3.41.64 $\int \frac{-21-3x+7x^2-x^3+e^x(5+x)+(5+2x-x^2)\log (x)+(5-e^x+4x-x^2+(-1-x)\log (x))\log (x^2)}{4x-5x^2+x^3+e^x(-x+x^2)+(-x+x^2)\log (x)+(-4x+e^{x}x+x^2+x\log (x))\log (x^2)}dx$

Optimal. Leaf size=32 $$-x+\log \left(\frac{(4-e^x-x-\log (x)){(-1+x+\log \left(x^2\right))}^2}{x}\right)$$

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Rubi [F]  time = 2.12, 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 = {} $$\begin{array}{c}\int \frac{-21-3x+7x^2-x^3+e^x(5+x)+(5+2x-x^2)\log (x)+(5-e^x+4x-x^2+(-1-x)\log (x))\log \left(x^2\right)}{4x-5x^2+x^3+e^x(-x+x^2)+(-x+x^2)\log (x)+(-4x+e^{x}x+x^2+x\log (x))\log \left(x^2\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-21 - 3*x + 7*x^2 - x^3 + E^x*(5 + x) + (5 + 2*x - x^2)*Log[x] + (5 - E^x + 4*x - x^2 + (-1 - x)*Log[x])*
Log[x^2])/(4*x - 5*x^2 + x^3 + E^x*(-x + x^2) + (-x + x^2)*Log[x] + (-4*x + E^x*x + x^2 + x*Log[x])*Log[x^2]),
x]

[Out]

-Log[x] + 2*Log[1 - x - Log[x^2]] + 5*Defer[Int][(-4 + E^x + x + Log[x])^(-1), x] + Defer[Int][1/(x*(-4 + E^x
+ x + Log[x])), x] - Defer[Int][x/(-4 + E^x + x + Log[x]), x] - Defer[Int][Log[x]/(-4 + E^x + x + Log[x]), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-21-3x+7x^2-x^3+e^x(5+x)+(5+2x-x^2)\log (x)+(5-e^x+4x-x^2+(-1-x)\log (x))\log \left(x^2\right)}{x(4-e^x-x-\log (x))(1-x-\log \left(x^2\right))}dx\\ & =\int (-\frac{-1-5x+x^2+x\log (x)}{x(-4+e^x+x+\log (x))}+\frac{5+x-\log \left(x^2\right)}{x(-1+x+\log \left(x^2\right))})dx\\ & =-\int \frac{-1-5x+x^2+x\log (x)}{x(-4+e^x+x+\log (x))}dx+\int \frac{5+x-\log \left(x^2\right)}{x(-1+x+\log \left(x^2\right))}dx\\ & =-\int (-\frac{5}{-4+e^x+x+\log (x)}-\frac{1}{x(-4+e^x+x+\log (x))}+\frac{x}{-4+e^x+x+\log (x)}+\frac{\log (x)}{-4+e^x+x+\log (x)})dx+\int (-\frac{1}{x}+\frac{2(2+x)}{x(-1+x+\log \left(x^2\right))})dx\\ & =-\log (x)+2\int \frac{2+x}{x(-1+x+\log \left(x^2\right))}dx+5\int \frac{1}{-4+e^x+x+\log (x)}dx+\int \frac{1}{x(-4+e^x+x+\log (x))}dx-\int \frac{x}{-4+e^x+x+\log (x)}dx-\int \frac{\log (x)}{-4+e^x+x+\log (x)}dx\\ & =-\log (x)+2\log (1-x-\log \left(x^2\right))+5\int \frac{1}{-4+e^x+x+\log (x)}dx+\int \frac{1}{x(-4+e^x+x+\log (x))}dx-\int \frac{x}{-4+e^x+x+\log (x)}dx-\int \frac{\log (x)}{-4+e^x+x+\log (x)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.09, size = 37, normalized size = 1.16 $$\begin{array}{c}-x-\log (x)+\log (4-e^x-x-\log (x))+2\log (1-x-\log \left(x^2\right))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-21 - 3*x + 7*x^2 - x^3 + E^x*(5 + x) + (5 + 2*x - x^2)*Log[x] + (5 - E^x + 4*x - x^2 + (-1 - x)*Lo
g[x])*Log[x^2])/(4*x - 5*x^2 + x^3 + E^x*(-x + x^2) + (-x + x^2)*Log[x] + (-4*x + E^x*x + x^2 + x*Log[x])*Log[
x^2]),x]

[Out]

-x - Log[x] + Log[4 - E^x - x - Log[x]] + 2*Log[1 - x - Log[x^2]]

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fricas [A]  time = 0.90, size = 26, normalized size = 0.81 $$\begin{array}{c}-x+\log (x+e^x+\log (x)-4)+2\log (x+2\log (x)-1)-\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*log(x)-exp(x)-x^2+4*x+5)*log(x^2)+(-x^2+2*x+5)*log(x)+(5+x)*exp(x)-x^3+7*x^2-3*x-21)/((x*lo
g(x)+exp(x)*x+x^2-4*x)*log(x^2)+log(x)*(x^2-x)+(x^2-x)*exp(x)+x^3-5*x^2+4*x),x, algorithm="fricas")

