64*x^3*log(x
- 4) + 20*x^4*log(x - 4) - 2*x^5*log(x - 4) - 224*x^2 + 76*x^3 - 8*x^4 + 16*x^2*log(x - 4)*exp(x - 5) + 8*x^3*
log(x - 4)*exp(x - 5) - 7*x^4*log(x - 4)*exp(x - 5) + x^5*log(x - 4)*exp(x - 5) + 256))

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sympy [A]  time = 0.60, size = 31, normalized size = 0.91 \begin {gather*} \frac {3 x^{2}}{x e^{- \frac {x^{2}}{4} + \frac {x e^{x - 5}}{4} + x} + x \log {\left (x - 4 \right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**5+9*x**4+12*x**3)*exp(x-5)+6*x**5-36*x**4+60*x**3-48*x**2)*exp(1/4*x*exp(x-5)-1/4*x**2+x)+(
12*x**3-48*x**2)*ln(x-4)-12*x**3+96*x**2-384*x)/((4*x**3-16*x**2)*exp(1/4*x*exp(x-5)-1/4*x**2+x)**2+((8*x**3-3
2*x**2)*ln(x-4)+32*x**2-128*x)*exp(1/4*x*exp(x-5)-1/4*x**2+x)+(4*x**3-16*x**2)*ln(x-4)**2+(32*x**2-128*x)*ln(x
-4)+64*x-256),x)

[Out]

3*x**2/(x*exp(-x**2/4 + x*exp(x - 5)/4 + x) + x*log(x - 4) + 4)

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