3.89.4 $\int \frac{-60x^2+40x^3-8x^4+x^5+e^4(x^3-x^4)+(-44x+60x^2-18x^3+2x^4+e^4(2x^2-2x^3))\log (-1+x)+(-16+24x-9x^2+x^3+e^4(x-x^2)){\log }^2(-1+x)}{-16x^3+24x^4-9x^5+x^6+(-32x^2+48x^3-18x^4+2x^5)\log (-1+x)+(-16x+24x^2-9x^3+x^4){\log }^2(-1+x)}dx$

Optimal. Leaf size=28 $$5+\frac{e^4+\frac{-8-x}{x+\log (-1+x)}}{-4+x}+\log (x)$$

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Rubi [F]  time = 2.98, 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 = {} $$\begin{array}{c}\int \frac{-60x^2+40x^3-8x^4+x^5+e^4(x^3-x^4)+(-44x+60x^2-18x^3+2x^4+e^4(2x^2-2x^3))\log (-1+x)+(-16+24x-9x^2+x^3+e^4(x-x^2)){\log }^2(-1+x)}{-16x^3+24x^4-9x^5+x^6+(-32x^2+48x^3-18x^4+2x^5)\log (-1+x)+(-16x+24x^2-9x^3+x^4){\log }^2(-1+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-60*x^2 + 40*x^3 - 8*x^4 + x^5 + E^4*(x^3 - x^4) + (-44*x + 60*x^2 - 18*x^3 + 2*x^4 + E^4*(2*x^2 - 2*x^3)
)*Log[-1 + x] + (-16 + 24*x - 9*x^2 + x^3 + E^4*(x - x^2))*Log[-1 + x]^2)/(-16*x^3 + 24*x^4 - 9*x^5 + x^6 + (-
32*x^2 + 48*x^3 - 18*x^4 + 2*x^5)*Log[-1 + x] + (-16*x + 24*x^2 - 9*x^3 + x^4)*Log[-1 + x]^2),x]

[Out]

-(E^4/(4 - x)) + Log[x] + Defer[Int][(x + Log[-1 + x])^(-2), x] + 16*Defer[Int][1/((-4 + x)*(x + Log[-1 + x])^
2), x] - 3*Defer[Int][1/((-1 + x)*(x + Log[-1 + x])^2), x] + 12*Defer[Int][1/((-4 + x)^2*(x + Log[-1 + x])), x
]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-x^2(-60+(40+e^4)x-(8+e^4)x^2+x^3)-2x(-22+(30+e^4)x-(9+e^4)x^2+x^3)\log (-1+x)-(-16+(24+e^4)x-(9+e^4)x^2+x^3){\log }^2(-1+x)}{(1-x)(4-x{)}^{2}x(x+\log (-1+x){)}^2}dx\\ & =\int (\frac{16-(8+e^4)x+x^2}{(4-x{)}^{2}x}+\frac{x(8+x)}{(-4+x)(-1+x)(x+\log (-1+x){)}^2}+\frac{12}{(-4+x{)}^2(x+\log (-1+x))})dx\\ & =12\int \frac{1}{(-4+x{)}^2(x+\log (-1+x))}dx+\int \frac{16-(8+e^4)x+x^2}{(4-x{)}^{2}x}dx+\int \frac{x(8+x)}{(-4+x)(-1+x)(x+\log (-1+x){)}^2}dx\\ & =12\int \frac{1}{(-4+x{)}^2(x+\log (-1+x))}dx+\int (-\frac{e^4}{(-4+x{)}^2}+\frac{1}{x})dx+\int (\frac{1}{(x+\log (-1+x){)}^2}+\frac{16}{(-4+x)(x+\log (-1+x){)}^2}-\frac{3}{(-1+x)(x+\log (-1+x){)}^2})dx\\ & =-\frac{e^4}{4-x}+\log (x)-3\int \frac{1}{(-1+x)(x+\log (-1+x){)}^2}dx+12\int \frac{1}{(-4+x{)}^2(x+\log (-1+x))}dx+16\int \frac{1}{(-4+x)(x+\log (-1+x){)}^2}dx+\int \frac{1}{(x+\log (-1+x){)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.10, size = 31, normalized size = 1.11 $$\begin{array}{c}\frac{e^4}{-4+x}+\frac{-8-x}{(-4+x)(x+\log (-1+x))}+\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-60*x^2 + 40*x^3 - 8*x^4 + x^5 + E^4*(x^3 - x^4) + (-44*x + 60*x^2 - 18*x^3 + 2*x^4 + E^4*(2*x^2 -
2*x^3))*Log[-1 + x] + (-16 + 24*x - 9*x^2 + x^3 + E^4*(x - x^2))*Log[-1 + x]^2)/(-16*x^3 + 24*x^4 - 9*x^5 + x^
6 + (-32*x^2 + 48*x^3 - 18*x^4 + 2*x^5)*Log[-1 + x] + (-16*x + 24*x^2 - 9*x^3 + x^4)*Log[-1 + x]^2),x]

[Out]

E^4/(-4 + x) + (-8 - x)/((-4 + x)*(x + Log[-1 + x])) + Log[x]

