3.18.73 \(\int \frac {-13+e^x (-3-3 x)}{9-9 e+13 x+3 e^x x} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {1}{1-e+x+\frac {1}{3} \left (\frac {4}{3}+e^x\right ) x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6684} \begin {gather*} -\log \left (3 e^x x+13 x+9 (1-e)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-13 + E^x*(-3 - 3*x))/(9 - 9*E + 13*x + 3*E^x*x),x]

[Out]

-Log[9*(1 - E) + 13*x + 3*E^x*x]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (9 (1-e)+13 x+3 e^x x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 17, normalized size = 0.81 \begin {gather*} -\log \left (9-9 e+13 x+3 e^x x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-13 + E^x*(-3 - 3*x))/(9 - 9*E + 13*x + 3*E^x*x),x]

[Out]

-Log[9 - 9*E + 13*x + 3*E^x*x]

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fricas [A]  time = 0.71, size = 26, normalized size = 1.24 \begin {gather*} -\log \relax (x) - \log \left (\frac {3 \, x e^{x} + 13 \, x - 9 \, e + 9}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-3)*exp(x)-13)/(3*exp(x)*x-9*exp(1)+13*x+9),x, algorithm="fricas")

[Out]

-log(x) - log((3*x*e^x + 13*x - 9*e + 9)/x)

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giac [A]  time = 0.30, size = 17, normalized size = 0.81 \begin {gather*} -\log \left (3 \, x e^{x} + 13 \, x - 9 \, e + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-3)*exp(x)-13)/(3*exp(x)*x-9*exp(1)+13*x+9),x, algorithm="giac")

[Out]

-log(3*x*e^x + 13*x - 9*e + 9)

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maple [A]  time = 0.06, size = 18, normalized size = 0.86

method	result	size
norman	\(-\ln \left (-3 \,{\mathrm e}^{x} x +9 \,{\mathrm e}-13 x -9\right )\)	\(18\)
risch	\(-\ln \relax (x )-\ln \left ({\mathrm e}^{x}-\frac {9 \,{\mathrm e}-13 x -9}{3 x}\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x-3)*exp(x)-13)/(3*exp(x)*x-9*exp(1)+13*x+9),x,method=_RETURNVERBOSE)

[Out]

-ln(-3*exp(x)*x+9*exp(1)-13*x-9)

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maxima [A]  time = 0.47, size = 27, normalized size = 1.29 \begin {gather*} -\log \relax (x) - \log \left (\frac {3 \, x e^{x} + 13 \, x - 9 \, e + 9}{3 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-3)*exp(x)-13)/(3*exp(x)*x-9*exp(1)+13*x+9),x, algorithm="maxima")

[Out]

-log(x) - log(1/3*(3*x*e^x + 13*x - 9*e + 9)/x)

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mupad [B]  time = 0.10, size = 17, normalized size = 0.81 \begin {gather*} -\ln \left (13\,x-9\,\mathrm {e}+3\,x\,{\mathrm {e}}^x+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(3*x + 3) + 13)/(13*x - 9*exp(1) + 3*x*exp(x) + 9),x)

[Out]

-log(13*x - 9*exp(1) + 3*x*exp(x) + 9)

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sympy [A]  time = 0.20, size = 22, normalized size = 1.05 \begin {gather*} - \log {\relax (x )} - \log {\left (e^{x} + \frac {13 x - 9 e + 9}{3 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-3)*exp(x)-13)/(3*exp(x)*x-9*exp(1)+13*x+9),x)

[Out]

-log(x) - log(exp(x) + (13*x - 9*E + 9)/(3*x))

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