3.64.2 \(\int \frac {e^x x^2+(4 x-2 e^x x-2 x \log (3)) \log (-2+e^x+\log (3))}{(-2+e^x+\log (3)) \log ^2(-2+e^x+\log (3))} \, dx\)

Optimal. Leaf size=17 \[ 7-\frac {x^2}{\log \left (-2+e^x+\log (3)\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 1.44, 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 x^2+\left (4 x-2 e^x x-2 x \log (3)\right ) \log \left (-2+e^x+\log (3)\right )}{\left (-2+e^x+\log (3)\right ) \log ^2\left (-2+e^x+\log (3)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 15, normalized size = 0.88 \begin {gather*} -\frac {x^2}{\log \left (-2+e^x+\log (3)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^2/Log[-2 + E^x + Log[3]])

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 14, normalized size = 0.82 \begin {gather*} -\frac {x^{2}}{\log \left (e^{x} + \log \relax (3) - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/log(e^x + log(3) - 2)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 14, normalized size = 0.82 \begin {gather*} -\frac {x^{2}}{\log \left (e^{x} + \log \relax (3) - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/log(e^x + log(3) - 2)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 15, normalized size = 0.88

method	result	size
norman	\(-\frac {x^{2}}{\ln \left ({\mathrm e}^{x}+\ln \relax (3)-2\right )}\)	\(15\)
risch	\(-\frac {x^{2}}{\ln \left ({\mathrm e}^{x}+\ln \relax (3)-2\right )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/ln(exp(x)+ln(3)-2)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 14, normalized size = 0.82 \begin {gather*} -\frac {x^{2}}{\log \left (e^{x} + \log \relax (3) - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/log(e^x + log(3) - 2)

________________________________________________________________________________________

mupad [B]  time = 4.51, size = 14, normalized size = 0.82 \begin {gather*} -\frac {x^2}{\ln \left (\ln \relax (3)+{\mathrm {e}}^x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/log(log(3) + exp(x) - 2)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 14, normalized size = 0.82 \begin {gather*} - \frac {x^{2}}{\log {\left (e^{x} - 2 + \log {\relax (3 )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2/log(exp(x) - 2 + log(3))

________________________________________________________________________________________