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

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

________________________________________________________________________________________

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 {gather*} \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2 \left (-60+\left (40+e^4\right ) x-\left (8+e^4\right ) x^2+x^3\right )-2 x \left (-22+\left (30+e^4\right ) x-\left (9+e^4\right ) x^2+x^3\right ) \log (-1+x)-\left (-16+\left (24+e^4\right ) x-\left (9+e^4\right ) x^2+x^3\right ) \log ^2(-1+x)}{(1-x) (4-x)^2 x (x+\log (-1+x))^2} \, dx\\ &=\int \left (\frac {16-\left (8+e^4\right ) 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))}\right ) \, dx\\ &=12 \int \frac {1}{(-4+x)^2 (x+\log (-1+x))} \, dx+\int \frac {16-\left (8+e^4\right ) 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 \left (-\frac {e^4}{(-4+x)^2}+\frac {1}{x}\right ) \, dx+\int \left (\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}\right ) \, 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 {aligned} \end {gather*}

________________________________________________________________________________________

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

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]

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 52, normalized size = 1.86 \begin {gather*} \frac {x e^{4} + e^{4} \log \left (x - 1\right ) + {\left (x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x\right )} \log \relax (x) - x - 8}{x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x} \end {gather*}

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)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 65, normalized size = 2.32 \begin {gather*} \frac {x^{2} \log \relax (x) + x \log \left (x - 1\right ) \log \relax (x) + x e^{4} + e^{4} \log \left (x - 1\right ) - 4 \, x \log \relax (x) - 4 \, \log \left (x - 1\right ) \log \relax (x) - x - 8}{x^{2} + x \log \left (x - 1\right ) - 4 \, x - 4 \, \log \left (x - 1\right )} \end {gather*}

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))

________________________________________________________________________________________

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

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

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))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 36, normalized size = 1.29 \begin {gather*} \frac {x {\left (e^{4} - 1\right )} + e^{4} \log \left (x - 1\right ) - 8}{x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x} + \log \relax (x) \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 29, normalized size = 1.04 \begin {gather*} \frac {- x - 8}{x^{2} - 4 x + \left (x - 4\right ) \log {\left (x - 1 \right )}} + \log {\relax (x )} + \frac {e^{4}}{x - 4} \end {gather*}

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)

________________________________________________________________________________________