3.85.87 $\int \frac{4-10x+4x^2+e^x(-1-2x+7x^2-4x^3)+(-8+9x+e^x(2+5x-6x^2-x^3))\log (x)+(4+e^x(-1-2x-2x^2)){\log }^2(x)-e^{x}x{\log }^3(x)}{x^2-x^3+e^x(4x-8x^2+4x^3)+(-4x+7x^2-4x^3+e^x(-7x+6x^2+x^3))\log (x)+(8x-8x^2+e^x(2x+2x^2)){\log }^2(x)+(-4x+e^{x}x){\log }^3(x)}dx$

Optimal. Leaf size=26 $$\log \left(\frac{1}{-4\log (x)+e^x(4+\log (x))-\frac{x}{-1+x+\log (x)}}\right)$$

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Rubi [F]  time = 52.73, 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 = {} $$\begin{array}{c}\int \frac{4-10x+4x^2+e^x(-1-2x+7x^2-4x^3)+(-8+9x+e^x(2+5x-6x^2-x^3))\log (x)+(4+e^x(-1-2x-2x^2)){\log }^2(x)-e^{x}x{\log }^3(x)}{x^2-x^3+e^x(4x-8x^2+4x^3)+(-4x+7x^2-4x^3+e^x(-7x+6x^2+x^3))\log (x)+(8x-8x^2+e^x(2x+2x^2)){\log }^2(x)+(-4x+e^{x}x){\log }^3(x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*(2 + 5*x - 6*x^2 - x^3))*Log[x] + (4
+ E^x*(-1 - 2*x - 2*x^2))*Log[x]^2 - E^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*x^2 -
4*x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2*x^2))*Log[x]^2 + (-4*x + E^x*x)*Log[x]^
3),x]

[Out]

-x - Log[4 + Log[x]] - 39*Defer[Int][1/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3
+ x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 16*Defer[Int][1/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x)
- x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 19*Defer[Int][x/((4 + Log[x])*(-1 + x + Log
[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][x^2/((4 +
Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] +
17*Defer[Int][Log[x]/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-
4 + E^x)*Log[x]^2)), x] - 32*Defer[Int][Log[x]/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*
x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] + 29*Defer[Int][(x*Log[x])/((4 + Log[x])*(-1 + x + Log[x])
*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 17*Defer[Int][(x^2*Log[x])
/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)),
x] + 29*Defer[Int][Log[x]^2/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log
[x] + (-4 + E^x)*Log[x]^2)), x] + 16*Defer[Int][Log[x]^2/(x*(4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x
+ (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 25*Defer[Int][(x*Log[x]^2)/((4 + Log[x])*(-1 +
x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][(
x^2*Log[x]^2)/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x
)*Log[x]^2)), x] - 8*Defer[Int][Log[x]^3/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*
(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 8*Defer[Int][(x*Log[x]^3)/((4 + Log[x])*(-1 + x + Log[x])*(4*E^x
*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x] - 4*Defer[Int][Log[x]^4/((4 + Log[x
])*(-1 + x + Log[x])*(4*E^x*(-1 + x) - x + (4 - 4*x + E^x*(3 + x))*Log[x] + (-4 + E^x)*Log[x]^2)), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-4+10x-4x^2+e^x(-1+x{)}^2(1+4x)+(8-9x+e^x(-2-5x+6x^2+x^3))\log (x)+(-4+e^x(1+2x+2x^2)){\log }^2(x)+e^{x}x{\log }^3(x)}{x(1-x-\log (x))(4e^x(-1+x)-x+(4-4x+e^x(3+x))\log (x)+(-4+e^x){\log }^2(x))}dx\\ & =\int (\frac{-1-4x-x\log (x)}{x(4+\log (x))}-\frac{-16+39x-19x^2+4x^3+32\log (x)-17x\log (x)-29x^2\log (x)+17x^3\log (x)-16{\log }^2(x)-29x{\log }^2(x)+25x^2{\log }^2(x)+4x^3{\log }^2(x)+8x{\log }^3(x)+8x^2{\log }^3(x)+4x{\log }^4(x)}{x(4+\log (x))(-1+x+\log (x))(-4e^x-x+4e^{x}x+4\log (x)+3e^x\log (x)-4x\log (x)+e^{x}x\log (x)-4{\log }^2(x)+e^x{\log }^2(x))})dx\\ & =\int \frac{-1-4x-x\log (x)}{x(4+\log (x))}dx-\int \frac{-16+39x-19x^2+4x^3+32\log (x)-17x\log (x)-29x^2\log (x)+17x^3\log (x)-16{\log }^2(x)-29x{\log }^2(x)+25x^2{\log }^2(x)+4x^3{\log }^2(x)+8x{\log }^3(x)+8x^2{\log }^3(x)+4x{\log }^4(x)}{x(4+\log (x))(-1+x+\log (x))(-4e^x-x+4e^{x}x+4\log (x)+3e^x\log (x)-4x\log (x)+e^{x}x\log (x)-4{\log }^2(x)+e^x{\log }^2(x))}dx\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [B]  time = 0.23, size = 66, normalized size = 2.54 $$\begin{array}{c}\log (1-x-\log (x))-\log (-4e^x-x+4e^{x}x+4\log (x)+3e^x\log (x)-4x\log (x)+e^{x}x\log (x)-4{\log }^2(x)+e^x{\log }^2(x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(4 - 10*x + 4*x^2 + E^x*(-1 - 2*x + 7*x^2 - 4*x^3) + (-8 + 9*x + E^x*(2 + 5*x - 6*x^2 - x^3))*Log[x]
+ (4 + E^x*(-1 - 2*x - 2*x^2))*Log[x]^2 - E^x*x*Log[x]^3)/(x^2 - x^3 + E^x*(4*x - 8*x^2 + 4*x^3) + (-4*x + 7*
x^2 - 4*x^3 + E^x*(-7*x + 6*x^2 + x^3))*Log[x] + (8*x - 8*x^2 + E^x*(2*x + 2*x^2))*Log[x]^2 + (-4*x + E^x*x)*L
og[x]^3),x]

