\(\int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx\) [1711]

Optimal result

Integrand size = 48, antiderivative size = 21 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=x-\frac {e^{x^2}}{(3-\log (6)) \log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=\int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{x (-3+\log (6)) \log ^2(x)} \, dx \\ & = \frac {\int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{x \log ^2(x)} \, dx}{-3+\log (6)} \\ & = \frac {\int \left (-3+\log (6)-\frac {e^{x^2}}{x \log ^2(x)}+\frac {2 e^{x^2} x}{\log (x)}\right ) \, dx}{-3+\log (6)} \\ & = x-\frac {2 \int \frac {e^{x^2} x}{\log (x)} \, dx}{3-\log (6)}-\frac {\int \frac {e^{x^2}}{x \log ^2(x)} \, dx}{-3+\log (6)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=x+\frac {e^{x^2}}{(-3+\log (6)) \log (x)} \]

[In]

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

[Out]

x + E^x^2/((-3 + Log[6])*Log[x])

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
default	\(x +\frac {{\mathrm e}^{x^{2}}}{\left (\ln \left (6\right )-3\right ) \ln \left (x \right )}\)	\(18\)
parts	\(x +\frac {{\mathrm e}^{x^{2}}}{\left (\ln \left (6\right )-3\right ) \ln \left (x \right )}\)	\(18\)
risch	\(x +\frac {{\mathrm e}^{x^{2}}}{\left (\ln \left (2\right )+\ln \left (3\right )-3\right ) \ln \left (x \right )}\)	\(20\)
norman	\(\frac {x \ln \left (x \right )+\frac {{\mathrm e}^{x^{2}}}{\ln \left (6\right )-3}}{\ln \left (x \right )}\)	\(22\)
parallelrisch	\(\frac {x \ln \left (x \right ) \ln \left (6\right )-3 x \ln \left (x \right )+{\mathrm e}^{x^{2}}}{\left (\ln \left (6\right )-3\right ) \ln \left (x \right )}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=\frac {{\left (x \log \left (6\right ) - 3 \, x\right )} \log \left (x\right ) + e^{\left (x^{2}\right )}}{{\left (\log \left (6\right ) - 3\right )} \log \left (x\right )} \]

[In]

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

[Out]

((x*log(6) - 3*x)*log(x) + e^(x^2))/((log(6) - 3)*log(x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=x + \frac {e^{x^{2}}}{- 3 \log {\left (x \right )} + \log {\left (6 \right )} \log {\left (x \right )}} \]

[In]

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

[Out]

x + exp(x**2)/(-3*log(x) + log(6)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=\frac {x {\left (\log \left (3\right ) + \log \left (2\right ) - 3\right )} \log \left (x\right ) + e^{\left (x^{2}\right )}}{{\left (\log \left (3\right ) + \log \left (2\right ) - 3\right )} \log \left (x\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=\frac {x \log \left (6\right ) \log \left (x\right ) - 3 \, x \log \left (x\right ) + e^{\left (x^{2}\right )}}{\log \left (6\right ) \log \left (x\right ) - 3 \, \log \left (x\right )} \]

[In]

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

[Out]

(x*log(6)*log(x) - 3*x*log(x) + e^(x^2))/(log(6)*log(x) - 3*log(x))

Mupad [B] (verification not implemented)

Time = 9.43 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-e^{x^2}+2 e^{x^2} x^2 \log (x)+(-3 x+x \log (6)) \log ^2(x)}{(-3 x+x \log (6)) \log ^2(x)} \, dx=x+\frac {{\mathrm {e}}^{x^2}}{\ln \left (x\right )\,\left (\ln \left (6\right )-3\right )} \]

[In]

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

[Out]

x + exp(x^2)/(log(x)*(log(6) - 3))