3.17.20 \(\int \frac {e^{-\frac {e^{4 x}-16 e^{3 x} \log (2 x)+e^{2 x} (96-2 x) \log ^2(2 x)+e^x (-256+16 x) \log ^3(2 x)+(256-32 x+x^2) \log ^4(2 x)}{\log ^4(2 x)}} (e^{4 x} (20 x-4 x^2)+(e^{3 x} (-240 x+48 x^2)+e^{4 x} (-20 x^2+4 x^3)) \log (2 x)+(e^{3 x} (240 x^2-48 x^3)+e^{2 x} (960 x-212 x^2+4 x^3)) \log ^2(2 x)+(e^x (-1280 x+336 x^2-16 x^3)+e^{2 x} (-950 x^2+210 x^3-4 x^4)) \log ^3(2 x)+e^x (1200 x^2-320 x^3+16 x^4) \log ^4(2 x)+(10 x+157 x^2-42 x^3+2 x^4) \log ^5(2 x))}{\log ^5(2 x)} \, dx\)
Optimal. Leaf size=34 \[ e^{-\left (x-\left (4-\frac {e^x}{\log (2 x)}\right )^2\right )^2} (5-x) x^2 \]
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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[(E^(4*x)*(20*x - 4*x^2) + (E^(3*x)*(-240*x + 48*x^2) + E^(4*x)*(-20*x^2 + 4*x^3))*Log[2*x] + (E^(3*x)*(240
*x^2 - 48*x^3) + E^(2*x)*(960*x - 212*x^2 + 4*x^3))*Log[2*x]^2 + (E^x*(-1280*x + 336*x^2 - 16*x^3) + E^(2*x)*(
-950*x^2 + 210*x^3 - 4*x^4))*Log[2*x]^3 + E^x*(1200*x^2 - 320*x^3 + 16*x^4)*Log[2*x]^4 + (10*x + 157*x^2 - 42*
x^3 + 2*x^4)*Log[2*x]^5)/(E^((E^(4*x) - 16*E^(3*x)*Log[2*x] + E^(2*x)*(96 - 2*x)*Log[2*x]^2 + E^x*(-256 + 16*x
)*Log[2*x]^3 + (256 - 32*x + x^2)*Log[2*x]^4)/Log[2*x]^4)*Log[2*x]^5),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [A] time = 0.44, size = 46, normalized size = 1.35 \begin {gather*} -e^{-\frac {\left (e^{2 x}-8 e^x \log (2 x)-(-16+x) \log ^2(2 x)\right )^2}{\log ^4(2 x)}} (-5+x) x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^(4*x)*(20*x - 4*x^2) + (E^(3*x)*(-240*x + 48*x^2) + E^(4*x)*(-20*x^2 + 4*x^3))*Log[2*x] + (E^(3*x
)*(240*x^2 - 48*x^3) + E^(2*x)*(960*x - 212*x^2 + 4*x^3))*Log[2*x]^2 + (E^x*(-1280*x + 336*x^2 - 16*x^3) + E^(
2*x)*(-950*x^2 + 210*x^3 - 4*x^4))*Log[2*x]^3 + E^x*(1200*x^2 - 320*x^3 + 16*x^4)*Log[2*x]^4 + (10*x + 157*x^2
- 42*x^3 + 2*x^4)*Log[2*x]^5)/(E^((E^(4*x) - 16*E^(3*x)*Log[2*x] + E^(2*x)*(96 - 2*x)*Log[2*x]^2 + E^x*(-256
+ 16*x)*Log[2*x]^3 + (256 - 32*x + x^2)*Log[2*x]^4)/Log[2*x]^4)*Log[2*x]^5),x]
[Out]
-(((-5 + x)*x^2)/E^((E^(2*x) - 8*E^x*Log[2*x] - (-16 + x)*Log[2*x]^2)^2/Log[2*x]^4))
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fricas [B] time = 0.62, size = 78, normalized size = 2.29 \begin {gather*} -{\left (x^{3} - 5 \, x^{2}\right )} e^{\left (-\frac {16 \, {\left (x - 16\right )} e^{x} \log \left (2 \, x\right )^{3} + {\left (x^{2} - 32 \, x + 256\right )} \log \left (2 \, x\right )^{4} - 2 \, {\left (x - 48\right )} e^{\left (2 \, x\right )} \log \left (2 \, x\right )^{2} - 16 \, e^{\left (3 \, x\right )} \log \left (2 \, x\right ) + e^{\left (4 \, x\right )}}{\log \left (2 \, x\right )^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^4-42*x^3+157*x^2+10*x)*log(2*x)^5+(16*x^4-320*x^3+1200*x^2)*exp(x)*log(2*x)^4+((-4*x^4+210*x^3
