3.91.44 \(\int \frac {e^{\frac {-5-3 x+3 (65536-512 x+x^2)^{2 \log ^2(x^2)}}{-x+(65536-512 x+x^2)^{2 \log ^2(x^2)}}} (1280 x-5 x^2+(65536-512 x+x^2)^{2 \log ^2(x^2)} (20 x \log ^2(x^2)+(-10240+40 x) \log (x^2) \log (65536-512 x+x^2)))}{-256 x^3+x^4+(65536-512 x+x^2)^{4 \log ^2(x^2)} (-256 x+x^2)+(65536-512 x+x^2)^{2 \log ^2(x^2)} (512 x^2-2 x^3)} \, dx\)

Optimal. Leaf size=26 \[ e^{3+\frac {5}{-\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}+x}} \]

________________________________________________________________________________________

Rubi [F]  time = 92.77, 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 {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left (1280 x-5 x^2+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )} \left (20 x \log ^2\left (x^2\right )+(-10240+40 x) \log \left (x^2\right ) \log \left (65536-512 x+x^2\right )\right )\right )}{-256 x^3+x^4+\left (65536-512 x+x^2\right )^{4 \log ^2\left (x^2\right )} \left (-256 x+x^2\right )+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )} \left (512 x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-5 - 3*x + 3*(65536 - 512*x + x^2)^(2*Log[x^2]^2))/(-x + (65536 - 512*x + x^2)^(2*Log[x^2]^2)))*(1280
*x - 5*x^2 + (65536 - 512*x + x^2)^(2*Log[x^2]^2)*(20*x*Log[x^2]^2 + (-10240 + 40*x)*Log[x^2]*Log[65536 - 512*
x + x^2])))/(-256*x^3 + x^4 + (65536 - 512*x + x^2)^(4*Log[x^2]^2)*(-256*x + x^2) + (65536 - 512*x + x^2)^(2*L
og[x^2]^2)*(512*x^2 - 2*x^3)),x]

[Out]

-5*Defer[Int][E^((-5 - 3*x + 3*(65536 - 512*x + x^2)^(2*Log[x^2]^2))/(-x + (65536 - 512*x + x^2)^(2*Log[x^2]^2
)))/(((-256 + x)^2)^(2*Log[x^2]^2) - x)^2, x] + 40*Defer[Int][(E^((-5 - 3*x + 3*(65536 - 512*x + x^2)^(2*Log[x
^2]^2))/(-x + (65536 - 512*x + x^2)^(2*Log[x^2]^2)))*((-256 + x)^2)^(2*Log[x^2]^2)*Log[(-256 + x)^2]*Log[x^2])
/((((-256 + x)^2)^(2*Log[x^2]^2) - x)^2*x), x] + 20*Defer[Int][(E^((-5 - 3*x + 3*(65536 - 512*x + x^2)^(2*Log[
x^2]^2))/(-x + (65536 - 512*x + x^2)^(2*Log[x^2]^2)))*((-256 + x)^2)^(2*Log[x^2]^2)*Log[x^2]^2)/((((-256 + x)^
2)^(2*Log[x^2]^2) - x)^2*(-256 + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left (-1280 x+5 x^2-\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )} \left (20 x \log ^2\left (x^2\right )+(-10240+40 x) \log \left (x^2\right ) \log \left (65536-512 x+x^2\right )\right )\right )}{(256-x) \left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 x} \, dx\\ &=\int \left (-\frac {5 \exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2}+\frac {20 \exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 (-256+x) x}\right ) \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2} \, dx\right )+20 \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 (-256+x) x} \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2} \, dx\right )+20 \int \left (\frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{256 \left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 (-256+x)}-\frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{256 \left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 x}\right ) \, dx\\ &=\frac {5}{64} \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 (-256+x)} \, dx-\frac {5}{64} \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right ) \left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )} \log \left (x^2\right ) \left (-512 \log \left ((-256+x)^2\right )+2 x \log \left ((-256+x)^2\right )+x \log \left (x^2\right )\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2 x} \, dx-5 \int \frac {\exp \left (\frac {-5-3 x+3 \left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}{-x+\left (65536-512 x+x^2\right )^{2 \log ^2\left (x^2\right )}}\right )}{\left (\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 26, normalized size = 1.00 \begin {gather*} e^{3-\frac {5}{\left ((-256+x)^2\right )^{2 \log ^2\left (x^2\right )}-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-5 - 3*x + 3*(65536 - 512*x + x^2)^(2*Log[x^2]^2))/(-x + (65536 - 512*x + x^2)^(2*Log[x^2]^2)))
*(1280*x - 5*x^2 + (65536 - 512*x + x^2)^(2*Log[x^2]^2)*(20*x*Log[x^2]^2 + (-10240 + 40*x)*Log[x^2]*Log[65536
- 512*x + x^2])))/(-256*x^3 + x^4 + (65536 - 512*x + x^2)^(4*Log[x^2]^2)*(-256*x + x^2) + (65536 - 512*x + x^2
)^(2*Log[x^2]^2)*(512*x^2 - 2*x^3)),x]

