3.93.77 \(\int \frac {e^x x^3+(324+324 x+e^{e^{4 x}} (324+324 x)) \log (1+e^{e^{4 x}})+2 e^x x^2 \log (x)+e^x x \log ^2(x)+e^{e^{4 x}} (-1296 e^{4 x} x^2+e^x x^3+(-1296 e^{4 x} x+2 e^x x^2) \log (x)+e^x x \log ^2(x))}{x^3+2 x^2 \log (x)+x \log ^2(x)+e^{e^{4 x}} (x^3+2 x^2 \log (x)+x \log ^2(x))} \, dx\)

Optimal. Leaf size=22 \[ e^x-\frac {324 \log \left (1+e^{e^{4 x}}\right )}{x+\log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 1.49, 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 {e^x x^3+\left (324+324 x+e^{e^{4 x}} (324+324 x)\right ) \log \left (1+e^{e^{4 x}}\right )+2 e^x x^2 \log (x)+e^x x \log ^2(x)+e^{e^{4 x}} \left (-1296 e^{4 x} x^2+e^x x^3+\left (-1296 e^{4 x} x+2 e^x x^2\right ) \log (x)+e^x x \log ^2(x)\right )}{x^3+2 x^2 \log (x)+x \log ^2(x)+e^{e^{4 x}} \left (x^3+2 x^2 \log (x)+x \log ^2(x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*x^3 + (324 + 324*x + E^E^(4*x)*(324 + 324*x))*Log[1 + E^E^(4*x)] + 2*E^x*x^2*Log[x] + E^x*x*Log[x]^2
+ E^E^(4*x)*(-1296*E^(4*x)*x^2 + E^x*x^3 + (-1296*E^(4*x)*x + 2*E^x*x^2)*Log[x] + E^x*x*Log[x]^2))/(x^3 + 2*x^
2*Log[x] + x*Log[x]^2 + E^E^(4*x)*(x^3 + 2*x^2*Log[x] + x*Log[x]^2)),x]

[Out]

E^x + 324*Defer[Int][Log[1 + E^E^(4*x)]/(x + Log[x])^2, x] + 324*Defer[Int][Log[1 + E^E^(4*x)]/(x*(x + Log[x])
^2), x] - 1296*Defer[Int][E^(E^(4*x) + 4*x)/((1 + E^E^(4*x))*(x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {324 (1+x) \log \left (1+e^{e^{4 x}}\right )}{x}-\frac {e^x (x+\log (x)) \left (1296 e^{e^{4 x}+3 x}-x-e^{e^{4 x}} x-\left (1+e^{e^{4 x}}\right ) \log (x)\right )}{1+e^{e^{4 x}}}}{(x+\log (x))^2} \, dx\\ &=\int \left (e^x+\frac {324 (1+x) \log \left (1+e^{e^{4 x}}\right )}{x (x+\log (x))^2}-\frac {1296 e^{e^{4 x}+4 x}}{\left (1+e^{e^{4 x}}\right ) (x+\log (x))}\right ) \, dx\\ &=324 \int \frac {(1+x) \log \left (1+e^{e^{4 x}}\right )}{x (x+\log (x))^2} \, dx-1296 \int \frac {e^{e^{4 x}+4 x}}{\left (1+e^{e^{4 x}}\right ) (x+\log (x))} \, dx+\int e^x \, dx\\ &=e^x+324 \int \left (\frac {\log \left (1+e^{e^{4 x}}\right )}{(x+\log (x))^2}+\frac {\log \left (1+e^{e^{4 x}}\right )}{x (x+\log (x))^2}\right ) \, dx-1296 \int \frac {e^{e^{4 x}+4 x}}{\left (1+e^{e^{4 x}}\right ) (x+\log (x))} \, dx\\ &=e^x+324 \int \frac {\log \left (1+e^{e^{4 x}}\right )}{(x+\log (x))^2} \, dx+324 \int \frac {\log \left (1+e^{e^{4 x}}\right )}{x (x+\log (x))^2} \, dx-1296 \int \frac {e^{e^{4 x}+4 x}}{\left (1+e^{e^{4 x}}\right ) (x+\log (x))} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 22, normalized size = 1.00 \begin {gather*} e^x-\frac {324 \log \left (1+e^{e^{4 x}}\right )}{x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^3 + (324 + 324*x + E^E^(4*x)*(324 + 324*x))*Log[1 + E^E^(4*x)] + 2*E^x*x^2*Log[x] + E^x*x*Log
[x]^2 + E^E^(4*x)*(-1296*E^(4*x)*x^2 + E^x*x^3 + (-1296*E^(4*x)*x + 2*E^x*x^2)*Log[x] + E^x*x*Log[x]^2))/(x^3
+ 2*x^2*Log[x] + x*Log[x]^2 + E^E^(4*x)*(x^3 + 2*x^2*Log[x] + x*Log[x]^2)),x]

