3.27.63 \(\int \frac {-12 x^2+3 x^3+e (-12 x^2+3 x^3)+(24 x-6 x^2+e (12 x-3 x^2)) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+(12 x^2-3 x^3+e (9 x^2-3 x^3)+(-24 x+6 x^2+e (-24 x+6 x^2)) \log (-4+x)+(12-3 x) \log ^2(-4+x)) \log (x)}{(-4 x^2+x^3+(8 x-2 x^2) \log (-4+x)+(-4+x) \log ^2(-4+x)) \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {3 x \left (-1+\frac {e x}{-x+\log (-4+x)}\right )}{\log (x)} \]

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Rubi [F]  time = 5.59, 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 {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x^2 + 3*x^3 + E*(-12*x^2 + 3*x^3) + (24*x - 6*x^2 + E*(12*x - 3*x^2))*Log[-4 + x] + (-12 + 3*x)*Log[-
4 + x]^2 + (12*x^2 - 3*x^3 + E*(9*x^2 - 3*x^3) + (-24*x + 6*x^2 + E*(-24*x + 6*x^2))*Log[-4 + x] + (12 - 3*x)*
Log[-4 + x]^2)*Log[x])/((-4*x^2 + x^3 + (8*x - 2*x^2)*Log[-4 + x] + (-4 + x)*Log[-4 + x]^2)*Log[x]^2),x]

[Out]

3*(1 + E)*Defer[Int][x/((x - Log[-4 + x])*Log[x]^2), x] - 3*Defer[Int][Log[-4 + x]/((x - Log[-4 + x])*Log[x]^2
), x] - 48*(1 + E)*Defer[Int][1/((x - Log[-4 + x])^2*Log[x]), x] + 12*(4 + 3*E)*Defer[Int][1/((x - Log[-4 + x]
)^2*Log[x]), x] - 192*(1 + E)*Defer[Int][1/((-4 + x)*(x - Log[-4 + x])^2*Log[x]), x] + 48*(4 + 3*E)*Defer[Int]
[1/((-4 + x)*(x - Log[-4 + x])^2*Log[x]), x] - 12*(1 + E)*Defer[Int][x/((x - Log[-4 + x])^2*Log[x]), x] + 3*(4
+ 3*E)*Defer[Int][x/((x - Log[-4 + x])^2*Log[x]), x] - 3*(1 + E)*Defer[Int][x^2/((x - Log[-4 + x])^2*Log[x]),
x] + 6*(1 + E)*Defer[Int][(x*Log[-4 + x])/((x - Log[-4 + x])^2*Log[x]), x] - 3*Defer[Int][Log[-4 + x]^2/((x -
Log[-4 + x])^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left ((-4+x) \log ^2(-4+x) (-1+\log (x))-(-4+x) x \log (-4+x) (-2-e+2 (1+e) \log (x))+x^2 (-((1+e) (-4+x))+(-4+e (-3+x)+x) \log (x))\right )}{(4-x) (x-\log (-4+x))^2 \log ^2(x)} \, dx\\ &=3 \int \frac {(-4+x) \log ^2(-4+x) (-1+\log (x))-(-4+x) x \log (-4+x) (-2-e+2 (1+e) \log (x))+x^2 (-((1+e) (-4+x))+(-4+e (-3+x)+x) \log (x))}{(4-x) (x-\log (-4+x))^2 \log ^2(x)} \, dx\\ &=3 \int \left (\frac {(1+e) x-\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)}+\frac {-4 \left (1+\frac {3 e}{4}\right ) x^2+(1+e) x^3+8 (1+e) x \log (-4+x)-2 (1+e) x^2 \log (-4+x)-4 \log ^2(-4+x)+x \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx\\ &=3 \int \frac {(1+e) x-\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+3 \int \frac {-4 \left (1+\frac {3 e}{4}\right ) x^2+(1+e) x^3+8 (1+e) x \log (-4+x)-2 (1+e) x^2 \log (-4+x)-4 \log ^2(-4+x)+x \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)} \, dx\\ &=3 \int \left (\frac {(1+e) x}{(x-\log (-4+x)) \log ^2(x)}-\frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)}\right ) \, dx+3 \int \frac {x^2 (-4+e (-3+x)+x)-2 (1+e) (-4+x) x \log (-4+x)+(-4+x) \log ^2(-4+x)}{(4-x) (x-\log (-4+x))^2 \log (x)} \, dx\\ &=3 \int \left (\frac {4 \left (1+\frac {3 e}{4}\right ) x^2}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {(1+e) x^3}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {8 (1+e) x \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {2 (1+e) x^2 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {4 \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}-\frac {x \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx-3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx\\ &=-\left (3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx\right )-3 \int \frac {x \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+12 \int \frac {\log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx-(3 (1+e)) \int \frac {x^3}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(6 (1+e)) \int \frac {x^2 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx-(24 (1+e)) \int \frac {x \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (4+3 e)) \int \frac {x^2}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx\\ &=-\left (3 \int \left (\frac {\log ^2(-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {4 \log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx\right )-3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx+12 \int \frac {\log ^2(-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx-(3 (1+e)) \int \left (\frac {16}{(x-\log (-4+x))^2 \log (x)}+\frac {64}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {4 x}{(x-\log (-4+x))^2 \log (x)}+\frac {x^2}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx+(6 (1+e)) \int \left (\frac {4 \log (-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {16 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {x \log (-4+x)}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx-(24 (1+e)) \int \left (\frac {\log (-4+x)}{(x-\log (-4+x))^2 \log (x)}+\frac {4 \log (-4+x)}{(-4+x) (x-\log (-4+x))^2 \log (x)}\right ) \, dx+(3 (4+3 e)) \int \left (\frac {4}{(x-\log (-4+x))^2 \log (x)}+\frac {16}{(-4+x) (x-\log (-4+x))^2 \log (x)}+\frac {x}{(x-\log (-4+x))^2 \log (x)}\right ) \, dx\\ &=-\left (3 \int \frac {\log (-4+x)}{(x-\log (-4+x)) \log ^2(x)} \, dx\right )-3 \int \frac {\log ^2(-4+x)}{(x-\log (-4+x))^2 \log (x)} \, dx+(3 (1+e)) \int \frac {x}{(x-\log (-4+x)) \log ^2(x)} \, dx-(3 (1+e)) \int \frac {x^2}{(x-\log (-4+x))^2 \log (x)} \, dx+(6 (1+e)) \int \frac {x \log (-4+x)}{(x-\log (-4+x))^2 \log (x)} \, dx-(12 (1+e)) \int \frac {x}{(x-\log (-4+x))^2 \log (x)} \, dx-(48 (1+e)) \int \frac {1}{(x-\log (-4+x))^2 \log (x)} \, dx-(192 (1+e)) \int \frac {1}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx+(3 (4+3 e)) \int \frac {x}{(x-\log (-4+x))^2 \log (x)} \, dx+(12 (4+3 e)) \int \frac {1}{(x-\log (-4+x))^2 \log (x)} \, dx+(48 (4+3 e)) \int \frac {1}{(-4+x) (x-\log (-4+x))^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 28, normalized size = 1.27 \begin {gather*} -\frac {3 x (x+e x-\log (-4+x))}{(x-\log (-4+x)) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^2 + 3*x^3 + E*(-12*x^2 + 3*x^3) + (24*x - 6*x^2 + E*(12*x - 3*x^2))*Log[-4 + x] + (-12 + 3*x)
*Log[-4 + x]^2 + (12*x^2 - 3*x^3 + E*(9*x^2 - 3*x^3) + (-24*x + 6*x^2 + E*(-24*x + 6*x^2))*Log[-4 + x] + (12 -
3*x)*Log[-4 + x]^2)*Log[x])/((-4*x^2 + x^3 + (8*x - 2*x^2)*Log[-4 + x] + (-4 + x)*Log[-4 + x]^2)*Log[x]^2),x]