[Out]

E^(3 - 5/(((-256 + x)^2)^(2*Log[x^2]^2) - x))

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 49, normalized size = 1.88 \begin {gather*} e^{\left (\frac {3 \, {\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}} - 3 \, x - 5}{{\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}} - x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x*log(x^2)^2+(40*x-10240)*log(x^2-512*x+65536)*log(x^2))*exp(log(x^2-512*x+65536)*log(x^2)^2)^2
-5*x^2+1280*x)*exp((3*exp(log(x^2-512*x+65536)*log(x^2)^2)^2-3*x-5)/(exp(log(x^2-512*x+65536)*log(x^2)^2)^2-x)
)/((x^2-256*x)*exp(log(x^2-512*x+65536)*log(x^2)^2)^4+(-2*x^3+512*x^2)*exp(log(x^2-512*x+65536)*log(x^2)^2)^2+
x^4-256*x^3),x, algorithm="fricas")

[Out]

e^((3*(x^2 - 512*x + 65536)^(2*log(x^2)^2) - 3*x - 5)/((x^2 - 512*x + 65536)^(2*log(x^2)^2) - x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, {\left (4 \, {\left (2 \, {\left (x - 256\right )} \log \left (x^{2} - 512 \, x + 65536\right ) \log \left (x^{2}\right ) + x \log \left (x^{2}\right )^{2}\right )} {\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}} - x^{2} + 256 \, x\right )} e^{\left (\frac {3 \, {\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}} - 3 \, x - 5}{{\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}} - x}\right )}}{x^{4} - 256 \, x^{3} + {\left (x^{2} - 256 \, x\right )} {\left (x^{2} - 512 \, x + 65536\right )}^{4 \, \log \left (x^{2}\right )^{2}} - 2 \, {\left (x^{3} - 256 \, x^{2}\right )} {\left (x^{2} - 512 \, x + 65536\right )}^{2 \, \log \left (x^{2}\right )^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x*log(x^2)^2+(40*x-10240)*log(x^2-512*x+65536)*log(x^2))*exp(log(x^2-512*x+65536)*log(x^2)^2)^2
-5*x^2+1280*x)*exp((3*exp(log(x^2-512*x+65536)*log(x^2)^2)^2-3*x-5)/(exp(log(x^2-512*x+65536)*log(x^2)^2)^2-x)
)/((x^2-256*x)*exp(log(x^2-512*x+65536)*log(x^2)^2)^4+(-2*x^3+512*x^2)*exp(log(x^2-512*x+65536)*log(x^2)^2)^2+
x^4-256*x^3),x, algorithm="giac")

[Out]

integrate(5*(4*(2*(x - 256)*log(x^2 - 512*x + 65536)*log(x^2) + x*log(x^2)^2)*(x^2 - 512*x + 65536)^(2*log(x^2
)^2) - x^2 + 256*x)*e^((3*(x^2 - 512*x + 65536)^(2*log(x^2)^2) - 3*x - 5)/((x^2 - 512*x + 65536)^(2*log(x^2)^2
) - x))/(x^4 - 256*x^3 + (x^2 - 256*x)*(x^2 - 512*x + 65536)^(4*log(x^2)^2) - 2*(x^3 - 256*x^2)*(x^2 - 512*x +
65536)^(2*log(x^2)^2)), x)

________________________________________________________________________________________

maple [C]  time = 0.89, size = 266, normalized size = 10.23

method	result	size
risch	\({\mathrm e}^{\frac {-3 \,{\mathrm e}^{-\frac {\left (i \pi \mathrm {csgn}\left (i \left (x -256\right )^{2}\right )^{3}-2 i \pi \mathrm {csgn}\left (i \left (x -256\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (x -256\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (x -256\right )^{2}\right ) \mathrm {csgn}\left (i \left (x -256\right )\right )^{2}-4 \ln \left (x -256\right )\right ) \left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 \ln \relax (x )\right )^{2}}{4}}+3 x +5}{-{\mathrm e}^{-\frac {\left (i \pi \mathrm {csgn}\left (i \left (x -256\right )^{2}\right )^{3}-2 i \pi \mathrm {csgn}\left (i \left (x -256\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (x -256\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (x -256\right )^{2}\right ) \mathrm {csgn}\left (i \left (x -256\right )\right )^{2}-4 \ln \left (x -256\right )\right ) \left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 \ln \relax (x )\right )^{2}}{4}}+x}}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x*ln(x^2)^2+(40*x-10240)*ln(x^2-512*x+65536)*ln(x^2))*exp(ln(x^2-512*x+65536)*ln(x^2)^2)^2-5*x^2+1280
*x)*exp((3*exp(ln(x^2-512*x+65536)*ln(x^2)^2)^2-3*x-5)/(exp(ln(x^2-512*x+65536)*ln(x^2)^2)^2-x))/((x^2-256*x)*
exp(ln(x^2-512*x+65536)*ln(x^2)^2)^4+(-2*x^3+512*x^2)*exp(ln(x^2-512*x+65536)*ln(x^2)^2)^2+x^4-256*x^3),x,meth
od=_RETURNVERBOSE)

