3.19.37 \(\int \frac {x^2 \log ^2(x)+e^{\frac {12}{\log (x)}} (-12+(-1+x) \log ^2(x))}{e^{\frac {12}{\log (x)}} x \log ^2(x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=18 \[ 1+x+\log \left (1+\frac {e^{\frac {12}{\log (x)}}}{x}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.24, 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 {x^2 \log ^2(x)+e^{\frac {12}{\log (x)}} \left (-12+(-1+x) \log ^2(x)\right )}{e^{\frac {12}{\log (x)}} x \log ^2(x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2*Log[x]^2 + E^(12/Log[x])*(-12 + (-1 + x)*Log[x]^2))/(E^(12/Log[x])*x*Log[x]^2 + x^2*Log[x]^2),x]

[Out]

x + 12/Log[x] - Log[x] + Defer[Int][(E^(12/Log[x]) + x)^(-1), x] + 12*Defer[Int][1/((E^(12/Log[x]) + x)*Log[x]
^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \log ^2(x)+e^{\frac {12}{\log (x)}} \left (-12+(-1+x) \log ^2(x)\right )}{x \left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)} \, dx\\ &=\int \left (\frac {12+\log ^2(x)}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)}+\frac {-12-\log ^2(x)+x \log ^2(x)}{x \log ^2(x)}\right ) \, dx\\ &=\int \frac {12+\log ^2(x)}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)} \, dx+\int \frac {-12-\log ^2(x)+x \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\int \left (1-\frac {1}{x}-\frac {12}{x \log ^2(x)}\right ) \, dx+\int \left (\frac {1}{e^{\frac {12}{\log (x)}}+x}+\frac {12}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)}\right ) \, dx\\ &=x-\log (x)-12 \int \frac {1}{x \log ^2(x)} \, dx+12 \int \frac {1}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)} \, dx+\int \frac {1}{e^{\frac {12}{\log (x)}}+x} \, dx\\ &=x-\log (x)+12 \int \frac {1}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)} \, dx-12 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+\int \frac {1}{e^{\frac {12}{\log (x)}}+x} \, dx\\ &=x+\frac {12}{\log (x)}-\log (x)+12 \int \frac {1}{\left (e^{\frac {12}{\log (x)}}+x\right ) \log ^2(x)} \, dx+\int \frac {1}{e^{\frac {12}{\log (x)}}+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 17, normalized size = 0.94 \begin {gather*} x-\log (x)+\log \left (e^{\frac {12}{\log (x)}}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Log[x]^2 + E^(12/Log[x])*(-12 + (-1 + x)*Log[x]^2))/(E^(12/Log[x])*x*Log[x]^2 + x^2*Log[x]^2),x
]

[Out]

x - Log[x] + Log[E^(12/Log[x]) + x]

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 16, normalized size = 0.89 \begin {gather*} x + \log \left (x + e^{\frac {12}{\log \relax (x)}}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*log(x)^2-12)*exp(12/log(x))+x^2*log(x)^2)/(x*log(x)^2*exp(12/log(x))+x^2*log(x)^2),x, algori
thm="fricas")

[Out]

x + log(x + e^(12/log(x))) - log(x)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 16, normalized size = 0.89 \begin {gather*} x + \log \left (x + e^{\frac {12}{\log \relax (x)}}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*log(x)^2-12)*exp(12/log(x))+x^2*log(x)^2)/(x*log(x)^2*exp(12/log(x))+x^2*log(x)^2),x, algori
thm="giac")

[Out]

x + log(x + e^(12/log(x))) - log(x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 17, normalized size = 0.94

method	result	size
norman	\(x -\ln \relax (x )+\ln \left ({\mathrm e}^{\frac {12}{\ln \relax (x )}}+x \right )\)	\(17\)
risch	\(x -\ln \relax (x )+\ln \left ({\mathrm e}^{\frac {12}{\ln \relax (x )}}+x \right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-1)*ln(x)^2-12)*exp(12/ln(x))+x^2*ln(x)^2)/(x*ln(x)^2*exp(12/ln(x))+x^2*ln(x)^2),x,method=_RETURNVERBO
SE)

[Out]

x-ln(x)+ln(exp(12/ln(x))+x)

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 16, normalized size = 0.89 \begin {gather*} x + \log \left (x + e^{\frac {12}{\log \relax (x)}}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*log(x)^2-12)*exp(12/log(x))+x^2*log(x)^2)/(x*log(x)^2*exp(12/log(x))+x^2*log(x)^2),x, algori
thm="maxima")

[Out]

x + log(x + e^(12/log(x))) - log(x)

________________________________________________________________________________________

mupad [B]  time = 1.34, size = 16, normalized size = 0.89 \begin {gather*} x+\ln \left (x+{\mathrm {e}}^{\frac {12}{\ln \relax (x)}}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(12/log(x))*(log(x)^2*(x - 1) - 12) + x^2*log(x)^2)/(x^2*log(x)^2 + x*exp(12/log(x))*log(x)^2),x)

[Out]

x + log(x + exp(12/log(x))) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 14, normalized size = 0.78 \begin {gather*} x - \log {\relax (x )} + \log {\left (x + e^{\frac {12}{\log {\relax (x )}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*ln(x)**2-12)*exp(12/ln(x))+x**2*ln(x)**2)/(x*ln(x)**2*exp(12/ln(x))+x**2*ln(x)**2),x)

[Out]

x - log(x) + log(x + exp(12/log(x)))

________________________________________________________________________________________