3.102.89 \(\int \frac {-1+4 x+4 x^2}{-2 x^2+4 x^3+e (-x+2 x^2)+(-x+2 x^2) \log (x)+(-2 x+4 x^2) \log (-1+2 x)} \, dx\)

Optimal. Leaf size=16 \[ \log (e+2 x+\log (x)+2 \log (-1+2 x)) \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6688, 6684} \begin {gather*} \log (2 x+\log (x)+2 \log (2 x-1)+e) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x + 4*x^2)/(-2*x^2 + 4*x^3 + E*(-x + 2*x^2) + (-x + 2*x^2)*Log[x] + (-2*x + 4*x^2)*Log[-1 + 2*x]),
x]

[Out]

Log[E + 2*x + Log[x] + 2*Log[-1 + 2*x]]

Rule 6684

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.54, size = 16, normalized size = 1.00 \begin {gather*} \log (e+2 x+\log (x)+2 \log (-1+2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 4*x + 4*x^2)/(-2*x^2 + 4*x^3 + E*(-x + 2*x^2) + (-x + 2*x^2)*Log[x] + (-2*x + 4*x^2)*Log[-1 +
2*x]),x]

[Out]

Log[E + 2*x + Log[x] + 2*Log[-1 + 2*x]]

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 17, normalized size = 1.06 \begin {gather*} \log \left (2 \, x + e + 2 \, \log \left (2 \, x - 1\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+4*x-1)/((4*x^2-2*x)*log(2*x-1)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="f
ricas")

[Out]

log(2*x + e + 2*log(2*x - 1) + log(x))

________________________________________________________________________________________

giac [A]  time = 0.16, size = 17, normalized size = 1.06 \begin {gather*} \log \left (2 \, x + e + 2 \, \log \left (2 \, x - 1\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+4*x-1)/((4*x^2-2*x)*log(2*x-1)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="g
iac")

[Out]

log(2*x + e + 2*log(2*x - 1) + log(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 18, normalized size = 1.12

method	result	size
risch	\(\ln \left (x +\frac {{\mathrm e}}{2}+\frac {\ln \relax (x )}{2}+\ln \left (2 x -1\right )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2+4*x-1)/((4*x^2-2*x)*ln(2*x-1)+(2*x^2-x)*ln(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x,method=_RETURNVERBOSE
)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 17, normalized size = 1.06 \begin {gather*} \log \left (x + \frac {1}{2} \, e + \log \left (2 \, x - 1\right ) + \frac {1}{2} \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2+4*x-1)/((4*x^2-2*x)*log(2*x-1)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="m
axima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 7.63, size = 17, normalized size = 1.06 \begin {gather*} \ln \left (2\,x+\mathrm {e}+2\,\ln \left (2\,x-1\right )+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 4*x^2 - 1)/(log(x)*(x - 2*x^2) + log(2*x - 1)*(2*x - 4*x^2) + exp(1)*(x - 2*x^2) + 2*x^2 - 4*x^3),
x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.39, size = 19, normalized size = 1.19 \begin {gather*} \log {\left (x + \frac {\log {\relax (x )}}{2} + \log {\left (2 x - 1 \right )} + \frac {e}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x)/2 + log(2*x - 1) + E/2)

________________________________________________________________________________________