[Out]

E^x - (324*Log[1 + E^E^(4*x)])/(x + Log[x])

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 27, normalized size = 1.23 \begin {gather*} \frac {x e^{x} + e^{x} \log \relax (x) - 324 \, \log \left (e^{\left (e^{\left (4 \, x\right )}\right )} + 1\right )}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x+324)*exp(exp(4*x))+324*x+324)*log(exp(exp(4*x))+1)+(x*exp(x)*log(x)^2+(-1296*x*exp(4*x)+2*e
xp(x)*x^2)*log(x)-1296*x^2*exp(4*x)+exp(x)*x^3)*exp(exp(4*x))+x*exp(x)*log(x)^2+2*x^2*exp(x)*log(x)+exp(x)*x^3
)/((x*log(x)^2+2*x^2*log(x)+x^3)*exp(exp(4*x))+x*log(x)^2+2*x^2*log(x)+x^3),x, algorithm="fricas")

[Out]

(x*e^x + e^x*log(x) - 324*log(e^(e^(4*x)) + 1))/(x + log(x))

________________________________________________________________________________________

giac [A]  time = 0.22, size = 35, normalized size = 1.59 \begin {gather*} \frac {x e^{x} + e^{x} \log \relax (x) - 324 \, \log \left ({\left (e^{\left (x + e^{\left (4 \, x\right )}\right )} + e^{x}\right )} e^{\left (-x\right )}\right )}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x+324)*exp(exp(4*x))+324*x+324)*log(exp(exp(4*x))+1)+(x*exp(x)*log(x)^2+(-1296*x*exp(4*x)+2*e
xp(x)*x^2)*log(x)-1296*x^2*exp(4*x)+exp(x)*x^3)*exp(exp(4*x))+x*exp(x)*log(x)^2+2*x^2*exp(x)*log(x)+exp(x)*x^3
)/((x*log(x)^2+2*x^2*log(x)+x^3)*exp(exp(4*x))+x*log(x)^2+2*x^2*log(x)+x^3),x, algorithm="giac")

[Out]

(x*e^x + e^x*log(x) - 324*log((e^(x + e^(4*x)) + e^x)*e^(-x)))/(x + log(x))

________________________________________________________________________________________

maple [A]  time = 0.07, size = 20, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{x}-\frac {324 \ln \left ({\mathrm e}^{{\mathrm e}^{4 x}}+1\right )}{x +\ln \relax (x )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((324*x+324)*exp(exp(4*x))+324*x+324)*ln(exp(exp(4*x))+1)+(x*exp(x)*ln(x)^2+(-1296*x*exp(4*x)+2*exp(x)*x^
2)*ln(x)-1296*x^2*exp(4*x)+exp(x)*x^3)*exp(exp(4*x))+x*exp(x)*ln(x)^2+2*x^2*exp(x)*ln(x)+exp(x)*x^3)/((x*ln(x)
^2+2*x^2*ln(x)+x^3)*exp(exp(4*x))+x*ln(x)^2+2*x^2*ln(x)+x^3),x,method=_RETURNVERBOSE)

