3.100.9 \(\int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2(-\frac {8}{-4+x})}} (72+e (12-3 x)-18 x+(108+18 e+90 x) \log (-\frac {8}{-4+x})+(-180+45 x) \log ^2(-\frac {8}{-4+x}))}{-4 x^2+x^3+(-24 x+6 x^2) \log ^2(-\frac {8}{-4+x})+(-36+9 x) \log ^4(-\frac {8}{-4+x})} \, dx\)

Optimal. Leaf size=31 \[ -3+e^{\frac {6+e+5 x}{\frac {x}{3}+\log ^2\left (\frac {8}{4-x}\right )}} \]

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Rubi [F]  time = 9.52, 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 {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (72+e (12-3 x)-18 x+(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )+(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{-4 x^2+x^3+\left (-24 x+6 x^2\right ) \log ^2\left (-\frac {8}{-4+x}\right )+(-36+9 x) \log ^4\left (-\frac {8}{-4+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*(72 + E*(12 - 3*x) - 18*x + (108 + 18*E + 90*x)*Log[-8/(
-4 + x)] + (-180 + 45*x)*Log[-8/(-4 + x)]^2))/(-4*x^2 + x^3 + (-24*x + 6*x^2)*Log[-8/(-4 + x)]^2 + (-36 + 9*x)
*Log[-8/(-4 + x)]^4),x]

[Out]

60*Defer[Int][E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))/(x + 3*Log[-8/(-4 + x)]^2)^2, x] - 3*(26 + E)*D
efer[Int][E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))/(x + 3*Log[-8/(-4 + x)]^2)^2, x] - 15*Defer[Int][(E
^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*x)/(x + 3*Log[-8/(-4 + x)]^2)^2, x] + 90*Defer[Int][(E^((18 +
3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*Log[-8/(-4 + x)])/(x + 3*Log[-8/(-4 + x)]^2)^2, x] + 18*(26 + E)*Defer
[Int][(E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*Log[-8/(-4 + x)])/((-4 + x)*(x + 3*Log[-8/(-4 + x)]^2)
^2), x] + 15*Defer[Int][E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))/(x + 3*Log[-8/(-4 + x)]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (-72-e (12-3 x)+18 x-(108+18 e+90 x) \log \left (-\frac {8}{-4+x}\right )-(-180+45 x) \log ^2\left (-\frac {8}{-4+x}\right )\right )}{(4-x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\\ &=\int \left (-\frac {3 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} (6+e+5 x) \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}+\frac {15 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}\right ) \, dx\\ &=-\left (3 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} (6+e+5 x) \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx\\ &=-\left (3 \int \left (\frac {5 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}+\frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} (26+e) \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}\right ) \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx\\ &=-\left (15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx-(3 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \left (-4+x-6 \log \left (-\frac {8}{-4+x}\right )\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\\ &=15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx-15 \int \left (-\frac {4 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}+\frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}-\frac {6 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}\right ) \, dx-(3 (26+e)) \int \left (-\frac {4 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}+\frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}-\frac {6 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}\right ) \, dx\\ &=-\left (15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx+60 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+90 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx-(3 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+(12 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+(18 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\\ &=-\left (15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx+60 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+90 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx-(3 (26+e)) \int \left (\frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}+\frac {4 e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2}\right ) \, dx+(12 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+(18 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\\ &=-\left (15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} x}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\right )+15 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )} \, dx+60 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+90 \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx-(3 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}}}{\left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx+(18 (26+e)) \int \frac {e^{\frac {18+3 e+15 x}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \log \left (-\frac {8}{-4+x}\right )}{(-4+x) \left (x+3 \log ^2\left (-\frac {8}{-4+x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 28, normalized size = 0.90 \begin {gather*} e^{\frac {3 (26+e+5 (-4+x))}{x+3 \log ^2\left (-\frac {8}{-4+x}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((18 + 3*E + 15*x)/(x + 3*Log[-8/(-4 + x)]^2))*(72 + E*(12 - 3*x) - 18*x + (108 + 18*E + 90*x)*Lo
g[-8/(-4 + x)] + (-180 + 45*x)*Log[-8/(-4 + x)]^2))/(-4*x^2 + x^3 + (-24*x + 6*x^2)*Log[-8/(-4 + x)]^2 + (-36
+ 9*x)*Log[-8/(-4 + x)]^4),x]

[Out]

E^((3*(26 + E + 5*(-4 + x)))/(x + 3*Log[-8/(-4 + x)]^2))

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fricas [A]  time = 0.84, size = 26, normalized size = 0.84 \begin {gather*} e^{\left (\frac {3 \, {\left (5 \, x + e + 6\right )}}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+(-3*x+12)*exp(1)-18*x+72)*exp((3*exp(
1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9*x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algori
thm="fricas")

[Out]

e^(3*(5*x + e + 6)/(3*log(-8/(x - 4))^2 + x))

