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

Optimal. Leaf size=21 \[ x-\frac {e^{x^2}}{(3-\log (6)) \log (x)} \]

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Rubi [F]  time = 0.38, 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^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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 18, normalized size = 0.86 \begin {gather*} x+\frac {e^{x^2}}{(-3+\log (6)) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.71, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\left (x \log \relax (6) - 3 \, x\right )} \log \relax (x) + e^{\left (x^{2}\right )}}{{\left (\log \relax (6) - 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.20, size = 29, normalized size = 1.38 \begin {gather*} \frac {x \log \relax (6) \log \relax (x) - 3 \, x \log \relax (x) + e^{\left (x^{2}\right )}}{\log \relax (6) \log \relax (x) - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.12, size = 18, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.64, size = 28, normalized size = 1.33 \begin {gather*} \frac {x {\left (\log \relax (3) + \log \relax (2) - 3\right )} \log \relax (x) + e^{\left (x^{2}\right )}}{{\left (\log \relax (3) + \log \relax (2) - 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.32, size = 17, normalized size = 0.81 \begin {gather*} x+\frac {{\mathrm {e}}^{x^2}}{\ln \relax (x)\,\left (\ln \relax (6)-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.27, size = 17, normalized size = 0.81 \begin {gather*} x + \frac {e^{x^{2}}}{- 3 \log {\relax (x )} + \log {\relax (6 )} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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