[Out]

exp(x)-324*ln(exp(exp(4*x))+1)/(x+ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 25, normalized size = 1.14 \begin {gather*} \frac {{\left (x + \log \relax (x)\right )} e^{x} - 324 \, \log \left (e^{\left (e^{\left (4 \, x\right )}\right )} + 1\right )}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x+324)*exp(exp(4*x))+324*x+324)*log(exp(exp(4*x))+1)+(x*exp(x)*log(x)^2+(-1296*x*exp(4*x)+2*e
xp(x)*x^2)*log(x)-1296*x^2*exp(4*x)+exp(x)*x^3)*exp(exp(4*x))+x*exp(x)*log(x)^2+2*x^2*exp(x)*log(x)+exp(x)*x^3
)/((x*log(x)^2+2*x^2*log(x)+x^3)*exp(exp(4*x))+x*log(x)^2+2*x^2*log(x)+x^3),x, algorithm="maxima")

[Out]

((x + log(x))*e^x - 324*log(e^(e^(4*x)) + 1))/(x + log(x))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^3\,{\mathrm {e}}^x+\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{4\,x}}+1\right )\,\left (324\,x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (324\,x+324\right )+324\right )+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (x^3\,{\mathrm {e}}^x-1296\,x^2\,{\mathrm {e}}^{4\,x}-\ln \relax (x)\,\left (1296\,x\,{\mathrm {e}}^{4\,x}-2\,x^2\,{\mathrm {e}}^x\right )+x\,{\mathrm {e}}^x\,{\ln \relax (x)}^2\right )+x\,{\mathrm {e}}^x\,{\ln \relax (x)}^2+2\,x^2\,{\mathrm {e}}^x\,\ln \relax (x)}{x\,{\ln \relax (x)}^2+2\,x^2\,\ln \relax (x)+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (x^3+2\,x^2\,\ln \relax (x)+x\,{\ln \relax (x)}^2\right )+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*exp(x) + log(exp(exp(4*x)) + 1)*(324*x + exp(exp(4*x))*(324*x + 324) + 324) + exp(exp(4*x))*(x^3*exp(
x) - 1296*x^2*exp(4*x) - log(x)*(1296*x*exp(4*x) - 2*x^2*exp(x)) + x*exp(x)*log(x)^2) + x*exp(x)*log(x)^2 + 2*
x^2*exp(x)*log(x))/(x*log(x)^2 + 2*x^2*log(x) + exp(exp(4*x))*(x*log(x)^2 + 2*x^2*log(x) + x^3) + x^3),x)

[Out]

int((x^3*exp(x) + log(exp(exp(4*x)) + 1)*(324*x + exp(exp(4*x))*(324*x + 324) + 324) + exp(exp(4*x))*(x^3*exp(
x) - 1296*x^2*exp(4*x) - log(x)*(1296*x*exp(4*x) - 2*x^2*exp(x)) + x*exp(x)*log(x)^2) + x*exp(x)*log(x)^2 + 2*
x^2*exp(x)*log(x))/(x*log(x)^2 + 2*x^2*log(x) + exp(exp(4*x))*(x*log(x)^2 + 2*x^2*log(x) + x^3) + x^3), x)

________________________________________________________________________________________

sympy [A]  time = 0.90, size = 19, normalized size = 0.86 \begin {gather*} e^{x} - \frac {324 \log {\left (e^{e^{4 x}} + 1 \right )}}{x + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x+324)*exp(exp(4*x))+324*x+324)*ln(exp(exp(4*x))+1)+(x*exp(x)*ln(x)**2+(-1296*x*exp(4*x)+2*ex
p(x)*x**2)*ln(x)-1296*x**2*exp(4*x)+exp(x)*x**3)*exp(exp(4*x))+x*exp(x)*ln(x)**2+2*x**2*exp(x)*ln(x)+exp(x)*x*
*3)/((x*ln(x)**2+2*x**2*ln(x)+x**3)*exp(exp(4*x))+x*ln(x)**2+2*x**2*ln(x)+x**3),x)

[Out]

exp(x) - 324*log(exp(exp(4*x)) + 1)/(x + log(x))

________________________________________________________________________________________