3.69.46 $\int \frac{-7+14x+2e^{-2x/7}x+7x\log (x)}{-7e^{-2x/7}x+7x^2+(-7x+7x^2)\log (x)}dx$

Optimal. Leaf size=19 $$\log (e^{-2x/7}+x(-1-\log (x))+\log (x))$$

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Rubi [F]  time = 3.11, 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 = {} $$\begin{array}{c}\int \frac{-7+14x+2e^{-2x/7}x+7x\log (x)}{-7e^{-2x/7}x+7x^2+(-7x+7x^2)\log (x)}dx\end{array}$$

Verification is not applicable to the result.

[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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{-1+2x+x\log (x)}{x(x-\log (x)+x\log (x))}+\frac{-7+14x+2x^2+5x\log (x)+2x^2\log (x)}{7x(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))})dx\\ & =\frac{1}{7}\int \frac{-7+14x+2x^2+5x\log (x)+2x^2\log (x)}{x(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx+\int \frac{-1+2x+x\log (x)}{x(x-\log (x)+x\log (x))}dx\\ & =\log (x-\log (x)+x\log (x))+\frac{1}{7}\int (\frac{14}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}-\frac{7}{x(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}+\frac{2x}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}+\frac{5\log (x)}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}+\frac{2x\log (x)}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))})dx\\ & =\log (x-\log (x)+x\log (x))+\frac{2}{7}\int \frac{x}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx+\frac{2}{7}\int \frac{x\log (x)}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx+\frac{5}{7}\int \frac{\log (x)}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx+2\int \frac{1}{(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx-\int \frac{1}{x(x-\log (x)+x\log (x))(-1+e^{2x/7}x-e^{2x/7}\log (x)+e^{2x/7}x\log (x))}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.65, size = 19, normalized size = 1.00 $$\begin{array}{c}\log (e^{-2x/7}-x+\log (x)-x\log (x))\end{array}$$

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.56, size = 26, normalized size = 1.37 $$\begin{array}{c}\log (x-1)+\log \left(\frac{(x-1)\log (x)+x-e^{(-\frac{2}{7}x)}}{x-1}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.21, size = 17, normalized size = 0.89 $$\begin{array}{c}\log (x\log (x)+x-e^{(-\frac{2}{7}x)}-\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.03, size = 18, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [B]  time = 0.40, size = 50, normalized size = 2.63 $$\begin{array}{c}-\frac{2}{7}x+\log (x-1)+\log \left(\frac{((x-1)\log (x)+x)e^{\left(\frac{2}{7}x\right)}-1}{(x-1)\log (x)+x}\right)+\log \left(\frac{(x-1)\log (x)+x}{x-1}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.28, size = 17, normalized size = 0.89 $$\begin{array}{c}\ln (x-{\mathrm{e}}^{-\frac{2x}{7}}-\ln (x)+x\ln (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.50, size = 19, normalized size = 1.00 $$\begin{array}{c}\log (-x\log (x)-x+\log (x)+e^{-\frac{2x}{7}})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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))

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