[Out]

(-3*x*(x + E*x - Log[-4 + x]))/((x - Log[-4 + x])*Log[x])

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fricas [A]  time = 0.49, size = 33, normalized size = 1.50 \begin {gather*} -\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{{\left (x - \log \left (x - 4\right )\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4)+(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)
*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+12*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x
-4)*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.36, size = 34, normalized size = 1.55 \begin {gather*} -\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{x \log \relax (x) - \log \left (x - 4\right ) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4)+(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)
*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+12*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x
-4)*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 30, normalized size = 1.36

method	result	size
risch	\(-\frac {3 x}{\ln \relax (x )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \relax (x ) \left (x -\ln \left (x -4\right )\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x+12)*ln(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*ln(x-4)+(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)*ln(x)+(
3*x-12)*ln(x-4)^2+((-3*x^2+12*x)*exp(1)-6*x^2+24*x)*ln(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x-4)*ln(x-4)
^2+(-2*x^2+8*x)*ln(x-4)+x^3-4*x^2)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.62, size = 33, normalized size = 1.50 \begin {gather*} -\frac {3 \, {\left (x^{2} {\left (e + 1\right )} - x \log \left (x - 4\right )\right )}}{x \log \relax (x) - \log \left (x - 4\right ) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4)+(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)
*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+12*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x
-4)*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.37, size = 402, normalized size = 18.27 \begin {gather*} \frac {\frac {3\,x\,\ln \relax (x)\,\left (75\,x-100\,\mathrm {e}+60\,x\,\mathrm {e}-15\,x^2\,\mathrm {e}+x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}-\frac {3\,x\,\left (20\,\mathrm {e}-10\,x-8\,x\,\mathrm {e}+x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \relax (x)}^2\,\left (75\,x-200\,\mathrm {e}+160\,x\,\mathrm {e}-30\,x^2\,\mathrm {e}+2\,x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}}{\ln \relax (x)}+\frac {\frac {3\,x^2\,\left (4\,\mathrm {e}-x\,\mathrm {e}-3\,\mathrm {e}\,\ln \relax (x)+x\,\mathrm {e}\,\ln \relax (x)\right )}{{\ln \relax (x)}^2\,\left (x-5\right )}+\frac {3\,\ln \left (x-4\right )\,\left (x\,\mathrm {e}-2\,x\,\mathrm {e}\,\ln \relax (x)\right )\,\left (x-4\right )}{{\ln \relax (x)}^2\,\left (x-5\right )}}{x-\ln \left (x-4\right )}+\ln \relax (x)\,\left (45\,\mathrm {e}+\frac {45}{2}\right )-\frac {225\,x\,\mathrm {e}-75\,x^2\,\mathrm {e}}{2\,x^3-30\,x^2+150\,x-250}+\frac {\frac {3\,x\,\mathrm {e}\,\left (x-4\right )}{x-5}-\frac {3\,x\,\ln \relax (x)\,\left (60\,\mathrm {e}-10\,x-28\,x\,\mathrm {e}+3\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \relax (x)}^2\,\left (40\,\mathrm {e}-10\,x-20\,x\,\mathrm {e}+2\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}}{{\ln \relax (x)}^2}-x\,\left (\frac {9\,\mathrm {e}}{2}+3\right )+\frac {\ln \relax (x)\,\left (\left (-3\,\mathrm {e}-\frac {3}{2}\right )\,x^4+\left (435\,\mathrm {e}+225\right )\,x^2+\left (-3075\,\mathrm {e}-1500\right )\,x+5625\,\mathrm {e}+\frac {5625}{2}\right )}{x^3-15\,x^2+75\,x-125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(log(x - 4)^2*(3*x - 12) + log(x - 4)*(24*x + exp(1)*(24*x - 6*x^2) - 6*x^2) - exp(1)*(9*x^2 - 3*
x^3) - 12*x^2 + 3*x^3) - log(x - 4)^2*(3*x - 12) - log(x - 4)*(24*x + exp(1)*(12*x - 3*x^2) - 6*x^2) + exp(1)*
(12*x^2 - 3*x^3) + 12*x^2 - 3*x^3)/(log(x)^2*(log(x - 4)*(8*x - 2*x^2) - 4*x^2 + x^3 + log(x - 4)^2*(x - 4))),
x)

[Out]

((3*x*log(x)*(75*x - 100*exp(1) + 60*x*exp(1) - 15*x^2*exp(1) + x^3*exp(1) - 15*x^2 + x^3 - 125))/(2*(x - 5)^3
) - (3*x*(20*exp(1) - 10*x - 8*x*exp(1) + x^2*exp(1) + x^2 + 25))/(2*(x - 5)^2) + (3*x*log(x)^2*(75*x - 200*ex
p(1) + 160*x*exp(1) - 30*x^2*exp(1) + 2*x^3*exp(1) - 15*x^2 + x^3 - 125))/(2*(x - 5)^3))/log(x) + ((3*x^2*(4*e
xp(1) - x*exp(1) - 3*exp(1)*log(x) + x*exp(1)*log(x)))/(log(x)^2*(x - 5)) + (3*log(x - 4)*(x*exp(1) - 2*x*exp(
1)*log(x))*(x - 4))/(log(x)^2*(x - 5)))/(x - log(x - 4)) + log(x)*(45*exp(1) + 45/2) - (225*x*exp(1) - 75*x^2*
exp(1))/(150*x - 30*x^2 + 2*x^3 - 250) + ((3*x*exp(1)*(x - 4))/(x - 5) - (3*x*log(x)*(60*exp(1) - 10*x - 28*x*
exp(1) + 3*x^2*exp(1) + x^2 + 25))/(2*(x - 5)^2) + (3*x*log(x)^2*(40*exp(1) - 10*x - 20*x*exp(1) + 2*x^2*exp(1
) + x^2 + 25))/(2*(x - 5)^2))/log(x)^2 - x*((9*exp(1))/2 + 3) + (log(x)*(5625*exp(1) - x^4*(3*exp(1) + 3/2) +
x^2*(435*exp(1) + 225) - x*(3075*exp(1) + 1500) + 5625/2))/(75*x - 15*x^2 + x^3 - 125)

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sympy [A]  time = 0.35, size = 27, normalized size = 1.23 \begin {gather*} \frac {3 e x^{2}}{- x \log {\relax (x )} + \log {\relax (x )} \log {\left (x - 4 \right )}} - \frac {3 x}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+12)*ln(x-4)**2+((6*x**2-24*x)*exp(1)+6*x**2-24*x)*ln(x-4)+(-3*x**3+9*x**2)*exp(1)-3*x**3+12*
x**2)*ln(x)+(3*x-12)*ln(x-4)**2+((-3*x**2+12*x)*exp(1)-6*x**2+24*x)*ln(x-4)+(3*x**3-12*x**2)*exp(1)+3*x**3-12*
x**2)/((x-4)*ln(x-4)**2+(-2*x**2+8*x)*ln(x-4)+x**3-4*x**2)/ln(x)**2,x)

[Out]

3*E*x**2/(-x*log(x) + log(x)*log(x - 4)) - 3*x/log(x)

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