[Out]

exp((-3*exp(-1/4*(I*Pi*csgn(I*(x-256)^2)^3-2*I*Pi*csgn(I*(x-256)^2)^2*csgn(I*(x-256))+I*Pi*csgn(I*(x-256)^2)*c
sgn(I*(x-256))^2-4*ln(x-256))*(-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x^2)^3
+4*ln(x))^2)+3*x+5)/(-exp(-1/4*(I*Pi*csgn(I*(x-256)^2)^3-2*I*Pi*csgn(I*(x-256)^2)^2*csgn(I*(x-256))+I*Pi*csgn(
I*(x-256)^2)*csgn(I*(x-256))^2-4*ln(x-256))*(-I*Pi*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi
*csgn(I*x^2)^3+4*ln(x))^2)+x))

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 21, normalized size = 0.81 \begin {gather*} e^{\left (-\frac {5}{{\left (x - 256\right )}^{16 \, \log \relax (x)^{2}} - x} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x*log(x^2)^2+(40*x-10240)*log(x^2-512*x+65536)*log(x^2))*exp(log(x^2-512*x+65536)*log(x^2)^2)^2
-5*x^2+1280*x)*exp((3*exp(log(x^2-512*x+65536)*log(x^2)^2)^2-3*x-5)/(exp(log(x^2-512*x+65536)*log(x^2)^2)^2-x)
)/((x^2-256*x)*exp(log(x^2-512*x+65536)*log(x^2)^2)^4+(-2*x^3+512*x^2)*exp(log(x^2-512*x+65536)*log(x^2)^2)^2+
x^4-256*x^3),x, algorithm="maxima")

[Out]

e^(-5/((x - 256)^(16*log(x)^2) - x) + 3)

________________________________________________________________________________________

mupad [B]  time = 7.41, size = 97, normalized size = 3.73 \begin {gather*} {\mathrm {e}}^{-\frac {3\,{\left (x^2-512\,x+65536\right )}^{2\,{\ln \left (x^2\right )}^2}}{x-{\left (x^2-512\,x+65536\right )}^{2\,{\ln \left (x^2\right )}^2}}}\,{\mathrm {e}}^{\frac {3\,x}{x-{\left (x^2-512\,x+65536\right )}^{2\,{\ln \left (x^2\right )}^2}}}\,{\mathrm {e}}^{\frac {5}{x-{\left (x^2-512\,x+65536\right )}^{2\,{\ln \left (x^2\right )}^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((3*x - 3*exp(2*log(x^2)^2*log(x^2 - 512*x + 65536)) + 5)/(x - exp(2*log(x^2)^2*log(x^2 - 512*x + 6553
6))))*(1280*x + exp(2*log(x^2)^2*log(x^2 - 512*x + 65536))*(20*x*log(x^2)^2 + log(x^2)*log(x^2 - 512*x + 65536
)*(40*x - 10240)) - 5*x^2))/(exp(2*log(x^2)^2*log(x^2 - 512*x + 65536))*(512*x^2 - 2*x^3) - exp(4*log(x^2)^2*l
og(x^2 - 512*x + 65536))*(256*x - x^2) - 256*x^3 + x^4),x)

[Out]

exp(-(3*(x^2 - 512*x + 65536)^(2*log(x^2)^2))/(x - (x^2 - 512*x + 65536)^(2*log(x^2)^2)))*exp((3*x)/(x - (x^2
- 512*x + 65536)^(2*log(x^2)^2)))*exp(5/(x - (x^2 - 512*x + 65536)^(2*log(x^2)^2)))

________________________________________________________________________________________

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(((20*x*ln(x**2)**2+(40*x-10240)*ln(x**2-512*x+65536)*ln(x**2))*exp(ln(x**2-512*x+65536)*ln(x**2)**2)
**2-5*x**2+1280*x)*exp((3*exp(ln(x**2-512*x+65536)*ln(x**2)**2)**2-3*x-5)/(exp(ln(x**2-512*x+65536)*ln(x**2)**
2)**2-x))/((x**2-256*x)*exp(ln(x**2-512*x+65536)*ln(x**2)**2)**4+(-2*x**3+512*x**2)*exp(ln(x**2-512*x+65536)*l
n(x**2)**2)**2+x**4-256*x**3),x)

[Out]

Timed out

________________________________________________________________________________________