3.102.89 \(\int \frac {-1+4 x+4 x^2}{-2 x^2+4 x^3+e (-x+2 x^2)+(-x+2 x^2) \log (x)+(-2 x+4 x^2) \log (-1+2 x)} \, dx\) [10189]

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

[Out]

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

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Rubi [A]
time = 0.22, 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 = {6820, 6816} \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 6816

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

Rule 6820

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*}

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Mathematica [A]
time = 0.07, 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]]

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Maple [A]
time = 1.12, size = 18, normalized size = 1.12

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

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

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Maxima [A]
time = 0.30, 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 \left (x\right )\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(-1+2*x)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="
maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 17, normalized size = 1.06 \begin {gather*} \log \left (2 \, x + e + 2 \, \log \left (2 \, x - 1\right ) + \log \left (x\right )\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(-1+2*x)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="
fricas")

[Out]

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

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Sympy [A]
time = 0.14, size = 19, normalized size = 1.19 \begin {gather*} \log {\left (x + \frac {\log {\left (x \right )}}{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(-1+2*x)+(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)

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Giac [A]
time = 0.40, size = 17, normalized size = 1.06 \begin {gather*} \log \left (2 \, x + e + 2 \, \log \left (2 \, x - 1\right ) + \log \left (x\right )\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(-1+2*x)+(2*x^2-x)*log(x)+(2*x^2-x)*exp(1)+4*x^3-2*x^2),x, algorithm="
giac")

[Out]

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

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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 \left (x\right )\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))

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