3.7.27 \(\int \frac {-1-e^3-6 x+3 x^2}{-x-2 x^2+3 x^3+e^3 (-x+x^2)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (5 e^5 \left (2 x-\left (-1+\frac {4+e^3}{1-x}\right ) x\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 20, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2074} \begin {gather*} -\log (1-x)+\log (x)+\log \left (3 x+e^3+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x] + Log[x] + Log[1 + E^3 + 3*x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 20, normalized size = 0.71 \begin {gather*} -\log (1-x)+\log (x)+\log \left (1+e^3+3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - x] + Log[x] + Log[1 + E^3 + 3*x]

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 19, normalized size = 0.68 \begin {gather*} \log \left (3 \, x^{2} + x e^{3} + x\right ) - \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)+3*x^2-6*x-1)/((x^2-x)*exp(3)+3*x^3-2*x^2-x),x, algorithm="fricas")

[Out]

log(3*x^2 + x*e^3 + x) - log(x - 1)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 20, normalized size = 0.71 \begin {gather*} \log \left ({\left | 3 \, x + e^{3} + 1 \right |}\right ) - \log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)+3*x^2-6*x-1)/((x^2-x)*exp(3)+3*x^3-2*x^2-x),x, algorithm="giac")

[Out]

log(abs(3*x + e^3 + 1)) - log(abs(x - 1)) + log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 18, normalized size = 0.64

method	result	size
default	\(\ln \relax (x )-\ln \left (x -1\right )+\ln \left (3 x +{\mathrm e}^{3}+1\right )\)	\(18\)
norman	\(\ln \relax (x )-\ln \left (x -1\right )+\ln \left (3 x +{\mathrm e}^{3}+1\right )\)	\(18\)
risch	\(-\ln \left (x -1\right )+\ln \left (3 x^{2}+x \left ({\mathrm e}^{3}+1\right )\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(3)+3*x^2-6*x-1)/((x^2-x)*exp(3)+3*x^3-2*x^2-x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 17, normalized size = 0.61 \begin {gather*} \log \left (3 \, x + e^{3} + 1\right ) - \log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(3)+3*x^2-6*x-1)/((x^2-x)*exp(3)+3*x^3-2*x^2-x),x, algorithm="maxima")

[Out]

log(3*x + e^3 + 1) - log(x - 1) + log(x)

________________________________________________________________________________________

mupad [B]  time = 0.72, size = 17, normalized size = 0.61 \begin {gather*} \ln \left (x\,\left (3\,x+{\mathrm {e}}^3+1\right )\right )-\ln \left (x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(3) - 3*x^2 + 1)/(x + exp(3)*(x - x^2) + 2*x^2 - 3*x^3),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 19, normalized size = 0.68 \begin {gather*} - \log {\left (x - 1 \right )} + \log {\left (x^{2} + x \left (\frac {1}{3} + \frac {e^{3}}{3}\right ) \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________