\(\int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} (-3+24 x^2) \log (x))}{x^2} \, dx\) [5130]

Optimal result

Integrand size = 60, antiderivative size = 21 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=x^{\frac {3 e^{e^3+4 x^2}}{x}}+\log (x) \]

[Out]

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

Rubi [F]

\[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=\int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx \]

[In]

Int[(x + x^((3*E^(E^3 + 4*x^2))/x)*(3*E^(E^3 + 4*x^2) + E^(E^3 + 4*x^2)*(-3 + 24*x^2)*Log[x]))/x^2,x]

[Out]

Log[x] + 3*Defer[Int][E^(E^3 + 4*x^2)*x^(-2 + (3*E^(E^3 + 4*x^2))/x), x] - 3*Log[x]*Defer[Int][E^(E^3 + 4*x^2)
*x^(-2 + (3*E^(E^3 + 4*x^2))/x), x] + 24*Log[x]*Defer[Int][E^(E^3 + 4*x^2)*x^((3*E^(E^3 + 4*x^2))/x), x] + 3*D
efer[Int][Defer[Int][E^(E^3 + 4*x^2)*x^(-2 + (3*E^(E^3 + 4*x^2))/x), x]/x, x] - 24*Defer[Int][Defer[Int][E^(E^
3 + 4*x^2)*x^((3*E^(E^3 + 4*x^2))/x), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x}+3 e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \left (1-\log (x)+8 x^2 \log (x)\right )\right ) \, dx \\ & = \log (x)+3 \int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \left (1-\log (x)+8 x^2 \log (x)\right ) \, dx \\ & = \log (x)+3 \int \left (e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}}-e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \log (x)+8 e^{e^3+4 x^2} x^{\frac {3 e^{e^3+4 x^2}}{x}} \log (x)\right ) \, dx \\ & = \log (x)+3 \int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \, dx-3 \int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \log (x) \, dx+24 \int e^{e^3+4 x^2} x^{\frac {3 e^{e^3+4 x^2}}{x}} \log (x) \, dx \\ & = \log (x)+3 \int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \, dx+3 \int \frac {\int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \, dx}{x} \, dx-24 \int \frac {\int e^{e^3+4 x^2} x^{\frac {3 e^{e^3+4 x^2}}{x}} \, dx}{x} \, dx-(3 \log (x)) \int e^{e^3+4 x^2} x^{-2+\frac {3 e^{e^3+4 x^2}}{x}} \, dx+(24 \log (x)) \int e^{e^3+4 x^2} x^{\frac {3 e^{e^3+4 x^2}}{x}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=x^{\frac {3 e^{e^3+4 x^2}}{x}}+\log (x) \]

[In]

Integrate[(x + x^((3*E^(E^3 + 4*x^2))/x)*(3*E^(E^3 + 4*x^2) + E^(E^3 + 4*x^2)*(-3 + 24*x^2)*Log[x]))/x^2,x]

[Out]

x^((3*E^(E^3 + 4*x^2))/x) + Log[x]

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
risch	\(\ln \left (x \right )+x^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{3}+4 x^{2}}}{x}}\)	\(20\)
parallelrisch	\(\ln \left (x \right )+{\mathrm e}^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{3}+4 x^{2}} \ln \left (x \right )}{x}}\)	\(21\)

[In]

int((((24*x^2-3)*exp(exp(3)+4*x^2)*ln(x)+3*exp(exp(3)+4*x^2))*exp(3*exp(exp(3)+4*x^2)*ln(x)/x)+x)/x^2,x,method
=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=x^{\frac {3 \, e^{\left (4 \, x^{2} + e^{3}\right )}}{x}} + \log \left (x\right ) \]

[In]

integrate((((24*x^2-3)*exp(exp(3)+4*x^2)*log(x)+3*exp(exp(3)+4*x^2))*exp(3*exp(exp(3)+4*x^2)*log(x)/x)+x)/x^2,
x, algorithm="fricas")

[Out]

x^(3*e^(4*x^2 + e^3)/x) + log(x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=e^{\frac {3 e^{4 x^{2} + e^{3}} \log {\left (x \right )}}{x}} + \log {\left (x \right )} \]

[In]

integrate((((24*x**2-3)*exp(exp(3)+4*x**2)*ln(x)+3*exp(exp(3)+4*x**2))*exp(3*exp(exp(3)+4*x**2)*ln(x)/x)+x)/x*
*2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=x^{\frac {3 \, e^{\left (4 \, x^{2} + e^{3}\right )}}{x}} + \log \left (x\right ) \]

[In]

integrate((((24*x^2-3)*exp(exp(3)+4*x^2)*log(x)+3*exp(exp(3)+4*x^2))*exp(3*exp(exp(3)+4*x^2)*log(x)/x)+x)/x^2,
x, algorithm="maxima")

[Out]

x^(3*e^(4*x^2 + e^3)/x) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=x^{\frac {3 \, e^{\left (4 \, x^{2} + e^{3}\right )}}{x}} + \log \left (x\right ) \]

[In]

integrate((((24*x^2-3)*exp(exp(3)+4*x^2)*log(x)+3*exp(exp(3)+4*x^2))*exp(3*exp(exp(3)+4*x^2)*log(x)/x)+x)/x^2,
x, algorithm="giac")

[Out]

x^(3*e^(4*x^2 + e^3)/x) + log(x)

Mupad [B] (verification not implemented)

Time = 11.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {x+x^{\frac {3 e^{e^3+4 x^2}}{x}} \left (3 e^{e^3+4 x^2}+e^{e^3+4 x^2} \left (-3+24 x^2\right ) \log (x)\right )}{x^2} \, dx=\ln \left (x\right )+x^{\frac {3\,{\mathrm {e}}^{4\,x^2+{\mathrm {e}}^3}}{x}} \]

[In]

int((x + exp((3*exp(exp(3) + 4*x^2)*log(x))/x)*(3*exp(exp(3) + 4*x^2) + exp(exp(3) + 4*x^2)*log(x)*(24*x^2 - 3
)))/x^2,x)

[Out]

log(x) + x^((3*exp(exp(3) + 4*x^2))/x)