3.78.12 \(\int \frac {e^{28+x-2 x^2+(-24+6 x) \log (\frac {\log (x)}{5})+(4-x) \log ^2(\frac {\log (x)}{5})} (-24+6 x+(x-4 x^2) \log (x)+(8-2 x+6 x \log (x)) \log (\frac {\log (x)}{5})-x \log (x) \log ^2(\frac {\log (x)}{5}))}{x \log (x)} \, dx\)

Optimal. Leaf size=24 \[ e^{(4-x) \left (-2+2 x+\left (-3+\log \left (\frac {\log (x)}{5}\right )\right )^2\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 12.53, 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 (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)*(-24 + 6*x + (x - 4*x^2)*Log[x]
+ (8 - 2*x + 6*x*Log[x])*Log[Log[x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*Log[x]),x]

[Out]

Defer[Int][E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2), x] - 4*Defer[Int][E^(28 +
x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)*x, x] + 6*Defer[Int][E^(28 + x - 2*x^2 + (-2
4 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)/Log[x], x] - 24*Defer[Int][E^(28 + x - 2*x^2 + (-24 + 6*x)*L
og[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)/(x*Log[x]), x] + 6*Defer[Int][E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[
x]/5] + (4 - x)*Log[Log[x]/5]^2)*Log[Log[x]/5], x] - 2*Defer[Int][(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/
5] + (4 - x)*Log[Log[x]/5]^2)*Log[Log[x]/5])/Log[x], x] + 8*Defer[Int][(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Lo
g[x]/5] + (4 - x)*Log[Log[x]/5]^2)*Log[Log[x]/5])/(x*Log[x]), x] - Defer[Int][E^(28 + x - 2*x^2 + (-24 + 6*x)*
Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)*Log[Log[x]/5]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \left (-24+6 x+x \log (x)-4 x^2 \log (x)\right )}{x \log (x)}+\frac {2 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) (4-x+3 x \log (x)) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)}-\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \, dx\\ &=2 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) (4-x+3 x \log (x)) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)} \, dx+\int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \left (-24+6 x+x \log (x)-4 x^2 \log (x)\right )}{x \log (x)} \, dx-\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right ) \, dx\\ &=2 \int \left (3 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )-\frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{\log (x)}+\frac {4 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)}\right ) \, dx+\int \left (\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right )-4 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) x+\frac {6 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) (-4+x)}{x \log (x)}\right ) \, dx-\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{\log (x)} \, dx\right )-4 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) x \, dx+6 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) (-4+x)}{x \log (x)} \, dx+6 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right ) \, dx+8 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)} \, dx+\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \, dx-\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{\log (x)} \, dx\right )-4 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) x \, dx+6 \int \left (\frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{\log (x)}-\frac {4 \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)}\right ) \, dx+6 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right ) \, dx+8 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)} \, dx+\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \, dx-\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{\log (x)} \, dx\right )-4 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) x \, dx+6 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{\log (x)} \, dx+6 \int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right ) \, dx+8 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)} \, dx-24 \int \frac {\exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx+\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \, dx-\int \exp \left (28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right ) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [F]  time = 1.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)*(-24 + 6*x + (x - 4*x^2)*L
og[x] + (8 - 2*x + 6*x*Log[x])*Log[Log[x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*Log[x]),x]

[Out]

Integrate[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5]^2)*(-24 + 6*x + (x - 4*x^2)*L
og[x] + (8 - 2*x + 6*x*Log[x])*Log[Log[x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*Log[x]), x]

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 31, normalized size = 1.29 \begin {gather*} e^{\left (-{\left (x - 4\right )} \log \left (\frac {1}{5} \, \log \relax (x)\right )^{2} - 2 \, x^{2} + 6 \, {\left (x - 4\right )} \log \left (\frac {1}{5} \, \log \relax (x)\right ) + x + 28\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+(-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)
*log(1/5*log(x))^2+(6*x-24)*log(1/5*log(x))-2*x^2+x+28)/x/log(x),x, algorithm="fricas")

[Out]

e^(-(x - 4)*log(1/5*log(x))^2 - 2*x^2 + 6*(x - 4)*log(1/5*log(x)) + x + 28)

________________________________________________________________________________________

giac [B]  time = 5.24, size = 43, normalized size = 1.79 \begin {gather*} e^{\left (-x \log \left (\frac {1}{5} \, \log \relax (x)\right )^{2} - 2 \, x^{2} + 6 \, x \log \left (\frac {1}{5} \, \log \relax (x)\right ) + 4 \, \log \left (\frac {1}{5} \, \log \relax (x)\right )^{2} + x - 24 \, \log \left (\frac {1}{5} \, \log \relax (x)\right ) + 28\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+(-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)
*log(1/5*log(x))^2+(6*x-24)*log(1/5*log(x))-2*x^2+x+28)/x/log(x),x, algorithm="giac")

