3.66.60 \(\int \frac {69 x-16 x^2+x^3+e^x (-69+16 x-x^2)+(216-123 x+20 x^2-x^3+e^x (-216+123 x-20 x^2+x^3)) \log (\frac {-27+12 x-x^2}{-8+x})}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} (-216+123 x-20 x^2+x^3)+e^x (432 x-246 x^2+40 x^3-2 x^4)} \, dx\)

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

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Rubi [F]  time = 30.23, 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 {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(69*x - 16*x^2 + x^3 + E^x*(-69 + 16*x - x^2) + (216 - 123*x + 20*x^2 - x^3 + E^x*(-216 + 123*x - 20*x^2 +
x^3))*Log[(-27 + 12*x - x^2)/(-8 + x)])/(-216*x^2 + 123*x^3 - 20*x^4 + x^5 + E^(2*x)*(-216 + 123*x - 20*x^2 +
x^3) + E^x*(432*x - 246*x^2 + 40*x^3 - 2*x^4)),x]

[Out]

-(Log[(27 - 12*x + x^2)/(8 - x)]*Defer[Int][(E^x - x)^(-2), x]) + Log[(27 - 12*x + x^2)/(8 - x)]*Defer[Int][(E
^x - x)^(-1), x] + 24*Defer[Int][1/((E^x - x)*(-9 + x)), x] + 306*Log[(27 - 12*x + x^2)/(8 - x)]*Defer[Int][1/
((E^x - x)*(-9 + x)), x] - (128*Defer[Int][1/((E^x - x)*(-8 + x)), x])/5 - (1496*Log[(27 - 12*x + x^2)/(8 - x)
]*Defer[Int][1/((E^x - x)*(-8 + x)), x])/5 + (8*Defer[Int][1/((E^x - x)*(-3 + x)), x])/5 + (66*Log[(27 - 12*x
+ x^2)/(8 - x)]*Defer[Int][1/((E^x - x)*(-3 + x)), x])/5 + Log[(27 - 12*x + x^2)/(8 - x)]*Defer[Int][x/(E^x -
x)^2, x] + 25*Defer[Int][1/((-9 + x)*(-E^x + x)), x] + 306*Log[(27 - 12*x + x^2)/(8 - x)]*Defer[Int][1/((-9 +
x)*(-E^x + x)), x] - (133*Defer[Int][1/((-8 + x)*(-E^x + x)), x])/5 - (1496*Log[(27 - 12*x + x^2)/(8 - x)]*Def
er[Int][1/((-8 + x)*(-E^x + x)), x])/5 + (13*Defer[Int][1/((-3 + x)*(-E^x + x)), x])/5 + (66*Log[(27 - 12*x +
x^2)/(8 - x)]*Defer[Int][1/((-3 + x)*(-E^x + x)), x])/5 + Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(-9 + x), x
] - Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(-8 + x), x] + Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(-3 + x),
x] - Defer[Int][Defer[Int][(E^x - x)^(-1), x]/(-9 + x), x] + Defer[Int][Defer[Int][(E^x - x)^(-1), x]/(-8 + x
), x] - Defer[Int][Defer[Int][(E^x - x)^(-1), x]/(-3 + x), x] - 306*Defer[Int][Defer[Int][1/((E^x - x)*(-9 + x
)), x]/(-9 + x), x] + 306*Defer[Int][Defer[Int][1/((E^x - x)*(-9 + x)), x]/(-8 + x), x] - 306*Defer[Int][Defer
[Int][1/((E^x - x)*(-9 + x)), x]/(-3 + x), x] + (1496*Defer[Int][Defer[Int][1/((E^x - x)*(-8 + x)), x]/(-9 + x
), x])/5 - (1496*Defer[Int][Defer[Int][1/((E^x - x)*(-8 + x)), x]/(-8 + x), x])/5 + (1496*Defer[Int][Defer[Int
][1/((E^x - x)*(-8 + x)), x]/(-3 + x), x])/5 - (66*Defer[Int][Defer[Int][1/((E^x - x)*(-3 + x)), x]/(-9 + x),
x])/5 + (66*Defer[Int][Defer[Int][1/((E^x - x)*(-3 + x)), x]/(-8 + x), x])/5 - (66*Defer[Int][Defer[Int][1/((E
^x - x)*(-3 + x)), x]/(-3 + x), x])/5 - Defer[Int][Defer[Int][x/(E^x - x)^2, x]/(-9 + x), x] + Defer[Int][Defe
r[Int][x/(E^x - x)^2, x]/(-8 + x), x] - Defer[Int][Defer[Int][x/(E^x - x)^2, x]/(-3 + x), x] - 306*Defer[Int][
Defer[Int][1/((-9 + x)*(-E^x + x)), x]/(-9 + x), x] + 306*Defer[Int][Defer[Int][1/((-9 + x)*(-E^x + x)), x]/(-
8 + x), x] - 306*Defer[Int][Defer[Int][1/((-9 + x)*(-E^x + x)), x]/(-3 + x), x] + (1496*Defer[Int][Defer[Int][
1/((-8 + x)*(-E^x + x)), x]/(-9 + x), x])/5 - (1496*Defer[Int][Defer[Int][1/((-8 + x)*(-E^x + x)), x]/(-8 + x)
, x])/5 + (1496*Defer[Int][Defer[Int][1/((-8 + x)*(-E^x + x)), x]/(-3 + x), x])/5 - (66*Defer[Int][Defer[Int][
1/((-3 + x)*(-E^x + x)), x]/(-9 + x), x])/5 + (66*Defer[Int][Defer[Int][1/((-3 + x)*(-E^x + x)), x]/(-8 + x),