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fricas [A]  time = 0.72, size = 52, normalized size = 1.86 $$\begin{array}{c}\frac{x{e}^4+e^4\log (x-1)+(x^2+(x-4)\log (x-1)-4x)\log (x)-x-8}{x^2+(x-4)\log (x-1)-4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(x-1)^2+((-2*x^3+2*x^2)*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log
(x-1)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(x-1)^2+(2*x^5-18*x^4+48*x^3-32*x
^2)*log(x-1)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="fricas")

[Out]

(x*e^4 + e^4*log(x - 1) + (x^2 + (x - 4)*log(x - 1) - 4*x)*log(x) - x - 8)/(x^2 + (x - 4)*log(x - 1) - 4*x)

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giac [B]  time = 0.19, size = 65, normalized size = 2.32 $$\begin{array}{c}\frac{x^2\log (x)+x\log (x-1)\log (x)+x{e}^4+e^4\log (x-1)-4x\log (x)-4\log (x-1)\log (x)-x-8}{x^2+x\log (x-1)-4x-4\log (x-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(x-1)^2+((-2*x^3+2*x^2)*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log
(x-1)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(x-1)^2+(2*x^5-18*x^4+48*x^3-32*x
^2)*log(x-1)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="giac")

[Out]

(x^2*log(x) + x*log(x - 1)*log(x) + x*e^4 + e^4*log(x - 1) - 4*x*log(x) - 4*log(x - 1)*log(x) - x - 8)/(x^2 +
x*log(x - 1) - 4*x - 4*log(x - 1))

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maple [A]  time = 0.22, size = 37, normalized size = 1.32

method	result	size
risch	$\frac{x\ln (x)-4\ln (x)+{\mathrm{e}}^4}{x-4}-\frac{x+8}{(x-4)(x+\ln (x-1))}$	$37$
norman	$\frac{-8+{\mathrm{e}}^4\ln (x-1)+({\mathrm{e}}^4-1)x}{\ln (x-1)x+x^2-4\ln (x-1)-4x}+\ln (x)$	$41$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*ln(x-1)^2+((-2*x^3+2*x^2)*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*ln(x-1)+(-
x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*ln(x-1)^2+(2*x^5-18*x^4+48*x^3-32*x^2)*ln(x-
1)+x^6-9*x^5+24*x^4-16*x^3),x,method=_RETURNVERBOSE)

[Out]

(x*ln(x)-4*ln(x)+exp(4))/(x-4)-(x+8)/(x-4)/(x+ln(x-1))

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maxima [A]  time = 0.43, size = 36, normalized size = 1.29 $$\begin{array}{c}\frac{x(e^4-1)+e^4\log (x-1)-8}{x^2+(x-4)\log (x-1)-4x}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(x-1)^2+((-2*x^3+2*x^2)*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log
(x-1)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(x-1)^2+(2*x^5-18*x^4+48*x^3-32*x
^2)*log(x-1)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="maxima")

[Out]

(x*(e^4 - 1) + e^4*log(x - 1) - 8)/(x^2 + (x - 4)*log(x - 1) - 4*x) + log(x)

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mupad [B]  time = 5.72, size = 78, normalized size = 2.79 $$\begin{array}{c}\ln (x)-\frac{\frac{x^2+16x-44}{{(x-4)}^2}+\frac{12\ln (x-1)(x-1)}{x{(x-4)}^2}}{x+\ln (x-1)}-\frac{-{\mathrm{e}}^4{x}^2+(4{\mathrm{e}}^4-12)x+12}{x^3-8x^2+16x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - 1)*(exp(4)*(2*x^2 - 2*x^3) - 44*x + 60*x^2 - 18*x^3 + 2*x^4) + log(x - 1)^2*(24*x + exp(4)*(x -
x^2) - 9*x^2 + x^3 - 16) + exp(4)*(x^3 - x^4) - 60*x^2 + 40*x^3 - 8*x^4 + x^5)/(16*x^3 - 24*x^4 + 9*x^5 - x^6
+ log(x - 1)^2*(16*x - 24*x^2 + 9*x^3 - x^4) + log(x - 1)*(32*x^2 - 48*x^3 + 18*x^4 - 2*x^5)),x)

[Out]

log(x) - ((16*x + x^2 - 44)/(x - 4)^2 + (12*log(x - 1)*(x - 1))/(x*(x - 4)^2))/(x + log(x - 1)) - (x*(4*exp(4)
- 12) - x^2*exp(4) + 12)/(16*x - 8*x^2 + x^3)

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sympy [A]  time = 0.27, size = 29, normalized size = 1.04 $$\begin{array}{c}\frac{-x-8}{x^2-4x+(x-4)\log (x-1)}+\log (x)+\frac{e^4}{x-4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+x)*exp(4)+x**3-9*x**2+24*x-16)*ln(x-1)**2+((-2*x**3+2*x**2)*exp(4)+2*x**4-18*x**3+60*x**2-4
4*x)*ln(x-1)+(-x**4+x**3)*exp(4)+x**5-8*x**4+40*x**3-60*x**2)/((x**4-9*x**3+24*x**2-16*x)*ln(x-1)**2+(2*x**5-1
8*x**4+48*x**3-32*x**2)*ln(x-1)+x**6-9*x**5+24*x**4-16*x**3),x)

[Out]

(-x - 8)/(x**2 - 4*x + (x - 4)*log(x - 1)) + log(x) + exp(4)/(x - 4)

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