-950*x^2)*exp(x)^2+(-16*x^3+336*x^2-1280*x)*exp(x))*log(2*x)^3+((-48*x^3+240*x^2)*exp(x)^3+(4*x^3-212*x^2+960*
x)*exp(x)^2)*log(2*x)^2+((4*x^3-20*x^2)*exp(x)^4+(48*x^2-240*x)*exp(x)^3)*log(2*x)+(-4*x^2+20*x)*exp(x)^4)/log
(2*x)^5/exp(((x^2-32*x+256)*log(2*x)^4+(16*x-256)*exp(x)*log(2*x)^3+(-2*x+96)*exp(x)^2*log(2*x)^2-16*exp(x)^3*
log(2*x)+exp(x)^4)/log(2*x)^4),x, algorithm="fricas")
[Out]
-(x^3 - 5*x^2)*e^(-(16*(x - 16)*e^x*log(2*x)^3 + (x^2 - 32*x + 256)*log(2*x)^4 - 2*(x - 48)*e^(2*x)*log(2*x)^2
- 16*e^(3*x)*log(2*x) + e^(4*x))/log(2*x)^4)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^4-42*x^3+157*x^2+10*x)*log(2*x)^5+(16*x^4-320*x^3+1200*x^2)*exp(x)*log(2*x)^4+((-4*x^4+210*x^3
-950*x^2)*exp(x)^2+(-16*x^3+336*x^2-1280*x)*exp(x))*log(2*x)^3+((-48*x^3+240*x^2)*exp(x)^3+(4*x^3-212*x^2+960*
x)*exp(x)^2)*log(2*x)^2+((4*x^3-20*x^2)*exp(x)^4+(48*x^2-240*x)*exp(x)^3)*log(2*x)+(-4*x^2+20*x)*exp(x)^4)/log
(2*x)^5/exp(((x^2-32*x+256)*log(2*x)^4+(16*x-256)*exp(x)*log(2*x)^3+(-2*x+96)*exp(x)^2*log(2*x)^2-16*exp(x)^3*
log(2*x)+exp(x)^4)/log(2*x)^4),x, algorithm="giac")
[Out]
undef
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maple [B] time = 0.54, size = 110, normalized size = 3.24
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result |
size |
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| risch |
\(\left (-x^{3}+5 x^{2}\right ) {\mathrm e}^{-\frac {\ln \left (2 x \right )^{4} x^{2}+16 \,{\mathrm e}^{x} \ln \left (2 x \right )^{3} x -32 \ln \left (2 x \right )^{4} x -256 \,{\mathrm e}^{x} \ln \left (2 x \right )^{3}+256 \ln \left (2 x \right )^{4}-2 \,{\mathrm e}^{2 x} \ln \left (2 x \right )^{2} x +96 \,{\mathrm e}^{2 x} \ln \left (2 x \right )^{2}-16 \,{\mathrm e}^{3 x} \ln \left (2 x \right )+{\mathrm e}^{4 x}}{\ln \left (2 x \right )^{4}}}\) |
\(110\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((2*x^4-42*x^3+157*x^2+10*x)*ln(2*x)^5+(16*x^4-320*x^3+1200*x^2)*exp(x)*ln(2*x)^4+((-4*x^4+210*x^3-950*x^2
)*exp(x)^2+(-16*x^3+336*x^2-1280*x)*exp(x))*ln(2*x)^3+((-48*x^3+240*x^2)*exp(x)^3+(4*x^3-212*x^2+960*x)*exp(x)
^2)*ln(2*x)^2+((4*x^3-20*x^2)*exp(x)^4+(48*x^2-240*x)*exp(x)^3)*ln(2*x)+(-4*x^2+20*x)*exp(x)^4)/ln(2*x)^5/exp(
