\(\int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+(-7 x+7 x^2) \log (x)} \, dx\) [6846]

Optimal result

Integrand size = 51, antiderivative size = 19 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\log \left (e^{-2 x/7}+x (-1-\log (x))+\log (x)\right ) \]

[Out]

ln(x*(-ln(x)-1)+exp(-2/7*x)+ln(x))

Rubi [F]

\[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx \]

[In]

Int[(-7 + 14*x + (2*x)/E^((2*x)/7) + 7*x*Log[x])/((-7*x)/E^((2*x)/7) + 7*x^2 + (-7*x + 7*x^2)*Log[x]),x]

[Out]

Log[x - Log[x] + x*Log[x]] + 2*Defer[Int][1/((x - Log[x] + x*Log[x])*(-1 + E^((2*x)/7)*x - E^((2*x)/7)*Log[x]
+ E^((2*x)/7)*x*Log[x])), x] - Defer[Int][1/(x*(x - Log[x] + x*Log[x])*(-1 + E^((2*x)/7)*x - E^((2*x)/7)*Log[x
] + E^((2*x)/7)*x*Log[x])), x] + (2*Defer[Int][x/((x - Log[x] + x*Log[x])*(-1 + E^((2*x)/7)*x - E^((2*x)/7)*Lo
g[x] + E^((2*x)/7)*x*Log[x])), x])/7 + (5*Defer[Int][Log[x]/((x - Log[x] + x*Log[x])*(-1 + E^((2*x)/7)*x - E^(
(2*x)/7)*Log[x] + E^((2*x)/7)*x*Log[x])), x])/7 + (2*Defer[Int][(x*Log[x])/((x - Log[x] + x*Log[x])*(-1 + E^((
2*x)/7)*x - E^((2*x)/7)*Log[x] + E^((2*x)/7)*x*Log[x])), x])/7

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1+2 x+x \log (x)}{x (x-\log (x)+x \log (x))}+\frac {-7+14 x+2 x^2+5 x \log (x)+2 x^2 \log (x)}{7 x (x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}\right ) \, dx \\ & = \frac {1}{7} \int \frac {-7+14 x+2 x^2+5 x \log (x)+2 x^2 \log (x)}{x (x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx+\int \frac {-1+2 x+x \log (x)}{x (x-\log (x)+x \log (x))} \, dx \\ & = \log (x-\log (x)+x \log (x))+\frac {1}{7} \int \left (\frac {14}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}-\frac {7}{x (x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}+\frac {2 x}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}+\frac {5 \log (x)}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}+\frac {2 x \log (x)}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )}\right ) \, dx \\ & = \log (x-\log (x)+x \log (x))+\frac {2}{7} \int \frac {x}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx+\frac {2}{7} \int \frac {x \log (x)}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx+\frac {5}{7} \int \frac {\log (x)}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx+2 \int \frac {1}{(x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx-\int \frac {1}{x (x-\log (x)+x \log (x)) \left (-1+e^{2 x/7} x-e^{2 x/7} \log (x)+e^{2 x/7} x \log (x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\log \left (e^{-2 x/7}-x+\log (x)-x \log (x)\right ) \]

[In]

Integrate[(-7 + 14*x + (2*x)/E^((2*x)/7) + 7*x*Log[x])/((-7*x)/E^((2*x)/7) + 7*x^2 + (-7*x + 7*x^2)*Log[x]),x]

[Out]

Log[E^((-2*x)/7) - x + Log[x] - x*Log[x]]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
norman	\(\ln \left (x \ln \left (x \right )-\ln \left (x \right )-{\mathrm e}^{-\frac {2 x}{7}}+x \right )\)	\(18\)
parallelrisch	\(\ln \left (x \ln \left (x \right )-\ln \left (x \right )-{\mathrm e}^{-\frac {2 x}{7}}+x \right )\)	\(18\)
risch	\(\ln \left (-1+x \right )+\ln \left (\ln \left (x \right )+\frac {x -{\mathrm e}^{-\frac {2 x}{7}}}{-1+x}\right )\)	\(24\)

[In]

int((7*x*ln(x)+2*x*exp(-2/7*x)+14*x-7)/((7*x^2-7*x)*ln(x)-7*x*exp(-2/7*x)+7*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(x*ln(x)-ln(x)-exp(-2/7*x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\log \left (x - 1\right ) + \log \left (\frac {{\left (x - 1\right )} \log \left (x\right ) + x - e^{\left (-\frac {2}{7} \, x\right )}}{x - 1}\right ) \]

[In]

integrate((7*x*log(x)+2*x*exp(-2/7*x)+14*x-7)/((7*x^2-7*x)*log(x)-7*x*exp(-2/7*x)+7*x^2),x, algorithm="fricas"
)

[Out]

log(x - 1) + log(((x - 1)*log(x) + x - e^(-2/7*x))/(x - 1))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\log {\left (- x \log {\left (x \right )} - x + \log {\left (x \right )} + e^{- \frac {2 x}{7}} \right )} \]

[In]

integrate((7*x*ln(x)+2*x*exp(-2/7*x)+14*x-7)/((7*x**2-7*x)*ln(x)-7*x*exp(-2/7*x)+7*x**2),x)

[Out]

log(-x*log(x) - x + log(x) + exp(-2*x/7))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (15) = 30\).

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=-\frac {2}{7} \, x + \log \left (x - 1\right ) + \log \left (\frac {{\left ({\left (x - 1\right )} \log \left (x\right ) + x\right )} e^{\left (\frac {2}{7} \, x\right )} - 1}{{\left (x - 1\right )} \log \left (x\right ) + x}\right ) + \log \left (\frac {{\left (x - 1\right )} \log \left (x\right ) + x}{x - 1}\right ) \]

[In]

integrate((7*x*log(x)+2*x*exp(-2/7*x)+14*x-7)/((7*x^2-7*x)*log(x)-7*x*exp(-2/7*x)+7*x^2),x, algorithm="maxima"
)

[Out]

-2/7*x + log(x - 1) + log((((x - 1)*log(x) + x)*e^(2/7*x) - 1)/((x - 1)*log(x) + x)) + log(((x - 1)*log(x) + x
)/(x - 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\log \left (x \log \left (x\right ) + x - e^{\left (-\frac {2}{7} \, x\right )} - \log \left (x\right )\right ) \]

[In]

integrate((7*x*log(x)+2*x*exp(-2/7*x)+14*x-7)/((7*x^2-7*x)*log(x)-7*x*exp(-2/7*x)+7*x^2),x, algorithm="giac")

[Out]

log(x*log(x) + x - e^(-2/7*x) - log(x))

Mupad [B] (verification not implemented)

Time = 12.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-7+14 x+2 e^{-2 x/7} x+7 x \log (x)}{-7 e^{-2 x/7} x+7 x^2+\left (-7 x+7 x^2\right ) \log (x)} \, dx=\ln \left (x-{\mathrm {e}}^{-\frac {2\,x}{7}}-\ln \left (x\right )+x\,\ln \left (x\right )\right ) \]

[In]

int(-(14*x + 2*x*exp(-(2*x)/7) + 7*x*log(x) - 7)/(7*x*exp(-(2*x)/7) + log(x)*(7*x - 7*x^2) - 7*x^2),x)

[Out]

log(x - exp(-(2*x)/7) - log(x) + x*log(x))