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

Optimal. Leaf size=29 \[ \log \left (e^x \left (5-e+e^4\right ) \left (-x+\log \left (\frac {4+\frac {4}{x}}{x}\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2-2 x-2 x^2-x^3+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x^2}\right )}{-x^2-x^3+\left (x+x^2\right ) \log \left (\frac {4+4 x}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + 2*Defer[Int][(x - Log[(4*(1 + x))/x^2])^(-1), x] + 2*Defer[Int][1/(x*(x - Log[(4*(1 + x))/x^2])), x] - Def
er[Int][1/((1 + x)*(x - Log[(4*(1 + x))/x^2])), x] + Defer[Int][(-x + Log[(4*(1 + x))/x^2])^(-1), x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 16, normalized size = 0.55 \begin {gather*} x+\log \left (x-\log \left (\frac {4 (1+x)}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + Log[x - Log[(4*(1 + x))/x^2]]

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 16, normalized size = 0.55 \begin {gather*} x + \log \left (-x + \log \left (\frac {4 \, {\left (x + 1\right )}}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(-x + log(4*(x + 1)/x^2))

________________________________________________________________________________________

giac [A]  time = 0.16, size = 16, normalized size = 0.55 \begin {gather*} x + \log \left (x - \log \left (\frac {4 \, {\left (x + 1\right )}}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - log(4*(x + 1)/x^2))

________________________________________________________________________________________

maple [A]  time = 0.07, size = 18, normalized size = 0.62

method	result	size
norman	\(x +\ln \left (x -\ln \left (\frac {4 x +4}{x^{2}}\right )\right )\)	\(18\)
risch	\(x +\ln \left (-x +\ln \left (\frac {4 x +4}{x^{2}}\right )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+ln(x-ln((4*x+4)/x^2))

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 19, normalized size = 0.66 \begin {gather*} x + \log \left (-x + 2 \, \log \relax (2) + \log \left (x + 1\right ) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.27, size = 17, normalized size = 0.59 \begin {gather*} x+\ln \left (x-\ln \left (\frac {4\,x+4}{x^2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x - log((4*x + 4)/x^2))

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 14, normalized size = 0.48 \begin {gather*} x + \log {\left (- x + \log {\left (\frac {4 x + 4}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(-x + log((4*x + 4)/x**2))

________________________________________________________________________________________