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

Optimal. Leaf size=19 \[ 16+\log \left (\frac {1}{5 \left (e^3-x\right ) x}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 0.74, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {628} \begin {gather*} -\log \left (e^3 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^3 + 2*x)/(E^3*x - x^2),x]

[Out]

-Log[E^3*x - x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (e^3 x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 \begin {gather*} -\log \left (e^3-x\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[E^3 - x] - Log[x]

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fricas [A]  time = 0.88, size = 12, normalized size = 0.63 \begin {gather*} -\log \left (x^{2} - x e^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x^2 - x*e^3)

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giac [A]  time = 0.14, size = 13, normalized size = 0.68 \begin {gather*} -\log \left ({\left | x^{2} - x e^{3} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x^2 - x*e^3))

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maple [A]  time = 0.11, size = 12, normalized size = 0.63

method	result	size
default	\(-\ln \left (x \left (-x +{\mathrm e}^{3}\right )\right )\)	\(12\)
risch	\(-\ln \left (-x \,{\mathrm e}^{3}+x^{2}\right )\)	\(13\)
norman	\(-\ln \relax (x )-\ln \left (-x +{\mathrm e}^{3}\right )\)	\(15\)
meijerg	\(-\ln \left (1-x \,{\mathrm e}^{-3}\right )-\ln \relax (x )+3-i \pi \)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.39, size = 12, normalized size = 0.63 \begin {gather*} -\log \left (x^{2} - x e^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x^2 - x*e^3)

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mupad [B]  time = 0.11, size = 11, normalized size = 0.58 \begin {gather*} -\ln \left (x\,\left (x-{\mathrm {e}}^3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x*(x - exp(3)))

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sympy [A]  time = 0.12, size = 10, normalized size = 0.53 \begin {gather*} - \log {\left (x^{2} - x e^{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x**2 - x*exp(3))

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