3.67.49 \(\int \frac {e^6 (-2+3 x)}{-4 x+4 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{4} e^6 \log \left (\frac {3}{10} e^2 x \left (-x+x^2\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 631} \begin {gather*} \frac {1}{4} e^6 \log (1-x)+\frac {1}{2} e^6 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^6*(-2 + 3*x))/(-4*x + 4*x^2),x]

[Out]

(E^6*Log[1 - x])/4 + (E^6*Log[x])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.91 \begin {gather*} e^6 \left (\frac {1}{4} \log (1-x)+\frac {\log (x)}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^6*(-2 + 3*x))/(-4*x + 4*x^2),x]

[Out]

E^6*(Log[1 - x]/4 + Log[x]/2)

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fricas [A]  time = 0.53, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{4} \, e^{6} \log \left (x - 1\right ) + \frac {1}{2} \, e^{6} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*e^6*log(x - 1) + 1/2*e^6*log(x)

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giac [A]  time = 0.14, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{4} \, {\left (\log \left ({\left | x - 1 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right )\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(abs(x - 1)) + 2*log(abs(x)))*e^6

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maple [A]  time = 0.08, size = 16, normalized size = 0.70

method	result	size
default	\(\frac {{\mathrm e}^{6} \left (2 \ln \relax (x )+\ln \left (x -1\right )\right )}{4}\)	\(16\)
risch	\(\frac {{\mathrm e}^{6} \ln \relax (x )}{2}+\frac {{\mathrm e}^{6} \ln \left (x -1\right )}{4}\)	\(16\)
norman	\(\frac {{\mathrm e}^{6} \ln \relax (x )}{2}+\frac {{\mathrm e}^{6} \ln \left (x -1\right )}{4}\)	\(20\)
meijerg	\(\frac {{\mathrm e}^{6} \left (\ln \relax (x )+i \pi -\ln \left (1-x \right )\right )}{2}+\frac {3 \,{\mathrm e}^{6} \ln \left (1-x \right )}{4}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(3)^2*(2*ln(x)+ln(x-1))

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maxima [A]  time = 0.42, size = 13, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, {\left (\log \left (x - 1\right ) + 2 \, \log \relax (x)\right )} e^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(x - 1) + 2*log(x))*e^6

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mupad [B]  time = 0.07, size = 15, normalized size = 0.65 \begin {gather*} \frac {\ln \left (x-1\right )\,{\mathrm {e}}^6}{4}+\frac {{\mathrm {e}}^6\,\ln \relax (x)}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6)*(3*x - 2))/(4*x - 4*x^2),x)

[Out]

(log(x - 1)*exp(6))/4 + (exp(6)*log(x))/2

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sympy [A]  time = 0.20, size = 17, normalized size = 0.74 \begin {gather*} \frac {e^{6} \log {\relax (x )}}{2} + \frac {e^{6} \log {\left (x - 1 \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(6)*log(x)/2 + exp(6)*log(x - 1)/4

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