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giac [B]  time = 2.45, size = 59, normalized size = 1.90 \begin {gather*} e^{\left (\frac {15 \, x}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x} + \frac {3 \, e}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x} + \frac {18}{3 \, \log \left (-\frac {8}{x - 4}\right )^{2} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+(-3*x+12)*exp(1)-18*x+72)*exp((3*exp(
1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9*x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algori
thm="giac")

[Out]

e^(15*x/(3*log(-8/(x - 4))^2 + x) + 3*e/(3*log(-8/(x - 4))^2 + x) + 18/(3*log(-8/(x - 4))^2 + x))

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maple [A]  time = 0.20, size = 27, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{\frac {3 \,{\mathrm e}+15 x +18}{3 \ln \left (-\frac {8}{x -4}\right )^{2}+x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((45*x-180)*ln(-8/(x-4))^2+(18*exp(1)+90*x+108)*ln(-8/(x-4))+(-3*x+12)*exp(1)-18*x+72)*exp((3*exp(1)+15*x+
18)/(3*ln(-8/(x-4))^2+x))/((9*x-36)*ln(-8/(x-4))^4+(6*x^2-24*x)*ln(-8/(x-4))^2+x^3-4*x^2),x,method=_RETURNVERB
OSE)

[Out]

exp(3*(exp(1)+5*x+6)/(3*ln(-8/(x-4))^2+x))

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maxima [B]  time = 0.63, size = 185, normalized size = 5.97 \begin {gather*} e^{\left (-\frac {405 \, \log \relax (2)^{2}}{27 \, \log \relax (2)^{2} - 18 \, \log \relax (2) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {270 \, \log \relax (2) \log \left (-x + 4\right )}{27 \, \log \relax (2)^{2} - 18 \, \log \relax (2) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} - \frac {45 \, \log \left (-x + 4\right )^{2}}{27 \, \log \relax (2)^{2} - 18 \, \log \relax (2) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {3 \, e}{27 \, \log \relax (2)^{2} - 18 \, \log \relax (2) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + \frac {18}{27 \, \log \relax (2)^{2} - 18 \, \log \relax (2) \log \left (-x + 4\right ) + 3 \, \log \left (-x + 4\right )^{2} + x} + 15\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x-180)*log(-8/(x-4))^2+(18*exp(1)+90*x+108)*log(-8/(x-4))+(-3*x+12)*exp(1)-18*x+72)*exp((3*exp(
1)+15*x+18)/(3*log(-8/(x-4))^2+x))/((9*x-36)*log(-8/(x-4))^4+(6*x^2-24*x)*log(-8/(x-4))^2+x^3-4*x^2),x, algori
thm="maxima")

[Out]

e^(-405*log(2)^2/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) + 270*log(2)*log(-x + 4)/(27*log(
2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) - 45*log(-x + 4)^2/(27*log(2)^2 - 18*log(2)*log(-x + 4) +
3*log(-x + 4)^2 + x) + 3*e/(27*log(2)^2 - 18*log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) + 18/(27*log(2)^2 - 18*
log(2)*log(-x + 4) + 3*log(-x + 4)^2 + x) + 15)

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mupad [B]  time = 6.45, size = 61, normalized size = 1.97 \begin {gather*} {\mathrm {e}}^{\frac {3\,\mathrm {e}}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}}\,{\mathrm {e}}^{\frac {15\,x}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}}\,{\mathrm {e}}^{\frac {18}{3\,{\ln \left (-\frac {8}{x-4}\right )}^2+x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((15*x + 3*exp(1) + 18)/(x + 3*log(-8/(x - 4))^2))*(log(-8/(x - 4))*(90*x + 18*exp(1) + 108) - 18*x +
log(-8/(x - 4))^2*(45*x - 180) - exp(1)*(3*x - 12) + 72))/(log(-8/(x - 4))^2*(24*x - 6*x^2) + 4*x^2 - x^3 - l
og(-8/(x - 4))^4*(9*x - 36)),x)

[Out]

exp((3*exp(1))/(x + 3*log(-8/(x - 4))^2))*exp((15*x)/(x + 3*log(-8/(x - 4))^2))*exp(18/(x + 3*log(-8/(x - 4))^
2))

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sympy [A]  time = 1.02, size = 24, normalized size = 0.77 \begin {gather*} e^{\frac {15 x + 3 e + 18}{x + 3 \log {\left (- \frac {8}{x - 4} \right )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((45*x-180)*ln(-8/(x-4))**2+(18*exp(1)+90*x+108)*ln(-8/(x-4))+(-3*x+12)*exp(1)-18*x+72)*exp((3*exp(1
)+15*x+18)/(3*ln(-8/(x-4))**2+x))/((9*x-36)*ln(-8/(x-4))**4+(6*x**2-24*x)*ln(-8/(x-4))**2+x**3-4*x**2),x)

[Out]

exp((15*x + 3*E + 18)/(x + 3*log(-8/(x - 4))**2))

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