((x^2-32*x+256)*ln(2*x)^4+(16*x-256)*exp(x)*ln(2*x)^3+(-2*x+96)*exp(x)^2*ln(2*x)^2-16*exp(x)^3*ln(2*x)+exp(x)^
4)/ln(2*x)^4),x,method=_RETURNVERBOSE)
[Out]
(-x^3+5*x^2)*exp(-(ln(2*x)^4*x^2+16*exp(x)*ln(2*x)^3*x-32*ln(2*x)^4*x-256*exp(x)*ln(2*x)^3+256*ln(2*x)^4-2*exp
(2*x)*ln(2*x)^2*x+96*exp(2*x)*ln(2*x)^2-16*exp(3*x)*ln(2*x)+exp(4*x))/ln(2*x)^4)
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maxima [B] time = 93.59, size = 168, normalized size = 4.94 \begin {gather*} -{\left (x^{3} - 5 \, x^{2}\right )} e^{\left (-x^{2} + 32 \, x + \frac {2 \, x e^{\left (2 \, x\right )}}{\log \relax (2)^{2} + 2 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2}} - \frac {16 \, x e^{x}}{\log \relax (2) + \log \relax (x)} - \frac {e^{\left (4 \, x\right )}}{\log \relax (2)^{4} + 4 \, \log \relax (2)^{3} \log \relax (x) + 6 \, \log \relax (2)^{2} \log \relax (x)^{2} + 4 \, \log \relax (2) \log \relax (x)^{3} + \log \relax (x)^{4}} + \frac {16 \, e^{\left (3 \, x\right )}}{\log \relax (2)^{3} + 3 \, \log \relax (2)^{2} \log \relax (x) + 3 \, \log \relax (2) \log \relax (x)^{2} + \log \relax (x)^{3}} - \frac {96 \, e^{\left (2 \, x\right )}}{\log \relax (2)^{2} + 2 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2}} + \frac {256 \, e^{x}}{\log \relax (2) + \log \relax (x)} - 256\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x^4-42*x^3+157*x^2+10*x)*log(2*x)^5+(16*x^4-320*x^3+1200*x^2)*exp(x)*log(2*x)^4+((-4*x^4+210*x^3
-950*x^2)*exp(x)^2+(-16*x^3+336*x^2-1280*x)*exp(x))*log(2*x)^3+((-48*x^3+240*x^2)*exp(x)^3+(4*x^3-212*x^2+960*
x)*exp(x)^2)*log(2*x)^2+((4*x^3-20*x^2)*exp(x)^4+(48*x^2-240*x)*exp(x)^3)*log(2*x)+(-4*x^2+20*x)*exp(x)^4)/log
(2*x)^5/exp(((x^2-32*x+256)*log(2*x)^4+(16*x-256)*exp(x)*log(2*x)^3+(-2*x+96)*exp(x)^2*log(2*x)^2-16*exp(x)^3*
log(2*x)+exp(x)^4)/log(2*x)^4),x, algorithm="maxima")
[Out]
-(x^3 - 5*x^2)*e^(-x^2 + 32*x + 2*x*e^(2*x)/(log(2)^2 + 2*log(2)*log(x) + log(x)^2) - 16*x*e^x/(log(2) + log(x
)) - e^(4*x)/(log(2)^4 + 4*log(2)^3*log(x) + 6*log(2)^2*log(x)^2 + 4*log(2)*log(x)^3 + log(x)^4) + 16*e^(3*x)/
(log(2)^3 + 3*log(2)^2*log(x) + 3*log(2)*log(x)^2 + log(x)^3) - 96*e^(2*x)/(log(2)^2 + 2*log(2)*log(x) + log(x
)^2) + 256*e^x/(log(2) + log(x)) - 256)