[Out]

Log[1 - x - Log[x]] - Log[-4*E^x - x + 4*E^x*x + 4*Log[x] + 3*E^x*Log[x] - 4*x*Log[x] + E^x*x*Log[x] - 4*Log[x
]^2 + E^x*Log[x]^2]

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fricas [B]  time = 0.82, size = 58, normalized size = 2.23 $$\begin{array}{c}\log (x+\log (x)-1)-\log \left(\frac{(e^x-4)\log (x{)}^2+4(x-1)e^x+((x+3)e^x-4x+4)\log (x)-x}{e^x-4}\right)-\log (e^x-4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="fricas")

[Out]

log(x + log(x) - 1) - log(((e^x - 4)*log(x)^2 + 4*(x - 1)*e^x + ((x + 3)*e^x - 4*x + 4)*log(x) - x)/(e^x - 4))
- log(e^x - 4)

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giac [B]  time = 0.69, size = 57, normalized size = 2.19 $$\begin{array}{c}-\log (x{e}^x\log (x)+e^x\log (x{)}^2+4x{e}^x-4x\log (x)+3e^x\log (x)-4\log (x{)}^2-x-4e^x+4\log (x))+\log (x+\log (x)-1)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="giac")

[Out]

-log(x*e^x*log(x) + e^x*log(x)^2 + 4*x*e^x - 4*x*log(x) + 3*e^x*log(x) - 4*log(x)^2 - x - 4*e^x + 4*log(x)) +
log(x + log(x) - 1)

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maple [B]  time = 0.06, size = 65, normalized size = 2.50

method	result	size
risch	$-\ln ({\mathrm{e}}^x-4)+\ln (-1+\ln (x)+x)-\ln (\ln (x{)}^2+\frac{({\mathrm{e}}^{x}x-4x+3{\mathrm{e}}^x+4)\ln (x)}{{\mathrm{e}}^x-4}+\frac{4{\mathrm{e}}^{x}x-x-4{\mathrm{e}}^x}{{\mathrm{e}}^x-4})$	$65$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*exp(x)*ln(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*ln(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*ln(x)+(-4*x^3+7*x^2
-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*ln(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*ln(x)^2+((x^3+6*x^2-7*x)*e
xp(x)-4*x^3+7*x^2-4*x)*ln(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(x)-4)+ln(-1+ln(x)+x)-ln(ln(x)^2+(exp(x)*x-4*x+3*exp(x)+4)/(exp(x)-4)*ln(x)+(4*exp(x)*x-x-4*exp(x))/(ex
p(x)-4))