x])/5 - (66*Defer[Int][Defer[Int][1/((-3 + x)*(-E^x + x)), x]/(-3 + x), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {\left (e^x-x\right ) \left (69-16 x+x^2\right )}{-216+123 x-20 x^2+x^3}+\left (-1+e^x\right ) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right )^2} \, dx\\ &=\int \left (\frac {(-1+x) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right )^2}+\frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right ) (-9+x) (-8+x) (-3+x)}\right ) \, dx\\ &=\int \frac {(-1+x) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right )^2} \, dx+\int \frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right ) (-9+x) (-8+x) (-3+x)} \, dx\\ &=-\left (\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx\right )+\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx+\int \frac {69-16 x+x^2-\left (-216+123 x-20 x^2+x^3\right ) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )} \, dx-\int \frac {\left (69-16 x+x^2\right ) \left (\int \frac {1}{\left (e^x-x\right )^2} \, dx-\int \frac {x}{\left (e^x-x\right )^2} \, dx\right )}{(8-x) \left (27-12 x+x^2\right )} \, dx\\ &=-\left (\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx\right )+\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx+\int \left (\frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{6 \left (e^x-x\right ) (-9+x)}-\frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{5 \left (e^x-x\right ) (-8+x)}+\frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{30 \left (e^x-x\right ) (-3+x)}\right ) \, dx-\int \left (-\frac {\left (69-16 x+x^2\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx}{(-9+x) (-8+x) (-3+x)}+\frac {\left (69-16 x+x^2\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx}{(-9+x) (-8+x) (-3+x)}\right ) \, dx\\ &=\frac {1}{30} \int \frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right ) (-3+x)} \, dx+\frac {1}{6} \int \frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right ) (-9+x)} \, dx-\frac {1}{5} \int \frac {-69+16 x-x^2-216 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+123 x \log \left (-\frac {27-12 x+x^2}{-8+x}\right )-20 x^2 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )+x^3 \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{\left (e^x-x\right ) (-8+x)} \, dx-\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx+\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx+\int \frac {\left (69-16 x+x^2\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx}{(-9+x) (-8+x) (-3+x)} \, dx-\int \frac {\left (69-16 x+x^2\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx}{(-9+x) (-8+x) (-3+x)} \, dx\\ &=\frac {1}{30} \int \frac {69-16 x+x^2-\left (-216+123 x-20 x^2+x^3\right ) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{(3-x) \left (e^x-x\right )} \, dx+\frac {1}{6} \int \frac {69-16 x+x^2-\left (-216+123 x-20 x^2+x^3\right ) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{(9-x) \left (e^x-x\right )} \, dx-\frac {1}{5} \int \frac {69-16 x+x^2-\left (-216+123 x-20 x^2+x^3\right ) \log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{(8-x) \left (e^x-x\right )} \, dx-\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {1}{\left (e^x-x\right )^2} \, dx+\log \left (-\frac {27-12 x+x^2}{-8+x}\right ) \int \frac {x}{\left (e^x-x\right )^2} \, dx+\int \left (\frac {\int \frac {1}{\left (e^x-x\right )^2} \, dx}{-9+x}-\frac {\int \frac {1}{\left (e^x-x\right )^2} \, dx}{-8+x}+\frac {\int \frac {1}{\left (e^x-x\right )^2} \, dx}{-3+x}\right ) \, dx-\int \left (\frac {\int \frac {x}{\left (e^x-x\right )^2} \, dx}{-9+x}-\frac {\int \frac {x}{\left (e^x-x\right )^2} \, dx}{-8+x}+\frac {\int \frac {x}{\left (e^x-x\right )^2} \, dx}{-3+x}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.43, size = 27, normalized size = 0.96 \begin {gather*} -\frac {\log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{e^x-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(69*x - 16*x^2 + x^3 + E^x*(-69 + 16*x - x^2) + (216 - 123*x + 20*x^2 - x^3 + E^x*(-216 + 123*x - 20
*x^2 + x^3))*Log[(-27 + 12*x - x^2)/(-8 + x)])/(-216*x^2 + 123*x^3 - 20*x^4 + x^5 + E^(2*x)*(-216 + 123*x - 20
*x^2 + x^3) + E^x*(432*x - 246*x^2 + 40*x^3 - 2*x^4)),x]