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {\left (x^2-32\,x+256\right )\,{\ln \left (2\,x\right )}^4+{\mathrm {e}}^x\,\left (16\,x-256\right )\,{\ln \left (2\,x\right )}^3-{\mathrm {e}}^{2\,x}\,\left (2\,x-96\right )\,{\ln \left (2\,x\right )}^2-16\,{\mathrm {e}}^{3\,x}\,\ln \left (2\,x\right )+{\mathrm {e}}^{4\,x}}{{\ln \left (2\,x\right )}^4}}\,\left (\left (2\,x^4-42\,x^3+157\,x^2+10\,x\right )\,{\ln \left (2\,x\right )}^5+{\mathrm {e}}^x\,\left (16\,x^4-320\,x^3+1200\,x^2\right )\,{\ln \left (2\,x\right )}^4+\left (-{\mathrm {e}}^{2\,x}\,\left (4\,x^4-210\,x^3+950\,x^2\right )-{\mathrm {e}}^x\,\left (16\,x^3-336\,x^2+1280\,x\right )\right )\,{\ln \left (2\,x\right )}^3+\left ({\mathrm {e}}^{2\,x}\,\left (4\,x^3-212\,x^2+960\,x\right )+{\mathrm {e}}^{3\,x}\,\left (240\,x^2-48\,x^3\right )\right )\,{\ln \left (2\,x\right )}^2+\left (-{\mathrm {e}}^{3\,x}\,\left (240\,x-48\,x^2\right )-{\mathrm {e}}^{4\,x}\,\left (20\,x^2-4\,x^3\right )\right )\,\ln \left (2\,x\right )+{\mathrm {e}}^{4\,x}\,\left (20\,x-4\,x^2\right )\right )}{{\ln \left (2\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(-(exp(4*x) + log(2*x)^4*(x^2 - 32*x + 256) - 16*log(2*x)*exp(3*x) + log(2*x)^3*exp(x)*(16*x - 256) -
log(2*x)^2*exp(2*x)*(2*x - 96))/log(2*x)^4)*(exp(4*x)*(20*x - 4*x^2) + log(2*x)^2*(exp(2*x)*(960*x - 212*x^2 +
4*x^3) + exp(3*x)*(240*x^2 - 48*x^3)) - log(2*x)*(exp(3*x)*(240*x - 48*x^2) + exp(4*x)*(20*x^2 - 4*x^3)) + lo
g(2*x)^5*(10*x + 157*x^2 - 42*x^3 + 2*x^4) - log(2*x)^3*(exp(2*x)*(950*x^2 - 210*x^3 + 4*x^4) + exp(x)*(1280*x
- 336*x^2 + 16*x^3)) + log(2*x)^4*exp(x)*(1200*x^2 - 320*x^3 + 16*x^4)))/log(2*x)^5,x)
[Out]
int((exp(-(exp(4*x) + log(2*x)^4*(x^2 - 32*x + 256) - 16*log(2*x)*exp(3*x) + log(2*x)^3*exp(x)*(16*x - 256) -
log(2*x)^2*exp(2*x)*(2*x - 96))/log(2*x)^4)*(exp(4*x)*(20*x - 4*x^2) + log(2*x)^2*(exp(2*x)*(960*x - 212*x^2 +
4*x^3) + exp(3*x)*(240*x^2 - 48*x^3)) - log(2*x)*(exp(3*x)*(240*x - 48*x^2) + exp(4*x)*(20*x^2 - 4*x^3)) + lo
g(2*x)^5*(10*x + 157*x^2 - 42*x^3 + 2*x^4) - log(2*x)^3*(exp(2*x)*(950*x^2 - 210*x^3 + 4*x^4) + exp(x)*(1280*x
- 336*x^2 + 16*x^3)) + log(2*x)^4*exp(x)*(1200*x^2 - 320*x^3 + 16*x^4)))/log(2*x)^5, x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((2*x**4-42*x**3+157*x**2+10*x)*ln(2*x)**5+(16*x**4-320*x**3+1200*x**2)*exp(x)*ln(2*x)**4+((-4*x**4+
210*x**3-950*x**2)*exp(x)**2+(-16*x**3+336*x**2-1280*x)*exp(x))*ln(2*x)**3+((-48*x**3+240*x**2)*exp(x)**3+(4*x
**3-212*x**2+960*x)*exp(x)**2)*ln(2*x)**2+((4*x**3-20*x**2)*exp(x)**4+(48*x**2-240*x)*exp(x)**3)*ln(2*x)+(-4*x
**2+20*x)*exp(x)**4)/ln(2*x)**5/exp(((x**2-32*x+256)*ln(2*x)**4+(16*x-256)*exp(x)*ln(2*x)**3+(-2*x+96)*exp(x)*
*2*ln(2*x)**2-16*exp(x)**3*ln(2*x)+exp(x)**4)/ln(2*x)**4),x)
[Out]
Timed out
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