3.42.20 \(\int \frac {e^x (-3-3 x+x^2)}{36-24 x+4 x^2+e^x (-3 x+x^2)} \, dx\)

Optimal. Leaf size=16 \[ \log \left (4-\frac {e^x x}{3-x}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.74, 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 \left (-3-3 x+x^2\right )}{36-24 x+4 x^2+e^x \left (-3 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-3 - 3*x + x^2))/(36 - 24*x + 4*x^2 + E^x*(-3*x + x^2)),x]

[Out]

-3*Defer[Int][E^x/((-3 + x)*(-12 + 4*x + E^x*x)), x] + Defer[Int][(E^x*x)/(-12 + 4*x + E^x*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-3-3 x+x^2\right )}{(3-x) \left (12-4 x-e^x x\right )} \, dx\\ &=\int \left (-\frac {3 e^x}{(-3+x) \left (-12+4 x+e^x x\right )}+\frac {e^x x}{-12+4 x+e^x x}\right ) \, dx\\ &=-\left (3 \int \frac {e^x}{(-3+x) \left (-12+4 x+e^x x\right )} \, dx\right )+\int \frac {e^x x}{-12+4 x+e^x x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 21, normalized size = 1.31 \begin {gather*} -\log (3-x)+\log \left (12-4 x-e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-3 - 3*x + x^2))/(36 - 24*x + 4*x^2 + E^x*(-3*x + x^2)),x]

[Out]

-Log[3 - x] + Log[12 - 4*x - E^x*x]

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 23, normalized size = 1.44 \begin {gather*} -\log \left (x - 3\right ) + \log \relax (x) + \log \left (\frac {x e^{x} + 4 \, x - 12}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-3*x-3)*exp(x)/((x^2-3*x)*exp(x)+4*x^2-24*x+36),x, algorithm="fricas")

[Out]

-log(x - 3) + log(x) + log((x*e^x + 4*x - 12)/x)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 17, normalized size = 1.06 \begin {gather*} \log \left (x e^{x} + 4 \, x - 12\right ) - \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-3*x-3)*exp(x)/((x^2-3*x)*exp(x)+4*x^2-24*x+36),x, algorithm="giac")

[Out]

log(x*e^x + 4*x - 12) - log(x - 3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 18, normalized size = 1.12

method	result	size
norman	\(-\ln \left (x -3\right )+\ln \left ({\mathrm e}^{x} x +4 x -12\right )\)	\(18\)
risch	\(-\ln \left (x -3\right )+\ln \relax (x )+\ln \left ({\mathrm e}^{x}+\frac {4 x -12}{x}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-3*x-3)*exp(x)/((x^2-3*x)*exp(x)+4*x^2-24*x+36),x,method=_RETURNVERBOSE)

[Out]

-ln(x-3)+ln(exp(x)*x+4*x-12)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 23, normalized size = 1.44 \begin {gather*} -\log \left (x - 3\right ) + \log \relax (x) + \log \left (\frac {x e^{x} + 4 \, x - 12}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-3*x-3)*exp(x)/((x^2-3*x)*exp(x)+4*x^2-24*x+36),x, algorithm="maxima")

[Out]

-log(x - 3) + log(x) + log((x*e^x + 4*x - 12)/x)

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 17, normalized size = 1.06 \begin {gather*} \ln \left (4\,x+x\,{\mathrm {e}}^x-12\right )-\ln \left (x-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(3*x - x^2 + 3))/(24*x + exp(x)*(3*x - x^2) - 4*x^2 - 36),x)

[Out]

log(4*x + x*exp(x) - 12) - log(x - 3)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 19, normalized size = 1.19 \begin {gather*} \log {\relax (x )} - \log {\left (x - 3 \right )} + \log {\left (e^{x} + \frac {4 x - 12}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-3*x-3)*exp(x)/((x**2-3*x)*exp(x)+4*x**2-24*x+36),x)

[Out]

log(x) - log(x - 3) + log(exp(x) + (4*x - 12)/x)

________________________________________________________________________________________