[Out]

-(Log[-((27 - 12*x + x^2)/(-8 + x))]/(E^x - x))

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fricas [A]  time = 1.07, size = 25, normalized size = 0.89 \begin {gather*} \frac {\log \left (-\frac {x^{2} - 12 \, x + 27}{x - 8}\right )}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(
x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-2
16*x^2),x, algorithm="fricas")

[Out]

log(-(x^2 - 12*x + 27)/(x - 8))/(x - e^x)

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giac [A]  time = 0.27, size = 25, normalized size = 0.89 \begin {gather*} \frac {\log \left (-\frac {x^{2} - 12 \, x + 27}{x - 8}\right )}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(
x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-2
16*x^2),x, algorithm="giac")

[Out]

log(-(x^2 - 12*x + 27)/(x - 8))/(x - e^x)

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maple [C]  time = 0.18, size = 196, normalized size = 7.00

method	result	size
risch	\(\frac {\ln \left (x^{2}-12 x +27\right )}{x -{\mathrm e}^{x}}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{-8+x}\right ) \mathrm {csgn}\left (i \left (x^{2}-12 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{-8+x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x^{2}-12 x +27\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )^{2}-2 i \pi +2 \ln \left (-8+x \right )}{2 \left (x -{\mathrm e}^{x}\right )}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*ln((-x^2+12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-
16*x^2+69*x)/((x^3-20*x^2+123*x-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216*x^2)
,x,method=_RETURNVERBOSE)

[Out]

1/(x-exp(x))*ln(x^2-12*x+27)-1/2*(I*Pi*csgn(I/(-8+x))*csgn(I*(x^2-12*x+27))*csgn(I/(-8+x)*(x^2-12*x+27))-I*Pi*
csgn(I/(-8+x))*csgn(I/(-8+x)*(x^2-12*x+27))^2-I*Pi*csgn(I*(x^2-12*x+27))*csgn(I/(-8+x)*(x^2-12*x+27))^2-I*Pi*c
sgn(I/(-8+x)*(x^2-12*x+27))^3+2*I*Pi*csgn(I/(-8+x)*(x^2-12*x+27))^2-2*I*Pi+2*ln(-8+x))/(x-exp(x))

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maxima [A]  time = 0.47, size = 26, normalized size = 0.93 \begin {gather*} \frac {\log \left (x - 3\right ) - \log \left (x - 8\right ) + \log \left (-x + 9\right )}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(
x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-2
16*x^2),x, algorithm="maxima")

[Out]

(log(x - 3) - log(x - 8) + log(-x + 9))/(x - e^x)

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mupad [B]  time = 0.29, size = 25, normalized size = 0.89 \begin {gather*} \frac {\ln \left (-\frac {x^2-12\,x+27}{x-8}\right )}{x-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((69*x - exp(x)*(x^2 - 16*x + 69) + log(-(x^2 - 12*x + 27)/(x - 8))*(exp(x)*(123*x - 20*x^2 + x^3 - 216) -
123*x + 20*x^2 - x^3 + 216) - 16*x^2 + x^3)/(exp(x)*(432*x - 246*x^2 + 40*x^3 - 2*x^4) - 216*x^2 + 123*x^3 - 2
0*x^4 + x^5 + exp(2*x)*(123*x - 20*x^2 + x^3 - 216)),x)

[Out]

log(-(x^2 - 12*x + 27)/(x - 8))/(x - exp(x))

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sympy [A]  time = 0.42, size = 19, normalized size = 0.68 \begin {gather*} - \frac {\log {\left (\frac {- x^{2} + 12 x - 27}{x - 8} \right )}}{- x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-20*x**2+123*x-216)*exp(x)-x**3+20*x**2-123*x+216)*ln((-x**2+12*x-27)/(-8+x))+(-x**2+16*x-69)
*exp(x)+x**3-16*x**2+69*x)/((x**3-20*x**2+123*x-216)*exp(x)**2+(-2*x**4+40*x**3-246*x**2+432*x)*exp(x)+x**5-20
*x**4+123*x**3-216*x**2),x)

[Out]

-log((-x**2 + 12*x - 27)/(x - 8))/(-x + exp(x))

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