[Out]

-x + log(x + e^x + log(x) - 4) + 2*log(x + 2*log(x) - 1) - log(x)

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giac [A]  time = 0.37, size = 28, normalized size = 0.88 $$\begin{array}{c}-x+\log (x+e^x+\log (x)-4)-\log (x)+2\log (-x-2\log (x)+1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*log(x)-exp(x)-x^2+4*x+5)*log(x^2)+(-x^2+2*x+5)*log(x)+(5+x)*exp(x)-x^3+7*x^2-3*x-21)/((x*lo
g(x)+exp(x)*x+x^2-4*x)*log(x^2)+log(x)*(x^2-x)+(x^2-x)*exp(x)+x^3-5*x^2+4*x),x, algorithm="giac")

[Out]

-x + log(x + e^x + log(x) - 4) - log(x) + 2*log(-x - 2*log(x) + 1)

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maple [A]  time = 0.25, size = 27, normalized size = 0.84

method	result	size
default	$-\ln (x)-x+2\ln (-1+x+\ln \left(x^2\right))+\ln (x+{\mathrm{e}}^x+\ln (x)-4)$	$27$
risch	$-x-\ln (x)+\ln (x+{\mathrm{e}}^x+\ln (x)-4)+2\ln (\ln (x)-\frac{i(\pi \mathrm{csgn}{\left(ix\right)}^2\mathrm{csgn}\left(i{x}^2\right)-2\pi \mathrm{csgn}\left(ix\right)\mathrm{csgn}{\left(i{x}^2\right)}^2+\pi \mathrm{csgn}{\left(i{x}^2\right)}^3+2ix-2i)}{4})$	$77$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x-1)*ln(x)-exp(x)-x^2+4*x+5)*ln(x^2)+(-x^2+2*x+5)*ln(x)+(5+x)*exp(x)-x^3+7*x^2-3*x-21)/((x*ln(x)+exp(x
)*x+x^2-4*x)*ln(x^2)+ln(x)*(x^2-x)+(x^2-x)*exp(x)+x^3-5*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)-x+2*ln(-1+x+ln(x^2))+ln(x+exp(x)+ln(x)-4)

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maxima [A]  time = 0.42, size = 26, normalized size = 0.81 $$\begin{array}{c}-x+\log (x+e^x+\log (x)-4)-\log (x)+2\log (\frac{1}{2}x+\log (x)-\frac{1}{2})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*log(x)-exp(x)-x^2+4*x+5)*log(x^2)+(-x^2+2*x+5)*log(x)+(5+x)*exp(x)-x^3+7*x^2-3*x-21)/((x*lo
g(x)+exp(x)*x+x^2-4*x)*log(x^2)+log(x)*(x^2-x)+(x^2-x)*exp(x)+x^3-5*x^2+4*x),x, algorithm="maxima")

[Out]

-x + log(x + e^x + log(x) - 4) - log(x) + 2*log(1/2*x + log(x) - 1/2)

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mupad [B]  time = 3.31, size = 60, normalized size = 1.88 $$\begin{array}{c}\ln \left(\frac{(x+2)(x+{\mathrm{e}}^x+\ln (x)-4)}{x}\right)-x-2\ln \left(\frac{x+x{\mathrm{e}}^x+1}{x}\right)-\ln (x+2)+2\ln \left(\frac{(x+\ln \left(x^2\right)-1)(x+x{\mathrm{e}}^x+1)}{x}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - exp(x)*(x + 5) + log(x^2)*(exp(x) - 4*x + log(x)*(x + 1) + x^2 - 5) - log(x)*(2*x - x^2 + 5) - 7*x
^2 + x^3 + 21)/(4*x + log(x^2)*(x*exp(x) - 4*x + x*log(x) + x^2) - exp(x)*(x - x^2) - log(x)*(x - x^2) - 5*x^2
+ x^3),x)

[Out]

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

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sympy [A]  time = 0.44, size = 29, normalized size = 0.91 $$\begin{array}{c}-x-\log (x)+2\log (\frac{x}{2}+\log (x)-\frac{1}{2})+\log (x+e^x+\log (x)-4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x-1)*ln(x)-exp(x)-x**2+4*x+5)*ln(x**2)+(-x**2+2*x+5)*ln(x)+(5+x)*exp(x)-x**3+7*x**2-3*x-21)/((x*
ln(x)+exp(x)*x+x**2-4*x)*ln(x**2)+ln(x)*(x**2-x)+(x**2-x)*exp(x)+x**3-5*x**2+4*x),x)

[Out]

-x - log(x) + 2*log(x/2 + log(x) - 1/2) + log(x + exp(x) + log(x) - 4)

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