3.34.80 \(\int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {-2+2 x-\log \left (2+e^x-2 x\right )}{6 e} \]

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Rubi [F]  time = 0.23, 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 {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + E^x - 4*x)/(6*E^(1 + x) + E*(12 - 12*x)),x]

[Out]

x/(6*E) + (2*Defer[Int][(2 + E^x - 2*x)^(-1), x])/(3*E) - Defer[Int][x/(2 + E^x - 2*x), x]/(3*E)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+e^x-4 x}{6 e \left (2+e^x-2 x\right )} \, dx\\ &=\frac {\int \frac {6+e^x-4 x}{2+e^x-2 x} \, dx}{6 e}\\ &=\frac {\int \left (1+\frac {2 (-2+x)}{-2-e^x+2 x}\right ) \, dx}{6 e}\\ &=\frac {x}{6 e}+\frac {\int \frac {-2+x}{-2-e^x+2 x} \, dx}{3 e}\\ &=\frac {x}{6 e}+\frac {\int \left (\frac {2}{2+e^x-2 x}-\frac {x}{2+e^x-2 x}\right ) \, dx}{3 e}\\ &=\frac {x}{6 e}-\frac {\int \frac {x}{2+e^x-2 x} \, dx}{3 e}+\frac {2 \int \frac {1}{2+e^x-2 x} \, dx}{3 e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 x-\log \left (2+e^x-2 x\right )}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + E^x - 4*x)/(6*E^(1 + x) + E*(12 - 12*x)),x]

[Out]

(2*x - Log[2 + E^x - 2*x])/(6*E)

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fricas [A]  time = 0.78, size = 23, normalized size = 1.00 \begin {gather*} \frac {1}{6} \, {\left (2 \, x - \log \left (-2 \, {\left (x - 1\right )} e + e^{\left (x + 1\right )}\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="fricas")

[Out]

1/6*(2*x - log(-2*(x - 1)*e + e^(x + 1)))*e^(-1)

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giac [A]  time = 0.14, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{6} \, {\left (2 \, x - \log \left (2 \, x - e^{x} - 2\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="giac")

[Out]

1/6*(2*x - log(2*x - e^x - 2))*e^(-1)

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maple [A]  time = 0.04, size = 19, normalized size = 0.83

method	result	size
risch	\(\frac {{\mathrm e}^{-1} x}{3}-\frac {{\mathrm e}^{-1} \ln \left ({\mathrm e}^{x}-2 x +2\right )}{6}\)	\(19\)
norman	\(\frac {{\mathrm e}^{-1} x}{3}-\frac {{\mathrm e}^{-1} \ln \left (2 x -2-{\mathrm e}^{x}\right )}{6}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.50, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{3} \, x e^{\left (-1\right )} - \frac {1}{6} \, e^{\left (-1\right )} \log \left (-2 \, x + e^{x} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="maxima")

[Out]

1/3*x*e^(-1) - 1/6*e^(-1)*log(-2*x + e^x + 2)

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mupad [B]  time = 0.11, size = 20, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {e}}^{-1}\,\left (2\,x-\ln \left (2\,x-{\mathrm {e}}^x-2\right )\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) - 4*x + 6)/(6*exp(1)*exp(x) - exp(1)*(12*x - 12)),x)

[Out]

(exp(-1)*(2*x - log(2*x - exp(x) - 2)))/6

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sympy [A]  time = 0.15, size = 20, normalized size = 0.87 \begin {gather*} \frac {x}{3 e} - \frac {\log {\left (- 2 x + e^{x} + 2 \right )}}{6 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x)

[Out]

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

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