[Out]

e^(-x*log(1/5*log(x))^2 - 2*x^2 + 6*x*log(1/5*log(x)) + 4*log(1/5*log(x))^2 + x - 24*log(1/5*log(x)) + 28)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 30, normalized size = 1.25

method	result	size
risch	\(\left (\frac {\ln \relax (x )}{5}\right )^{6 x -24} {\mathrm e}^{-\left (x -4\right ) \left (\ln \left (\frac {\ln \relax (x )}{5}\right )^{2}+2 x +7\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x)*ln(1/5*ln(x))^2+(6*x*ln(x)-2*x+8)*ln(1/5*ln(x))+(-4*x^2+x)*ln(x)+6*x-24)*exp((-x+4)*ln(1/5*ln(x)
)^2+(6*x-24)*ln(1/5*ln(x))-2*x^2+x+28)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

(1/5*ln(x))^(6*x-24)*exp(-(x-4)*(ln(1/5*ln(x))^2+2*x+7))

________________________________________________________________________________________

maxima [B]  time = 0.61, size = 69, normalized size = 2.88 \begin {gather*} \frac {59604644775390625 \, e^{\left (-x \log \relax (5)^{2} + 2 \, x \log \relax (5) \log \left (\log \relax (x)\right ) - x \log \left (\log \relax (x)\right )^{2} - 2 \, x^{2} - 6 \, x \log \relax (5) + 4 \, \log \relax (5)^{2} + 6 \, x \log \left (\log \relax (x)\right ) - 8 \, \log \relax (5) \log \left (\log \relax (x)\right ) + 4 \, \log \left (\log \relax (x)\right )^{2} + x + 28\right )}}{\log \relax (x)^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+(-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)
*log(1/5*log(x))^2+(6*x-24)*log(1/5*log(x))-2*x^2+x+28)/x/log(x),x, algorithm="maxima")

[Out]

59604644775390625*e^(-x*log(5)^2 + 2*x*log(5)*log(log(x)) - x*log(log(x))^2 - 2*x^2 - 6*x*log(5) + 4*log(5)^2
+ 6*x*log(log(x)) - 8*log(5)*log(log(x)) + 4*log(log(x))^2 + x + 28)/log(x)^24

________________________________________________________________________________________

mupad [B]  time = 5.14, size = 78, normalized size = 3.25 \begin {gather*} \frac {59604644775390625\,{\mathrm {e}}^{-x\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{28}\,{\mathrm {e}}^{4\,{\ln \left (\ln \relax (x)\right )}^2}\,{\mathrm {e}}^{4\,{\ln \relax (5)}^2}\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-x\,{\ln \left (\ln \relax (x)\right )}^2}\,{\ln \relax (x)}^{6\,x}\,{\ln \relax (x)}^{2\,x\,\ln \relax (5)}}{5^{6\,x}\,{\ln \relax (x)}^{8\,\ln \relax (5)}\,{\ln \relax (x)}^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + log(log(x)/5)*(6*x - 24) - log(log(x)/5)^2*(x - 4) - 2*x^2 + 28)*(6*x + log(log(x)/5)*(6*x*log(x)
- 2*x + 8) + log(x)*(x - 4*x^2) - x*log(log(x)/5)^2*log(x) - 24))/(x*log(x)),x)

[Out]

(59604644775390625*exp(-x*log(5)^2)*exp(28)*exp(4*log(log(x))^2)*exp(4*log(5)^2)*exp(-2*x^2)*exp(x)*exp(-x*log
(log(x))^2)*log(x)^(6*x)*log(x)^(2*x*log(5)))/(5^(6*x)*log(x)^(8*log(5))*log(x)^24)

________________________________________________________________________________________

sympy [A]  time = 0.88, size = 32, normalized size = 1.33 \begin {gather*} e^{- 2 x^{2} + x + \left (4 - x\right ) \log {\left (\frac {\log {\relax (x )}}{5} \right )}^{2} + \left (6 x - 24\right ) \log {\left (\frac {\log {\relax (x )}}{5} \right )} + 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x)*ln(1/5*ln(x))**2+(6*x*ln(x)-2*x+8)*ln(1/5*ln(x))+(-4*x**2+x)*ln(x)+6*x-24)*exp((-x+4)*ln(1
/5*ln(x))**2+(6*x-24)*ln(1/5*ln(x))-2*x**2+x+28)/x/ln(x),x)

[Out]

exp(-2*x**2 + x + (4 - x)*log(log(x)/5)**2 + (6*x - 24)*log(log(x)/5) + 28)

________________________________________________________________________________________