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maxima [B]  time = 0.45, size = 64, normalized size = 2.46 $$\begin{array}{c}-\log \left(\frac{((x+3)\log (x)+\log (x{)}^2+4x-4)e^x-4(x-1)\log (x)-4\log (x{)}^2-x}{(x+3)\log (x)+\log (x{)}^2+4x-4}\right)-\log (\log (x)+4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*log(x)^3+((-2*x^2-2*x-1)*exp(x)+4)*log(x)^2+((-x^3-6*x^2+5*x+2)*exp(x)+9*x-8)*log(x)+(-4*
x^3+7*x^2-2*x-1)*exp(x)+4*x^2-10*x+4)/((exp(x)*x-4*x)*log(x)^3+((2*x^2+2*x)*exp(x)-8*x^2+8*x)*log(x)^2+((x^3+6
*x^2-7*x)*exp(x)-4*x^3+7*x^2-4*x)*log(x)+(4*x^3-8*x^2+4*x)*exp(x)-x^3+x^2),x, algorithm="maxima")

[Out]

-log((((x + 3)*log(x) + log(x)^2 + 4*x - 4)*e^x - 4*(x - 1)*log(x) - 4*log(x)^2 - x)/((x + 3)*log(x) + log(x)^
2 + 4*x - 4)) - log(log(x) + 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\int \frac{10x-\ln (x)(9x+{\mathrm{e}}^x(-x^3-6x^2+5x+2)-8)+{\ln (x)}^2({\mathrm{e}}^x(2x^2+2x+1)-4)-4x^2+{\mathrm{e}}^x(4x^3-7x^2+2x+1)+x{\mathrm{e}}^x{\ln (x)}^3-4}{\ln (x)(4x-{\mathrm{e}}^x(x^3+6x^2-7x)-7x^2+4x^3)+{\ln (x)}^3(4x-x{\mathrm{e}}^x)-{\ln (x)}^2(8x+{\mathrm{e}}^x(2x^2+2x)-8x^2)-x^2+x^3-{\mathrm{e}}^x(4x^3-8x^2+4x)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*(exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^
2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^
2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2) - 8*x^2) - x^2 + x^3 - exp(x)*(4
*x - 8*x^2 + 4*x^3)),x)

[Out]

int((10*x - log(x)*(9*x + exp(x)*(5*x - 6*x^2 - x^3 + 2) - 8) + log(x)^2*(exp(x)*(2*x + 2*x^2 + 1) - 4) - 4*x^
2 + exp(x)*(2*x - 7*x^2 + 4*x^3 + 1) + x*exp(x)*log(x)^3 - 4)/(log(x)*(4*x - exp(x)*(6*x^2 - 7*x + x^3) - 7*x^
2 + 4*x^3) + log(x)^3*(4*x - x*exp(x)) - log(x)^2*(8*x + exp(x)*(2*x + 2*x^2) - 8*x^2) - x^2 + x^3 - exp(x)*(4
*x - 8*x^2 + 4*x^3)), x)

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sympy [B]  time = 4.01, size = 53, normalized size = 2.04 $$\begin{array}{c}-\log (\frac{-4x\log (x)-x-4\log {(x)}^2+4\log (x)}{x\log (x)+4x+\log {(x)}^2+3\log (x)-4}+e^x)-\log (\log (x)+4)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(x)*ln(x)**3+((-2*x**2-2*x-1)*exp(x)+4)*ln(x)**2+((-x**3-6*x**2+5*x+2)*exp(x)+9*x-8)*ln(x)+(-
4*x**3+7*x**2-2*x-1)*exp(x)+4*x**2-10*x+4)/((exp(x)*x-4*x)*ln(x)**3+((2*x**2+2*x)*exp(x)-8*x**2+8*x)*ln(x)**2+
((x**3+6*x**2-7*x)*exp(x)-4*x**3+7*x**2-4*x)*ln(x)+(4*x**3-8*x**2+4*x)*exp(x)-x**3+x**2),x)

[Out]

-log((-4*x*log(x) - x - 4*log(x)**2 + 4*log(x))/(x*log(x) + 4*x + log(x)**2 + 3*log(x) - 4) + exp(x)) - log(lo
g(x) + 4)

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