3.14.10 \(\int \frac {-1+x+e^x (2 x^2+x^3)+6 x \log (\frac {e^{x+e^x x^2}}{x})}{x \log (\frac {e^{x+e^x x^2}}{x})} \, dx\)

Optimal. Leaf size=28 \[ 4+x+x \left (5+\frac {\log \left (\log \left (\frac {e^{x+e^x x^2}}{x}\right )\right )}{x}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.96, 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 {-1+x+e^x \left (2 x^2+x^3\right )+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + x + E^x*(2*x^2 + x^3) + 6*x*Log[E^(x + E^x*x^2)/x])/(x*Log[E^(x + E^x*x^2)/x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^x x (2+x)}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )}+\frac {-1+x+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}\right ) \, dx\\ &=\int \frac {e^x x (2+x)}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \frac {-1+x+6 x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx\\ &=\int \left (6+\frac {-1+x}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}\right ) \, dx+\int \left (\frac {2 e^x x}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )}+\frac {e^x x^2}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )}\right ) \, dx\\ &=6 x+2 \int \frac {e^x x}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \frac {-1+x}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \frac {e^x x^2}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx\\ &=6 x+2 \int \frac {e^x x}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \left (\frac {1}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )}-\frac {1}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )}\right ) \, dx+\int \frac {e^x x^2}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx\\ &=6 x+2 \int \frac {e^x x}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \frac {1}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx-\int \frac {1}{x \log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx+\int \frac {e^x x^2}{\log \left (\frac {e^{x+e^x x^2}}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.22, size = 21, normalized size = 0.75 \begin {gather*} 6 x+\log \left (\log \left (\frac {e^{x+e^x x^2}}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x + E^x*(2*x^2 + x^3) + 6*x*Log[E^(x + E^x*x^2)/x])/(x*Log[E^(x + E^x*x^2)/x]),x]

[Out]

6*x + Log[Log[E^(x + E^x*x^2)/x]]

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 19, normalized size = 0.68 \begin {gather*} 6 \, x + \log \left (\log \left (\frac {e^{\left (x^{2} e^{x} + x\right )}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp(exp(x)*x^2+x)/x),x, algorithm="frica
s")

[Out]

6*x + log(log(e^(x^2*e^x + x)/x))

________________________________________________________________________________________

giac [A]  time = 0.38, size = 18, normalized size = 0.64 \begin {gather*} 6 \, x + \log \left (-x^{2} e^{x} - x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp(exp(x)*x^2+x)/x),x, algorithm="giac"
)

[Out]

6*x + log(-x^2*e^x - x + log(x))

________________________________________________________________________________________

maple [A]  time = 0.30, size = 20, normalized size = 0.71

method	result	size
default	\(6 x +\ln \left (\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{x} x^{2}+x}}{x}\right )\right )\)	\(20\)
risch	\(6 x +\ln \left (\ln \left ({\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}}{x}\right )^{3}-2 i \ln \relax (x )\right )}{2}\right )\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*ln(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/ln(exp(exp(x)*x^2+x)/x),x,method=_RETURNVERBOSE)

[Out]

6*x+ln(ln(exp(exp(x)*x^2+x)/x))

________________________________________________________________________________________

maxima [A]  time = 0.84, size = 25, normalized size = 0.89 \begin {gather*} 6 \, x + 2 \, \log \relax (x) + \log \left (\frac {x^{2} e^{x} + x - \log \relax (x)}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*log(exp(exp(x)*x^2+x)/x)+(x^3+2*x^2)*exp(x)+x-1)/x/log(exp(exp(x)*x^2+x)/x),x, algorithm="maxim
a")

[Out]

6*x + 2*log(x) + log((x^2*e^x + x - log(x))/x^2)

________________________________________________________________________________________

mupad [B]  time = 1.00, size = 17, normalized size = 0.61 \begin {gather*} 6\,x+\ln \left (x+\ln \left (\frac {1}{x}\right )+x^2\,{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(x)*(2*x^2 + x^3) + 6*x*log(exp(x + x^2*exp(x))/x) - 1)/(x*log(exp(x + x^2*exp(x))/x)),x)

[Out]

6*x + log(x + log(1/x) + x^2*exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.32, size = 17, normalized size = 0.61 \begin {gather*} 6 x + \log {\left (\log {\left (\frac {e^{x^{2} e^{x} + x}}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*ln(exp(exp(x)*x**2+x)/x)+(x**3+2*x**2)*exp(x)+x-1)/x/ln(exp(exp(x)*x**2+x)/x),x)

[Out]

6*x + log(log(exp(x**2*exp(x) + x)/x))

________________________________________________________________________________________